[{"data":1,"prerenderedAt":27516},["ShallowReactive",2],{"locale-switcher-blog-es-null":3,"blogindex-es":4},null,[5,13125,23678,24115,24292,24556,26665,26920],{"id":6,"title":7,"_locale":8,"body":9,"date":12969,"description":12970,"extension":12971,"meta":12972,"navigation":13119,"path":13120,"published":13119,"seo":13121,"slug":13122,"stem":13123,"__hash__":13124},"blog\u002Fblog\u002Fpi-coin-flips-catalan-numbers.es.md","Lanzando una moneda hasta que obtenemos PI","es",{"type":10,"value":11,"toc":12953},"minimal",[12,16,177,216,225,228,250,253,560,565,581,586,589,630,850,854,861,906,1264,1268,1275,1282,1285,1295,1375,1501,1504,1510,1629,1666,1804,2152,2156,2163,2167,3545,3548,3971,4069,4072,4076,4214,4243,4246,4278,4551,4861,4864,5373,5880,6144,6151,6155,6357,6361,6420,7223,7282,7723,7726,9161,9334,9338,9341,9601,9866,10387,10390,11163,11170,11672,11761,11816,12184,12296,12436,12439,12747,12792,12795,12831,12837,12841,12844,12875,12907,12910,12942,12945],[13,14,7],"h1",{"id":15},"lanzando-una-moneda-hasta-que-obtenemos-pi",[17,18,19,20,27,28,33,34,176],"p",{},"Matt Parker, del canal ",[21,22,26],"a",{"href":23,"rel":24},"https:\u002F\u002Fwww.youtube.com\u002F@standupmaths",[25],"nofollow","Stand-up Maths",", publicó recientemente ",[21,29,32],{"href":30,"rel":31},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=kahGSss6SsU",[25],"un vídeo"," con una premisa tan simple de enunciar como difícil de creer: si lanzamos una moneda repetidamente hasta que haya más caras que cruces, y calculamos la proporción media de caras respecto al total de lanzamientos... el resultado es ",[35,36,39,69],"span",{"className":37},[38],"katex",[35,40,43],{"className":41},[42],"katex-mathml",[44,45,47],"math",{"xmlns":46},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[48,49,50,64],"semantics",{},[51,52,53],"mrow",{},[54,55,56,60],"mfrac",{},[57,58,59],"mi",{},"π",[61,62,63],"mn",{},"4",[65,66,68],"annotation",{"encoding":67},"application\u002Fx-tex","\\frac{\\pi}{4}",[35,70,74],{"className":71,"ariaHidden":73},[72],"katex-html","true",[35,75,78,83],{"className":76},[77],"base",[35,79],{"className":80,"style":82},[81],"strut","height:1.0404em;vertical-align:-0.345em;",[35,84,87,92,172],{"className":85},[86],"mord",[35,88],{"className":89},[90,91],"mopen","nulldelimiter",[35,93,95],{"className":94},[54],[35,96,100,163],{"className":97},[98,99],"vlist-t","vlist-t2",[35,101,104,158],{"className":102},[103],"vlist-r",[35,105,109,130,141],{"className":106,"style":108},[107],"vlist","height:0.6954em;",[35,110,112,117],{"style":111},"top:-2.655em;",[35,113],{"className":114,"style":116},[115],"pstrut","height:3em;",[35,118,124],{"className":119},[120,121,122,123],"sizing","reset-size6","size3","mtight",[35,125,127],{"className":126},[86,123],[35,128,63],{"className":129},[86,123],[35,131,133,136],{"style":132},"top:-3.23em;",[35,134],{"className":135,"style":116},[115],[35,137],{"className":138,"style":140},[139],"frac-line","border-bottom-width:0.04em;",[35,142,144,147],{"style":143},"top:-3.394em;",[35,145],{"className":146,"style":116},[115],[35,148,150],{"className":149},[120,121,122,123],[35,151,153],{"className":152},[86,123],[35,154,59],{"className":155,"style":157},[86,156,123],"mathnormal","margin-right:0.0359em;",[35,159,162],{"className":160},[161],"vlist-s","​",[35,164,166],{"className":165},[103],[35,167,170],{"className":168,"style":169},[107],"height:0.345em;",[35,171],{},[35,173],{"className":174},[175,91],"mclose",".",[17,178,179,180,210,211,215],{},"Sí, ",[35,181,183,197],{"className":182},[38],[35,184,186],{"className":185},[42],[44,187,188],{"xmlns":46},[48,189,190,194],{},[51,191,192],{},[57,193,59],{},[65,195,196],{"encoding":67},"\\pi",[35,198,200],{"className":199,"ariaHidden":73},[72],[35,201,203,207],{"className":202},[77],[35,204],{"className":205,"style":206},[81],"height:0.4306em;",[35,208,59],{"className":209,"style":157},[86,156],". ",[212,213,214],"em",{},"El número de los círculos",". Escondido dentro de lanzamientos de moneda.",[17,217,218,219,224],{},"En nuestro ",[21,220,223],{"href":221,"rel":222},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ftriangulo-de-tartaglia-combinatoria-y-probabilidad\u002F",[25],"artículo anterior sobre el triángulo de Tartaglia"," vimos cómo un triángulo de sumas sencillas contenía, en su interior, la combinatoria, el álgebra y la probabilidad. Hoy vamos a tirar del mismo hilo. Y nos va a llevar a un lugar que nadie esperaba.",[17,226,227],{},"Las reglas del experimento son sencillas:",[229,230,231,235,243],"ol",{},[232,233,234],"li",{},"Lanzamos una moneda justa repetidamente.",[232,236,237,238,242],{},"Nos detenemos ",[239,240,241],"strong",{},"en el momento exacto"," en que el número de caras supera al de cruces.",[232,244,245,246,249],{},"Anotamos la ",[239,247,248],{},"proporción de caras"," respecto al total de lanzamientos de esa secuencia.",[17,251,252],{},"La pregunta del artículo será: si repetimos este proceso infinitas veces y calculamos la media de todas esas proporciones... ¿qué obtenemos?",[17,254,255,256,385,386,485,486,530,531,559],{},"Antes de responder, pensemos en qué rango puede moverse esa proporción. En el mejor de los casos, el primer lanzamiento es cara: paramos inmediatamente y la proporción es ",[35,257,259,284],{"className":258},[38],[35,260,262],{"className":261},[42],[44,263,264],{"xmlns":46},[48,265,266,281],{},[51,267,268,275,279],{},[54,269,270,273],{},[61,271,272],{},"1",[61,274,272],{},[276,277,278],"mo",{},"=",[61,280,272],{},[65,282,283],{"encoding":67},"\\frac{1}{1} = 1",[35,285,287,375],{"className":286,"ariaHidden":73},[72],[35,288,290,294,363,368,372],{"className":289},[77],[35,291],{"className":292,"style":293},[81],"height:1.1901em;vertical-align:-0.345em;",[35,295,297,300,360],{"className":296},[86],[35,298],{"className":299},[90,91],[35,301,303],{"className":302},[54],[35,304,306,352],{"className":305},[98,99],[35,307,309,349],{"className":308},[103],[35,310,313,327,335],{"className":311,"style":312},[107],"height:0.8451em;",[35,314,315,318],{"style":111},[35,316],{"className":317,"style":116},[115],[35,319,321],{"className":320},[120,121,122,123],[35,322,324],{"className":323},[86,123],[35,325,272],{"className":326},[86,123],[35,328,329,332],{"style":132},[35,330],{"className":331,"style":116},[115],[35,333],{"className":334,"style":140},[139],[35,336,337,340],{"style":143},[35,338],{"className":339,"style":116},[115],[35,341,343],{"className":342},[120,121,122,123],[35,344,346],{"className":345},[86,123],[35,347,272],{"className":348},[86,123],[35,350,162],{"className":351},[161],[35,353,355],{"className":354},[103],[35,356,358],{"className":357,"style":169},[107],[35,359],{},[35,361],{"className":362},[175,91],[35,364],{"className":365,"style":367},[366],"mspace","margin-right:0.2778em;",[35,369,278],{"className":370},[371],"mrel",[35,373],{"className":374,"style":367},[366],[35,376,378,382],{"className":377},[77],[35,379],{"className":380,"style":381},[81],"height:0.6444em;",[35,383,272],{"className":384},[86]," (el 100 % son caras). En el peor, necesitamos miles de lanzamientos hasta que las caras superen a las cruces por uno, y la proporción será apenas superior a ",[35,387,389,408],{"className":388},[38],[35,390,392],{"className":391},[42],[44,393,394],{"xmlns":46},[48,395,396,405],{},[51,397,398],{},[54,399,400,402],{},[61,401,272],{},[61,403,404],{},"2",[65,406,407],{"encoding":67},"\\frac{1}{2}",[35,409,411],{"className":410,"ariaHidden":73},[72],[35,412,414,417],{"className":413},[77],[35,415],{"className":416,"style":293},[81],[35,418,420,423,482],{"className":419},[86],[35,421],{"className":422},[90,91],[35,424,426],{"className":425},[54],[35,427,429,474],{"className":428},[98,99],[35,430,432,471],{"className":431},[103],[35,433,435,449,457],{"className":434,"style":312},[107],[35,436,437,440],{"style":111},[35,438],{"className":439,"style":116},[115],[35,441,443],{"className":442},[120,121,122,123],[35,444,446],{"className":445},[86,123],[35,447,404],{"className":448},[86,123],[35,450,451,454],{"style":132},[35,452],{"className":453,"style":116},[115],[35,455],{"className":456,"style":140},[139],[35,458,459,462],{"style":143},[35,460],{"className":461,"style":116},[115],[35,463,465],{"className":464},[120,121,122,123],[35,466,468],{"className":467},[86,123],[35,469,272],{"className":470},[86,123],[35,472,162],{"className":473},[161],[35,475,477],{"className":476},[103],[35,478,480],{"className":479,"style":169},[107],[35,481],{},[35,483],{"className":484},[175,91],". Así que la media estará en algún punto entre ",[35,487,489,504],{"className":488},[38],[35,490,492],{"className":491},[42],[44,493,494],{"xmlns":46},[48,495,496,501],{},[51,497,498],{},[61,499,500],{},"0,5",[65,502,503],{"encoding":67},"0{,}5",[35,505,507],{"className":506,"ariaHidden":73},[72],[35,508,510,514,518,526],{"className":509},[77],[35,511],{"className":512,"style":513},[81],"height:0.8389em;vertical-align:-0.1944em;",[35,515,517],{"className":516},[86],"0",[35,519,521],{"className":520},[86],[35,522,525],{"className":523},[524],"mpunct",",",[35,527,529],{"className":528},[86],"5"," y ",[35,532,534,547],{"className":533},[38],[35,535,537],{"className":536},[42],[44,538,539],{"xmlns":46},[48,540,541,545],{},[51,542,543],{},[61,544,272],{},[65,546,272],{"encoding":67},[35,548,550],{"className":549,"ariaHidden":73},[72],[35,551,553,556],{"className":552},[77],[35,554],{"className":555,"style":381},[81],[35,557,272],{"className":558},[86],". Veamos exactamente dónde.",[561,562,564],"h2",{"id":563},"construyendo-las-secuencias-paso-a-paso","Construyendo las secuencias, paso a paso",[17,566,567,568,571,572,575,576,176],{},"Vamos a construir las secuencias posibles empezando por las más cortas. Usaremos ",[239,569,570],{},"O"," para cara y ",[239,573,574],{},"X"," para cruz — la misma notación que usamos en ",[21,577,580],{"href":578,"rel":579},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fplay\u002F",[25],"nuestra herramienta de lanzamiento",[582,583,585],"h3",{"id":584},"longitud-1","Longitud 1",[17,587,588],{},"La secuencia más corta posible: lanzamos una vez y sale cara.",[17,590,593],{"className":591},[592],"math-block",[35,594,596,612],{"className":595},[38],[35,597,599],{"className":598},[42],[44,600,601],{"xmlns":46},[48,602,603,609],{},[51,604,605],{},[606,607,570],"mtext",{"mathvariant":608},"bold",[65,610,611],{"encoding":67},"\\textbf{O}",[35,613,615],{"className":614,"ariaHidden":73},[72],[35,616,618,622],{"className":617},[77],[35,619],{"className":620,"style":621},[81],"height:0.6861em;",[35,623,626],{"className":624},[86,625],"text",[35,627,570],{"className":628},[86,629],"textbf",[17,631,632,633,752,753,176],{},"Paramos. Tenemos 1 cara y 0 cruces. La proporción es ",[35,634,636,657],{"className":635},[38],[35,637,639],{"className":638},[42],[44,640,641],{"xmlns":46},[48,642,643,655],{},[51,644,645,651,653],{},[54,646,647,649],{},[61,648,272],{},[61,650,272],{},[276,652,278],{},[61,654,272],{},[65,656,283],{"encoding":67},[35,658,660,743],{"className":659,"ariaHidden":73},[72],[35,661,663,666,734,737,740],{"className":662},[77],[35,664],{"className":665,"style":293},[81],[35,667,669,672,731],{"className":668},[86],[35,670],{"className":671},[90,91],[35,673,675],{"className":674},[54],[35,676,678,723],{"className":677},[98,99],[35,679,681,720],{"className":680},[103],[35,682,684,698,706],{"className":683,"style":312},[107],[35,685,686,689],{"style":111},[35,687],{"className":688,"style":116},[115],[35,690,692],{"className":691},[120,121,122,123],[35,693,695],{"className":694},[86,123],[35,696,272],{"className":697},[86,123],[35,699,700,703],{"style":132},[35,701],{"className":702,"style":116},[115],[35,704],{"className":705,"style":140},[139],[35,707,708,711],{"style":143},[35,709],{"className":710,"style":116},[115],[35,712,714],{"className":713},[120,121,122,123],[35,715,717],{"className":716},[86,123],[35,718,272],{"className":719},[86,123],[35,721,162],{"className":722},[161],[35,724,726],{"className":725},[103],[35,727,729],{"className":728,"style":169},[107],[35,730],{},[35,732],{"className":733},[175,91],[35,735],{"className":736,"style":367},[366],[35,738,278],{"className":739},[371],[35,741],{"className":742,"style":367},[366],[35,744,746,749],{"className":745},[77],[35,747],{"className":748,"style":381},[81],[35,750,272],{"className":751},[86]," y la probabilidad de que esto ocurra es ",[35,754,756,773],{"className":755},[38],[35,757,759],{"className":758},[42],[44,760,761],{"xmlns":46},[48,762,763,771],{},[51,764,765],{},[54,766,767,769],{},[61,768,272],{},[61,770,404],{},[65,772,407],{"encoding":67},[35,774,776],{"className":775,"ariaHidden":73},[72],[35,777,779,782],{"className":778},[77],[35,780],{"className":781,"style":293},[81],[35,783,785,788,847],{"className":784},[86],[35,786],{"className":787},[90,91],[35,789,791],{"className":790},[54],[35,792,794,839],{"className":793},[98,99],[35,795,797,836],{"className":796},[103],[35,798,800,814,822],{"className":799,"style":312},[107],[35,801,802,805],{"style":111},[35,803],{"className":804,"style":116},[115],[35,806,808],{"className":807},[120,121,122,123],[35,809,811],{"className":810},[86,123],[35,812,404],{"className":813},[86,123],[35,815,816,819],{"style":132},[35,817],{"className":818,"style":116},[115],[35,820],{"className":821,"style":140},[139],[35,823,824,827],{"style":143},[35,825],{"className":826,"style":116},[115],[35,828,830],{"className":829},[120,121,122,123],[35,831,833],{"className":832},[86,123],[35,834,272],{"className":835},[86,123],[35,837,162],{"className":838},[161],[35,840,842],{"className":841},[103],[35,843,845],{"className":844,"style":169},[107],[35,846],{},[35,848],{"className":849},[175,91],[582,851,853],{"id":852},"longitud-3","Longitud 3",[17,855,856,857,860],{},"Si el primer lanzamiento es cruz, no podemos parar. Y con sólo dos lanzamientos, lo mejor que podemos lograr es empatar (1 cara, 1 cruz) — empatar no basta, necesitamos que las caras ",[212,858,859],{},"superen"," a las cruces. Así que la siguiente secuencia posible tiene longitud 3:",[17,862,864],{"className":863},[592],[35,865,867,890],{"className":866},[38],[35,868,870],{"className":869},[42],[44,871,872],{"xmlns":46},[48,873,874,887],{},[51,875,876,878,881,883,885],{},[606,877,574],{"mathvariant":608},[606,879,880],{}," ",[606,882,570],{"mathvariant":608},[606,884,880],{},[606,886,570],{"mathvariant":608},[65,888,889],{"encoding":67},"\\textbf{X O O}",[35,891,893],{"className":892,"ariaHidden":73},[72],[35,894,896,899],{"className":895},[77],[35,897],{"className":898,"style":621},[81],[35,900,902],{"className":901},[86,625],[35,903,905],{"className":904},[86,629],"X O O",[17,907,908,909,1008,1009,1263],{},"Cruz, cara, cara. El tercer lanzamiento nos da 2 caras frente a 1 cruz. Proporción: ",[35,910,912,931],{"className":911},[38],[35,913,915],{"className":914},[42],[44,916,917],{"xmlns":46},[48,918,919,928],{},[51,920,921],{},[54,922,923,925],{},[61,924,404],{},[61,926,927],{},"3",[65,929,930],{"encoding":67},"\\frac{2}{3}",[35,932,934],{"className":933,"ariaHidden":73},[72],[35,935,937,940],{"className":936},[77],[35,938],{"className":939,"style":293},[81],[35,941,943,946,1005],{"className":942},[86],[35,944],{"className":945},[90,91],[35,947,949],{"className":948},[54],[35,950,952,997],{"className":951},[98,99],[35,953,955,994],{"className":954},[103],[35,956,958,972,980],{"className":957,"style":312},[107],[35,959,960,963],{"style":111},[35,961],{"className":962,"style":116},[115],[35,964,966],{"className":965},[120,121,122,123],[35,967,969],{"className":968},[86,123],[35,970,927],{"className":971},[86,123],[35,973,974,977],{"style":132},[35,975],{"className":976,"style":116},[115],[35,978],{"className":979,"style":140},[139],[35,981,982,985],{"style":143},[35,983],{"className":984,"style":116},[115],[35,986,988],{"className":987},[120,121,122,123],[35,989,991],{"className":990},[86,123],[35,992,404],{"className":993},[86,123],[35,995,162],{"className":996},[161],[35,998,1000],{"className":999},[103],[35,1001,1003],{"className":1002,"style":169},[107],[35,1004],{},[35,1006],{"className":1007},[175,91]," (2 caras de 3 lanzamientos). Probabilidad: ",[35,1010,1012,1052],{"className":1011},[38],[35,1013,1015],{"className":1014},[42],[44,1016,1017],{"xmlns":46},[48,1018,1019,1049],{},[51,1020,1021,1040,1042],{},[1022,1023,1024,1038],"msup",{},[51,1025,1026,1029,1035],{},[276,1027,1028],{"fence":73},"(",[54,1030,1031,1033],{},[61,1032,272],{},[61,1034,404],{},[276,1036,1037],{"fence":73},")",[61,1039,927],{},[276,1041,278],{},[54,1043,1044,1046],{},[61,1045,272],{},[61,1047,1048],{},"8",[65,1050,1051],{"encoding":67},"\\left(\\frac{1}{2}\\right)^3 = \\frac{1}{8}",[35,1053,1055,1189],{"className":1054,"ariaHidden":73},[72],[35,1056,1058,1062,1180,1183,1186],{"className":1057},[77],[35,1059],{"className":1060,"style":1061},[81],"height:1.404em;vertical-align:-0.35em;",[35,1063,1066,1153],{"className":1064},[1065],"minner",[35,1067,1069,1079,1147],{"className":1068},[1065],[35,1070,1074],{"className":1071,"style":1073},[90,1072],"delimcenter","top:0em;",[35,1075,1028],{"className":1076},[1077,1078],"delimsizing","size1",[35,1080,1082,1085,1144],{"className":1081},[86],[35,1083],{"className":1084},[90,91],[35,1086,1088],{"className":1087},[54],[35,1089,1091,1136],{"className":1090},[98,99],[35,1092,1094,1133],{"className":1093},[103],[35,1095,1097,1111,1119],{"className":1096,"style":312},[107],[35,1098,1099,1102],{"style":111},[35,1100],{"className":1101,"style":116},[115],[35,1103,1105],{"className":1104},[120,121,122,123],[35,1106,1108],{"className":1107},[86,123],[35,1109,404],{"className":1110},[86,123],[35,1112,1113,1116],{"style":132},[35,1114],{"className":1115,"style":116},[115],[35,1117],{"className":1118,"style":140},[139],[35,1120,1121,1124],{"style":143},[35,1122],{"className":1123,"style":116},[115],[35,1125,1127],{"className":1126},[120,121,122,123],[35,1128,1130],{"className":1129},[86,123],[35,1131,272],{"className":1132},[86,123],[35,1134,162],{"className":1135},[161],[35,1137,1139],{"className":1138},[103],[35,1140,1142],{"className":1141,"style":169},[107],[35,1143],{},[35,1145],{"className":1146},[175,91],[35,1148,1150],{"className":1149,"style":1073},[175,1072],[35,1151,1037],{"className":1152},[1077,1078],[35,1154,1157],{"className":1155},[1156],"msupsub",[35,1158,1160],{"className":1159},[98],[35,1161,1163],{"className":1162},[103],[35,1164,1167],{"className":1165,"style":1166},[107],"height:1.054em;",[35,1168,1170,1174],{"style":1169},"top:-3.3029em;margin-right:0.05em;",[35,1171],{"className":1172,"style":1173},[115],"height:2.7em;",[35,1175,1177],{"className":1176},[120,121,122,123],[35,1178,927],{"className":1179},[86,123],[35,1181],{"className":1182,"style":367},[366],[35,1184,278],{"className":1185},[371],[35,1187],{"className":1188,"style":367},[366],[35,1190,1192,1195],{"className":1191},[77],[35,1193],{"className":1194,"style":293},[81],[35,1196,1198,1201,1260],{"className":1197},[86],[35,1199],{"className":1200},[90,91],[35,1202,1204],{"className":1203},[54],[35,1205,1207,1252],{"className":1206},[98,99],[35,1208,1210,1249],{"className":1209},[103],[35,1211,1213,1227,1235],{"className":1212,"style":312},[107],[35,1214,1215,1218],{"style":111},[35,1216],{"className":1217,"style":116},[115],[35,1219,1221],{"className":1220},[120,121,122,123],[35,1222,1224],{"className":1223},[86,123],[35,1225,1048],{"className":1226},[86,123],[35,1228,1229,1232],{"style":132},[35,1230],{"className":1231,"style":116},[115],[35,1233],{"className":1234,"style":140},[139],[35,1236,1237,1240],{"style":143},[35,1238],{"className":1239,"style":116},[115],[35,1241,1243],{"className":1242},[120,121,122,123],[35,1244,1246],{"className":1245},[86,123],[35,1247,272],{"className":1248},[86,123],[35,1250,162],{"className":1251},[161],[35,1253,1255],{"className":1254},[103],[35,1256,1258],{"className":1257,"style":169},[107],[35,1259],{},[35,1261],{"className":1262},[175,91],". Y es la única secuencia posible de esta longitud.",[582,1265,1267],{"id":1266},"longitud-5","Longitud 5",[17,1269,1270,1271,1274],{},"Aquí se pone interesante. Necesitamos 3 caras y 2 cruces, y el quinto lanzamiento (el que nos detiene) tiene que ser cara. Además — y esto es clave — ",[239,1272,1273],{},"en ningún momento antes del final podemos haber tenido más caras que cruces",", porque si eso hubiera ocurrido, nos habríamos detenido antes.",[17,1276,1277,1278,1281],{},"No basta con que la secuencia termine con más caras que cruces; tiene que ser ",[239,1279,1280],{},"la primera vez"," que ocurre.",[17,1283,1284],{},"¿Cuántas secuencias cumplen estas condiciones? Sólo dos:",[1286,1287,1292],"pre",{"className":1288,"code":1290,"language":625,"meta":1291},[1289],"language-text","X X O O O\nX O X O O\n","",[1293,1294,1290],"code",{"__ignoreMap":1291},[17,1296,1297,1298,1301,1302,1338,1339,1374],{},"Comprobemos la primera: ",[239,1299,1300],{},"XXOOO",". Si contamos O como ",[35,1303,1305,1322],{"className":1304},[38],[35,1306,1308],{"className":1307},[42],[44,1309,1310],{"xmlns":46},[48,1311,1312,1319],{},[51,1313,1314,1317],{},[276,1315,1316],{},"+",[61,1318,272],{},[65,1320,1321],{"encoding":67},"+1",[35,1323,1325],{"className":1324,"ariaHidden":73},[72],[35,1326,1328,1332,1335],{"className":1327},[77],[35,1329],{"className":1330,"style":1331},[81],"height:0.7278em;vertical-align:-0.0833em;",[35,1333,1316],{"className":1334},[86],[35,1336,272],{"className":1337},[86]," y X como ",[35,1340,1342,1359],{"className":1341},[38],[35,1343,1345],{"className":1344},[42],[44,1346,1347],{"xmlns":46},[48,1348,1349,1356],{},[51,1350,1351,1354],{},[276,1352,1353],{},"−",[61,1355,272],{},[65,1357,1358],{"encoding":67},"-1",[35,1360,1362],{"className":1361,"ariaHidden":73},[72],[35,1363,1365,1368,1371],{"className":1364},[77],[35,1366],{"className":1367,"style":1331},[81],[35,1369,1353],{"className":1370},[86],[35,1372,272],{"className":1373},[86],", las sumas parciales son:",[17,1376,1378],{"className":1377},[592],[35,1379,1381,1428],{"className":1380},[38],[35,1382,1384],{"className":1383},[42],[44,1385,1386],{"xmlns":46},[48,1387,1388,1425],{},[51,1389,1390,1392,1394,1396,1399,1401,1403,1405,1407,1409,1411,1413,1415,1417,1419,1421,1423],{},[276,1391,1353],{},[61,1393,272],{},[276,1395,525],{"separator":73},[606,1397,1398],{},"  ",[276,1400,1353],{},[61,1402,404],{},[276,1404,525],{"separator":73},[606,1406,1398],{},[276,1408,1353],{},[61,1410,272],{},[276,1412,525],{"separator":73},[606,1414,1398],{},[61,1416,517],{},[276,1418,525],{"separator":73},[606,1420,1398],{},[276,1422,1316],{},[61,1424,272],{},[65,1426,1427],{"encoding":67},"-1,\\; -2,\\; -1,\\; 0,\\; +1",[35,1429,1431],{"className":1430,"ariaHidden":73},[72],[35,1432,1434,1437,1440,1443,1446,1449,1453,1456,1459,1462,1465,1468,1471,1474,1477,1480,1483,1486,1489,1492,1495,1498],{"className":1433},[77],[35,1435],{"className":1436,"style":513},[81],[35,1438,1353],{"className":1439},[86],[35,1441,272],{"className":1442},[86],[35,1444,525],{"className":1445},[524],[35,1447],{"className":1448,"style":367},[366],[35,1450],{"className":1451,"style":1452},[366],"margin-right:0.1667em;",[35,1454,1353],{"className":1455},[86],[35,1457,404],{"className":1458},[86],[35,1460,525],{"className":1461},[524],[35,1463],{"className":1464,"style":367},[366],[35,1466],{"className":1467,"style":1452},[366],[35,1469,1353],{"className":1470},[86],[35,1472,272],{"className":1473},[86],[35,1475,525],{"className":1476},[524],[35,1478],{"className":1479,"style":367},[366],[35,1481],{"className":1482,"style":1452},[366],[35,1484,517],{"className":1485},[86],[35,1487,525],{"className":1488},[524],[35,1490],{"className":1491,"style":367},[366],[35,1493],{"className":1494,"style":1452},[366],[35,1496,1316],{"className":1497},[86],[35,1499,272],{"className":1500},[86],[17,1502,1503],{},"Nunca supera el cero hasta el quinto lanzamiento. ✅",[17,1505,1506,1507,176],{},"Ahora la segunda: ",[239,1508,1509],{},"XOXOO",[17,1511,1513],{"className":1512},[592],[35,1514,1516,1560],{"className":1515},[38],[35,1517,1519],{"className":1518},[42],[44,1520,1521],{"xmlns":46},[48,1522,1523,1557],{},[51,1524,1525,1527,1529,1531,1533,1535,1537,1539,1541,1543,1545,1547,1549,1551,1553,1555],{},[276,1526,1353],{},[61,1528,272],{},[276,1530,525],{"separator":73},[606,1532,1398],{},[61,1534,517],{},[276,1536,525],{"separator":73},[606,1538,1398],{},[276,1540,1353],{},[61,1542,272],{},[276,1544,525],{"separator":73},[606,1546,1398],{},[61,1548,517],{},[276,1550,525],{"separator":73},[606,1552,1398],{},[276,1554,1316],{},[61,1556,272],{},[65,1558,1559],{"encoding":67},"-1,\\; 0,\\; -1,\\; 0,\\; +1",[35,1561,1563],{"className":1562,"ariaHidden":73},[72],[35,1564,1566,1569,1572,1575,1578,1581,1584,1587,1590,1593,1596,1599,1602,1605,1608,1611,1614,1617,1620,1623,1626],{"className":1565},[77],[35,1567],{"className":1568,"style":513},[81],[35,1570,1353],{"className":1571},[86],[35,1573,272],{"className":1574},[86],[35,1576,525],{"className":1577},[524],[35,1579],{"className":1580,"style":367},[366],[35,1582],{"className":1583,"style":1452},[366],[35,1585,517],{"className":1586},[86],[35,1588,525],{"className":1589},[524],[35,1591],{"className":1592,"style":367},[366],[35,1594],{"className":1595,"style":1452},[366],[35,1597,1353],{"className":1598},[86],[35,1600,272],{"className":1601},[86],[35,1603,525],{"className":1604},[524],[35,1606],{"className":1607,"style":367},[366],[35,1609],{"className":1610,"style":1452},[366],[35,1612,517],{"className":1613},[86],[35,1615,525],{"className":1616},[524],[35,1618],{"className":1619,"style":367},[366],[35,1621],{"className":1622,"style":1452},[366],[35,1624,1316],{"className":1625},[86],[35,1627,272],{"className":1628},[86],[17,1630,1631,1632,1665],{},"Vuelve a cero en el segundo y cuarto lanzamiento, pero nunca lo supera. Es la primera vez que llega a ",[35,1633,1635,1650],{"className":1634},[38],[35,1636,1638],{"className":1637},[42],[44,1639,1640],{"xmlns":46},[48,1641,1642,1648],{},[51,1643,1644,1646],{},[276,1645,1316],{},[61,1647,272],{},[65,1649,1321],{"encoding":67},[35,1651,1653],{"className":1652,"ariaHidden":73},[72],[35,1654,1656,1659,1662],{"className":1655},[77],[35,1657],{"className":1658,"style":1331},[81],[35,1660,1316],{"className":1661},[86],[35,1663,272],{"className":1664},[86],". ✅",[17,1667,1668,1669,1672,1673,1769,1770,1803],{},"¿Y qué pasa con ",[239,1670,1671],{},"OXXOO","? Veamos: ",[35,1674,1676,1712],{"className":1675},[38],[35,1677,1679],{"className":1678},[42],[44,1680,1681],{"xmlns":46},[48,1682,1683,1709],{},[51,1684,1685,1687,1689,1691,1693,1695,1697,1699,1701,1703,1705,1707],{},[276,1686,1316],{},[61,1688,272],{},[276,1690,525],{"separator":73},[61,1692,517],{},[276,1694,525],{"separator":73},[276,1696,1353],{},[61,1698,272],{},[276,1700,525],{"separator":73},[61,1702,517],{},[276,1704,525],{"separator":73},[276,1706,1316],{},[61,1708,272],{},[65,1710,1711],{"encoding":67},"+1, 0, -1, 0, +1",[35,1713,1715],{"className":1714,"ariaHidden":73},[72],[35,1716,1718,1721,1724,1727,1730,1733,1736,1739,1742,1745,1748,1751,1754,1757,1760,1763,1766],{"className":1717},[77],[35,1719],{"className":1720,"style":513},[81],[35,1722,1316],{"className":1723},[86],[35,1725,272],{"className":1726},[86],[35,1728,525],{"className":1729},[524],[35,1731],{"className":1732,"style":1452},[366],[35,1734,517],{"className":1735},[86],[35,1737,525],{"className":1738},[524],[35,1740],{"className":1741,"style":1452},[366],[35,1743,1353],{"className":1744},[86],[35,1746,272],{"className":1747},[86],[35,1749,525],{"className":1750},[524],[35,1752],{"className":1753,"style":1452},[366],[35,1755,517],{"className":1756},[86],[35,1758,525],{"className":1759},[524],[35,1761],{"className":1762,"style":1452},[366],[35,1764,1316],{"className":1765},[86],[35,1767,272],{"className":1768},[86],". El primer lanzamiento ya nos pone en ",[35,1771,1773,1788],{"className":1772},[38],[35,1774,1776],{"className":1775},[42],[44,1777,1778],{"xmlns":46},[48,1779,1780,1786],{},[51,1781,1782,1784],{},[276,1783,1316],{},[61,1785,272],{},[65,1787,1321],{"encoding":67},[35,1789,1791],{"className":1790,"ariaHidden":73},[72],[35,1792,1794,1797,1800],{"className":1793},[77],[35,1795],{"className":1796,"style":1331},[81],[35,1798,1316],{"className":1799},[86],[35,1801,272],{"className":1802},[86],": nos habríamos detenido ahí. No es válida. ❌",[17,1805,1806,1807,1810,1811,2052,2053,2151],{},"Así que hay exactamente ",[239,1808,1809],{},"2 secuencias"," de longitud 5, cada una con probabilidad ",[35,1812,1814,1851],{"className":1813},[38],[35,1815,1817],{"className":1816},[42],[44,1818,1819],{"xmlns":46},[48,1820,1821,1848],{},[51,1822,1823,1839,1841],{},[1022,1824,1825,1837],{},[51,1826,1827,1829,1835],{},[276,1828,1028],{"fence":73},[54,1830,1831,1833],{},[61,1832,272],{},[61,1834,404],{},[276,1836,1037],{"fence":73},[61,1838,529],{},[276,1840,278],{},[54,1842,1843,1845],{},[61,1844,272],{},[61,1846,1847],{},"32",[65,1849,1850],{"encoding":67},"\\left(\\frac{1}{2}\\right)^5 = \\frac{1}{32}",[35,1852,1854,1978],{"className":1853,"ariaHidden":73},[72],[35,1855,1857,1860,1969,1972,1975],{"className":1856},[77],[35,1858],{"className":1859,"style":1061},[81],[35,1861,1863,1946],{"className":1862},[1065],[35,1864,1866,1872,1940],{"className":1865},[1065],[35,1867,1869],{"className":1868,"style":1073},[90,1072],[35,1870,1028],{"className":1871},[1077,1078],[35,1873,1875,1878,1937],{"className":1874},[86],[35,1876],{"className":1877},[90,91],[35,1879,1881],{"className":1880},[54],[35,1882,1884,1929],{"className":1883},[98,99],[35,1885,1887,1926],{"className":1886},[103],[35,1888,1890,1904,1912],{"className":1889,"style":312},[107],[35,1891,1892,1895],{"style":111},[35,1893],{"className":1894,"style":116},[115],[35,1896,1898],{"className":1897},[120,121,122,123],[35,1899,1901],{"className":1900},[86,123],[35,1902,404],{"className":1903},[86,123],[35,1905,1906,1909],{"style":132},[35,1907],{"className":1908,"style":116},[115],[35,1910],{"className":1911,"style":140},[139],[35,1913,1914,1917],{"style":143},[35,1915],{"className":1916,"style":116},[115],[35,1918,1920],{"className":1919},[120,121,122,123],[35,1921,1923],{"className":1922},[86,123],[35,1924,272],{"className":1925},[86,123],[35,1927,162],{"className":1928},[161],[35,1930,1932],{"className":1931},[103],[35,1933,1935],{"className":1934,"style":169},[107],[35,1936],{},[35,1938],{"className":1939},[175,91],[35,1941,1943],{"className":1942,"style":1073},[175,1072],[35,1944,1037],{"className":1945},[1077,1078],[35,1947,1949],{"className":1948},[1156],[35,1950,1952],{"className":1951},[98],[35,1953,1955],{"className":1954},[103],[35,1956,1958],{"className":1957,"style":1166},[107],[35,1959,1960,1963],{"style":1169},[35,1961],{"className":1962,"style":1173},[115],[35,1964,1966],{"className":1965},[120,121,122,123],[35,1967,529],{"className":1968},[86,123],[35,1970],{"className":1971,"style":367},[366],[35,1973,278],{"className":1974},[371],[35,1976],{"className":1977,"style":367},[366],[35,1979,1981,1984],{"className":1980},[77],[35,1982],{"className":1983,"style":293},[81],[35,1985,1987,1990,2049],{"className":1986},[86],[35,1988],{"className":1989},[90,91],[35,1991,1993],{"className":1992},[54],[35,1994,1996,2041],{"className":1995},[98,99],[35,1997,1999,2038],{"className":1998},[103],[35,2000,2002,2016,2024],{"className":2001,"style":312},[107],[35,2003,2004,2007],{"style":111},[35,2005],{"className":2006,"style":116},[115],[35,2008,2010],{"className":2009},[120,121,122,123],[35,2011,2013],{"className":2012},[86,123],[35,2014,1847],{"className":2015},[86,123],[35,2017,2018,2021],{"style":132},[35,2019],{"className":2020,"style":116},[115],[35,2022],{"className":2023,"style":140},[139],[35,2025,2026,2029],{"style":143},[35,2027],{"className":2028,"style":116},[115],[35,2030,2032],{"className":2031},[120,121,122,123],[35,2033,2035],{"className":2034},[86,123],[35,2036,272],{"className":2037},[86,123],[35,2039,162],{"className":2040},[161],[35,2042,2044],{"className":2043},[103],[35,2045,2047],{"className":2046,"style":169},[107],[35,2048],{},[35,2050],{"className":2051},[175,91]," y proporción ",[35,2054,2056,2074],{"className":2055},[38],[35,2057,2059],{"className":2058},[42],[44,2060,2061],{"xmlns":46},[48,2062,2063,2071],{},[51,2064,2065],{},[54,2066,2067,2069],{},[61,2068,927],{},[61,2070,529],{},[65,2072,2073],{"encoding":67},"\\frac{3}{5}",[35,2075,2077],{"className":2076,"ariaHidden":73},[72],[35,2078,2080,2083],{"className":2079},[77],[35,2081],{"className":2082,"style":293},[81],[35,2084,2086,2089,2148],{"className":2085},[86],[35,2087],{"className":2088},[90,91],[35,2090,2092],{"className":2091},[54],[35,2093,2095,2140],{"className":2094},[98,99],[35,2096,2098,2137],{"className":2097},[103],[35,2099,2101,2115,2123],{"className":2100,"style":312},[107],[35,2102,2103,2106],{"style":111},[35,2104],{"className":2105,"style":116},[115],[35,2107,2109],{"className":2108},[120,121,122,123],[35,2110,2112],{"className":2111},[86,123],[35,2113,529],{"className":2114},[86,123],[35,2116,2117,2120],{"style":132},[35,2118],{"className":2119,"style":116},[115],[35,2121],{"className":2122,"style":140},[139],[35,2124,2125,2128],{"style":143},[35,2126],{"className":2127,"style":116},[115],[35,2129,2131],{"className":2130},[120,121,122,123],[35,2132,2134],{"className":2133},[86,123],[35,2135,927],{"className":2136},[86,123],[35,2138,162],{"className":2139},[161],[35,2141,2143],{"className":2142},[103],[35,2144,2146],{"className":2145,"style":169},[107],[35,2147],{},[35,2149],{"className":2150},[175,91]," (3 caras de 5 lanzamientos).",[582,2153,2155],{"id":2154},"longitud-7","Longitud 7",[17,2157,2158,2159,2162],{},"El mismo razonamiento se aplica a secuencias de longitud 7 (4 caras, 3 cruces). Si las enumeramos todas — y créeme, lleva un rato — encontramos exactamente ",[239,2160,2161],{},"5 secuencias"," válidas.",[582,2164,2166],{"id":2165},"la-tabla-resumen","La tabla resumen",[2168,2169,2170,2249],"table",{},[2171,2172,2173],"thead",{},[2174,2175,2176,2240,2243,2246],"tr",{},[2177,2178,2180,2181,1037],"th",{"align":2179},"center","Longitud (",[35,2182,2184,2205],{"className":2183},[38],[35,2185,2187],{"className":2186},[42],[44,2188,2189],{"xmlns":46},[48,2190,2191,2202],{},[51,2192,2193,2195,2198,2200],{},[61,2194,404],{},[57,2196,2197],{},"n",[276,2199,1316],{},[61,2201,272],{},[65,2203,2204],{"encoding":67},"2n+1",[35,2206,2208,2231],{"className":2207,"ariaHidden":73},[72],[35,2209,2211,2214,2217,2220,2224,2228],{"className":2210},[77],[35,2212],{"className":2213,"style":1331},[81],[35,2215,404],{"className":2216},[86],[35,2218,2197],{"className":2219},[86,156],[35,2221],{"className":2222,"style":2223},[366],"margin-right:0.2222em;",[35,2225,1316],{"className":2226},[2227],"mbin",[35,2229],{"className":2230,"style":2223},[366],[35,2232,2234,2237],{"className":2233},[77],[35,2235],{"className":2236,"style":381},[81],[35,2238,272],{"className":2239},[86],[2177,2241,2242],{"align":2179},"Proporción de caras",[2177,2244,2245],{"align":2179},"Secuencias válidas",[2177,2247,2248],{"align":2179},"Probabilidad de cada una",[2250,2251,2252,2510,2767,3024,3284],"tbody",{},[2174,2253,2254,2309,2409,2411],{},[2255,2256,2257,2258,1037],"td",{"align":2179},"1 (",[35,2259,2261,2279],{"className":2260},[38],[35,2262,2264],{"className":2263},[42],[44,2265,2266],{"xmlns":46},[48,2267,2268,2276],{},[51,2269,2270,2272,2274],{},[57,2271,2197],{},[276,2273,278],{},[61,2275,517],{},[65,2277,2278],{"encoding":67},"n=0",[35,2280,2282,2300],{"className":2281,"ariaHidden":73},[72],[35,2283,2285,2288,2291,2294,2297],{"className":2284},[77],[35,2286],{"className":2287,"style":206},[81],[35,2289,2197],{"className":2290},[86,156],[35,2292],{"className":2293,"style":367},[366],[35,2295,278],{"className":2296},[371],[35,2298],{"className":2299,"style":367},[366],[35,2301,2303,2306],{"className":2302},[77],[35,2304],{"className":2305,"style":381},[81],[35,2307,517],{"className":2308},[86],[2255,2310,2311],{"align":2179},[35,2312,2314,2332],{"className":2313},[38],[35,2315,2317],{"className":2316},[42],[44,2318,2319],{"xmlns":46},[48,2320,2321,2329],{},[51,2322,2323],{},[54,2324,2325,2327],{},[61,2326,272],{},[61,2328,272],{},[65,2330,2331],{"encoding":67},"\\frac{1}{1}",[35,2333,2335],{"className":2334,"ariaHidden":73},[72],[35,2336,2338,2341],{"className":2337},[77],[35,2339],{"className":2340,"style":293},[81],[35,2342,2344,2347,2406],{"className":2343},[86],[35,2345],{"className":2346},[90,91],[35,2348,2350],{"className":2349},[54],[35,2351,2353,2398],{"className":2352},[98,99],[35,2354,2356,2395],{"className":2355},[103],[35,2357,2359,2373,2381],{"className":2358,"style":312},[107],[35,2360,2361,2364],{"style":111},[35,2362],{"className":2363,"style":116},[115],[35,2365,2367],{"className":2366},[120,121,122,123],[35,2368,2370],{"className":2369},[86,123],[35,2371,272],{"className":2372},[86,123],[35,2374,2375,2378],{"style":132},[35,2376],{"className":2377,"style":116},[115],[35,2379],{"className":2380,"style":140},[139],[35,2382,2383,2386],{"style":143},[35,2384],{"className":2385,"style":116},[115],[35,2387,2389],{"className":2388},[120,121,122,123],[35,2390,2392],{"className":2391},[86,123],[35,2393,272],{"className":2394},[86,123],[35,2396,162],{"className":2397},[161],[35,2399,2401],{"className":2400},[103],[35,2402,2404],{"className":2403,"style":169},[107],[35,2405],{},[35,2407],{"className":2408},[175,91],[2255,2410,272],{"align":2179},[2255,2412,2413],{"align":2179},[35,2414,2416,2433],{"className":2415},[38],[35,2417,2419],{"className":2418},[42],[44,2420,2421],{"xmlns":46},[48,2422,2423,2431],{},[51,2424,2425],{},[54,2426,2427,2429],{},[61,2428,272],{},[61,2430,404],{},[65,2432,407],{"encoding":67},[35,2434,2436],{"className":2435,"ariaHidden":73},[72],[35,2437,2439,2442],{"className":2438},[77],[35,2440],{"className":2441,"style":293},[81],[35,2443,2445,2448,2507],{"className":2444},[86],[35,2446],{"className":2447},[90,91],[35,2449,2451],{"className":2450},[54],[35,2452,2454,2499],{"className":2453},[98,99],[35,2455,2457,2496],{"className":2456},[103],[35,2458,2460,2474,2482],{"className":2459,"style":312},[107],[35,2461,2462,2465],{"style":111},[35,2463],{"className":2464,"style":116},[115],[35,2466,2468],{"className":2467},[120,121,122,123],[35,2469,2471],{"className":2470},[86,123],[35,2472,404],{"className":2473},[86,123],[35,2475,2476,2479],{"style":132},[35,2477],{"className":2478,"style":116},[115],[35,2480],{"className":2481,"style":140},[139],[35,2483,2484,2487],{"style":143},[35,2485],{"className":2486,"style":116},[115],[35,2488,2490],{"className":2489},[120,121,122,123],[35,2491,2493],{"className":2492},[86,123],[35,2494,272],{"className":2495},[86,123],[35,2497,162],{"className":2498},[161],[35,2500,2502],{"className":2501},[103],[35,2503,2505],{"className":2504,"style":169},[107],[35,2506],{},[35,2508],{"className":2509},[175,91],[2174,2511,2512,2566,2665,2667],{},[2255,2513,2514,2515,1037],{"align":2179},"3 (",[35,2516,2518,2536],{"className":2517},[38],[35,2519,2521],{"className":2520},[42],[44,2522,2523],{"xmlns":46},[48,2524,2525,2533],{},[51,2526,2527,2529,2531],{},[57,2528,2197],{},[276,2530,278],{},[61,2532,272],{},[65,2534,2535],{"encoding":67},"n=1",[35,2537,2539,2557],{"className":2538,"ariaHidden":73},[72],[35,2540,2542,2545,2548,2551,2554],{"className":2541},[77],[35,2543],{"className":2544,"style":206},[81],[35,2546,2197],{"className":2547},[86,156],[35,2549],{"className":2550,"style":367},[366],[35,2552,278],{"className":2553},[371],[35,2555],{"className":2556,"style":367},[366],[35,2558,2560,2563],{"className":2559},[77],[35,2561],{"className":2562,"style":381},[81],[35,2564,272],{"className":2565},[86],[2255,2567,2568],{"align":2179},[35,2569,2571,2588],{"className":2570},[38],[35,2572,2574],{"className":2573},[42],[44,2575,2576],{"xmlns":46},[48,2577,2578,2586],{},[51,2579,2580],{},[54,2581,2582,2584],{},[61,2583,404],{},[61,2585,927],{},[65,2587,930],{"encoding":67},[35,2589,2591],{"className":2590,"ariaHidden":73},[72],[35,2592,2594,2597],{"className":2593},[77],[35,2595],{"className":2596,"style":293},[81],[35,2598,2600,2603,2662],{"className":2599},[86],[35,2601],{"className":2602},[90,91],[35,2604,2606],{"className":2605},[54],[35,2607,2609,2654],{"className":2608},[98,99],[35,2610,2612,2651],{"className":2611},[103],[35,2613,2615,2629,2637],{"className":2614,"style":312},[107],[35,2616,2617,2620],{"style":111},[35,2618],{"className":2619,"style":116},[115],[35,2621,2623],{"className":2622},[120,121,122,123],[35,2624,2626],{"className":2625},[86,123],[35,2627,927],{"className":2628},[86,123],[35,2630,2631,2634],{"style":132},[35,2632],{"className":2633,"style":116},[115],[35,2635],{"className":2636,"style":140},[139],[35,2638,2639,2642],{"style":143},[35,2640],{"className":2641,"style":116},[115],[35,2643,2645],{"className":2644},[120,121,122,123],[35,2646,2648],{"className":2647},[86,123],[35,2649,404],{"className":2650},[86,123],[35,2652,162],{"className":2653},[161],[35,2655,2657],{"className":2656},[103],[35,2658,2660],{"className":2659,"style":169},[107],[35,2661],{},[35,2663],{"className":2664},[175,91],[2255,2666,272],{"align":2179},[2255,2668,2669],{"align":2179},[35,2670,2672,2690],{"className":2671},[38],[35,2673,2675],{"className":2674},[42],[44,2676,2677],{"xmlns":46},[48,2678,2679,2687],{},[51,2680,2681],{},[54,2682,2683,2685],{},[61,2684,272],{},[61,2686,1048],{},[65,2688,2689],{"encoding":67},"\\frac{1}{8}",[35,2691,2693],{"className":2692,"ariaHidden":73},[72],[35,2694,2696,2699],{"className":2695},[77],[35,2697],{"className":2698,"style":293},[81],[35,2700,2702,2705,2764],{"className":2701},[86],[35,2703],{"className":2704},[90,91],[35,2706,2708],{"className":2707},[54],[35,2709,2711,2756],{"className":2710},[98,99],[35,2712,2714,2753],{"className":2713},[103],[35,2715,2717,2731,2739],{"className":2716,"style":312},[107],[35,2718,2719,2722],{"style":111},[35,2720],{"className":2721,"style":116},[115],[35,2723,2725],{"className":2724},[120,121,122,123],[35,2726,2728],{"className":2727},[86,123],[35,2729,1048],{"className":2730},[86,123],[35,2732,2733,2736],{"style":132},[35,2734],{"className":2735,"style":116},[115],[35,2737],{"className":2738,"style":140},[139],[35,2740,2741,2744],{"style":143},[35,2742],{"className":2743,"style":116},[115],[35,2745,2747],{"className":2746},[120,121,122,123],[35,2748,2750],{"className":2749},[86,123],[35,2751,272],{"className":2752},[86,123],[35,2754,162],{"className":2755},[161],[35,2757,2759],{"className":2758},[103],[35,2760,2762],{"className":2761,"style":169},[107],[35,2763],{},[35,2765],{"className":2766},[175,91],[2174,2768,2769,2823,2922,2924],{},[2255,2770,2771,2772,1037],{"align":2179},"5 (",[35,2773,2775,2793],{"className":2774},[38],[35,2776,2778],{"className":2777},[42],[44,2779,2780],{"xmlns":46},[48,2781,2782,2790],{},[51,2783,2784,2786,2788],{},[57,2785,2197],{},[276,2787,278],{},[61,2789,404],{},[65,2791,2792],{"encoding":67},"n=2",[35,2794,2796,2814],{"className":2795,"ariaHidden":73},[72],[35,2797,2799,2802,2805,2808,2811],{"className":2798},[77],[35,2800],{"className":2801,"style":206},[81],[35,2803,2197],{"className":2804},[86,156],[35,2806],{"className":2807,"style":367},[366],[35,2809,278],{"className":2810},[371],[35,2812],{"className":2813,"style":367},[366],[35,2815,2817,2820],{"className":2816},[77],[35,2818],{"className":2819,"style":381},[81],[35,2821,404],{"className":2822},[86],[2255,2824,2825],{"align":2179},[35,2826,2828,2845],{"className":2827},[38],[35,2829,2831],{"className":2830},[42],[44,2832,2833],{"xmlns":46},[48,2834,2835,2843],{},[51,2836,2837],{},[54,2838,2839,2841],{},[61,2840,927],{},[61,2842,529],{},[65,2844,2073],{"encoding":67},[35,2846,2848],{"className":2847,"ariaHidden":73},[72],[35,2849,2851,2854],{"className":2850},[77],[35,2852],{"className":2853,"style":293},[81],[35,2855,2857,2860,2919],{"className":2856},[86],[35,2858],{"className":2859},[90,91],[35,2861,2863],{"className":2862},[54],[35,2864,2866,2911],{"className":2865},[98,99],[35,2867,2869,2908],{"className":2868},[103],[35,2870,2872,2886,2894],{"className":2871,"style":312},[107],[35,2873,2874,2877],{"style":111},[35,2875],{"className":2876,"style":116},[115],[35,2878,2880],{"className":2879},[120,121,122,123],[35,2881,2883],{"className":2882},[86,123],[35,2884,529],{"className":2885},[86,123],[35,2887,2888,2891],{"style":132},[35,2889],{"className":2890,"style":116},[115],[35,2892],{"className":2893,"style":140},[139],[35,2895,2896,2899],{"style":143},[35,2897],{"className":2898,"style":116},[115],[35,2900,2902],{"className":2901},[120,121,122,123],[35,2903,2905],{"className":2904},[86,123],[35,2906,927],{"className":2907},[86,123],[35,2909,162],{"className":2910},[161],[35,2912,2914],{"className":2913},[103],[35,2915,2917],{"className":2916,"style":169},[107],[35,2918],{},[35,2920],{"className":2921},[175,91],[2255,2923,404],{"align":2179},[2255,2925,2926],{"align":2179},[35,2927,2929,2947],{"className":2928},[38],[35,2930,2932],{"className":2931},[42],[44,2933,2934],{"xmlns":46},[48,2935,2936,2944],{},[51,2937,2938],{},[54,2939,2940,2942],{},[61,2941,272],{},[61,2943,1847],{},[65,2945,2946],{"encoding":67},"\\frac{1}{32}",[35,2948,2950],{"className":2949,"ariaHidden":73},[72],[35,2951,2953,2956],{"className":2952},[77],[35,2954],{"className":2955,"style":293},[81],[35,2957,2959,2962,3021],{"className":2958},[86],[35,2960],{"className":2961},[90,91],[35,2963,2965],{"className":2964},[54],[35,2966,2968,3013],{"className":2967},[98,99],[35,2969,2971,3010],{"className":2970},[103],[35,2972,2974,2988,2996],{"className":2973,"style":312},[107],[35,2975,2976,2979],{"style":111},[35,2977],{"className":2978,"style":116},[115],[35,2980,2982],{"className":2981},[120,121,122,123],[35,2983,2985],{"className":2984},[86,123],[35,2986,1847],{"className":2987},[86,123],[35,2989,2990,2993],{"style":132},[35,2991],{"className":2992,"style":116},[115],[35,2994],{"className":2995,"style":140},[139],[35,2997,2998,3001],{"style":143},[35,2999],{"className":3000,"style":116},[115],[35,3002,3004],{"className":3003},[120,121,122,123],[35,3005,3007],{"className":3006},[86,123],[35,3008,272],{"className":3009},[86,123],[35,3011,162],{"className":3012},[161],[35,3014,3016],{"className":3015},[103],[35,3017,3019],{"className":3018,"style":169},[107],[35,3020],{},[35,3022],{"className":3023},[175,91],[2174,3025,3026,3080,3181,3183],{},[2255,3027,3028,3029,1037],{"align":2179},"7 (",[35,3030,3032,3050],{"className":3031},[38],[35,3033,3035],{"className":3034},[42],[44,3036,3037],{"xmlns":46},[48,3038,3039,3047],{},[51,3040,3041,3043,3045],{},[57,3042,2197],{},[276,3044,278],{},[61,3046,927],{},[65,3048,3049],{"encoding":67},"n=3",[35,3051,3053,3071],{"className":3052,"ariaHidden":73},[72],[35,3054,3056,3059,3062,3065,3068],{"className":3055},[77],[35,3057],{"className":3058,"style":206},[81],[35,3060,2197],{"className":3061},[86,156],[35,3063],{"className":3064,"style":367},[366],[35,3066,278],{"className":3067},[371],[35,3069],{"className":3070,"style":367},[366],[35,3072,3074,3077],{"className":3073},[77],[35,3075],{"className":3076,"style":381},[81],[35,3078,927],{"className":3079},[86],[2255,3081,3082],{"align":2179},[35,3083,3085,3104],{"className":3084},[38],[35,3086,3088],{"className":3087},[42],[44,3089,3090],{"xmlns":46},[48,3091,3092,3101],{},[51,3093,3094],{},[54,3095,3096,3098],{},[61,3097,63],{},[61,3099,3100],{},"7",[65,3102,3103],{"encoding":67},"\\frac{4}{7}",[35,3105,3107],{"className":3106,"ariaHidden":73},[72],[35,3108,3110,3113],{"className":3109},[77],[35,3111],{"className":3112,"style":293},[81],[35,3114,3116,3119,3178],{"className":3115},[86],[35,3117],{"className":3118},[90,91],[35,3120,3122],{"className":3121},[54],[35,3123,3125,3170],{"className":3124},[98,99],[35,3126,3128,3167],{"className":3127},[103],[35,3129,3131,3145,3153],{"className":3130,"style":312},[107],[35,3132,3133,3136],{"style":111},[35,3134],{"className":3135,"style":116},[115],[35,3137,3139],{"className":3138},[120,121,122,123],[35,3140,3142],{"className":3141},[86,123],[35,3143,3100],{"className":3144},[86,123],[35,3146,3147,3150],{"style":132},[35,3148],{"className":3149,"style":116},[115],[35,3151],{"className":3152,"style":140},[139],[35,3154,3155,3158],{"style":143},[35,3156],{"className":3157,"style":116},[115],[35,3159,3161],{"className":3160},[120,121,122,123],[35,3162,3164],{"className":3163},[86,123],[35,3165,63],{"className":3166},[86,123],[35,3168,162],{"className":3169},[161],[35,3171,3173],{"className":3172},[103],[35,3174,3176],{"className":3175,"style":169},[107],[35,3177],{},[35,3179],{"className":3180},[175,91],[2255,3182,529],{"align":2179},[2255,3184,3185],{"align":2179},[35,3186,3188,3207],{"className":3187},[38],[35,3189,3191],{"className":3190},[42],[44,3192,3193],{"xmlns":46},[48,3194,3195,3204],{},[51,3196,3197],{},[54,3198,3199,3201],{},[61,3200,272],{},[61,3202,3203],{},"128",[65,3205,3206],{"encoding":67},"\\frac{1}{128}",[35,3208,3210],{"className":3209,"ariaHidden":73},[72],[35,3211,3213,3216],{"className":3212},[77],[35,3214],{"className":3215,"style":293},[81],[35,3217,3219,3222,3281],{"className":3218},[86],[35,3220],{"className":3221},[90,91],[35,3223,3225],{"className":3224},[54],[35,3226,3228,3273],{"className":3227},[98,99],[35,3229,3231,3270],{"className":3230},[103],[35,3232,3234,3248,3256],{"className":3233,"style":312},[107],[35,3235,3236,3239],{"style":111},[35,3237],{"className":3238,"style":116},[115],[35,3240,3242],{"className":3241},[120,121,122,123],[35,3243,3245],{"className":3244},[86,123],[35,3246,3203],{"className":3247},[86,123],[35,3249,3250,3253],{"style":132},[35,3251],{"className":3252,"style":116},[115],[35,3254],{"className":3255,"style":140},[139],[35,3257,3258,3261],{"style":143},[35,3259],{"className":3260,"style":116},[115],[35,3262,3264],{"className":3263},[120,121,122,123],[35,3265,3267],{"className":3266},[86,123],[35,3268,272],{"className":3269},[86,123],[35,3271,162],{"className":3272},[161],[35,3274,3276],{"className":3275},[103],[35,3277,3279],{"className":3278,"style":169},[107],[35,3280],{},[35,3282],{"className":3283},[175,91],[2174,3285,3286,3340,3441,3444],{},[2255,3287,3288,3289,1037],{"align":2179},"9 (",[35,3290,3292,3310],{"className":3291},[38],[35,3293,3295],{"className":3294},[42],[44,3296,3297],{"xmlns":46},[48,3298,3299,3307],{},[51,3300,3301,3303,3305],{},[57,3302,2197],{},[276,3304,278],{},[61,3306,63],{},[65,3308,3309],{"encoding":67},"n=4",[35,3311,3313,3331],{"className":3312,"ariaHidden":73},[72],[35,3314,3316,3319,3322,3325,3328],{"className":3315},[77],[35,3317],{"className":3318,"style":206},[81],[35,3320,2197],{"className":3321},[86,156],[35,3323],{"className":3324,"style":367},[366],[35,3326,278],{"className":3327},[371],[35,3329],{"className":3330,"style":367},[366],[35,3332,3334,3337],{"className":3333},[77],[35,3335],{"className":3336,"style":381},[81],[35,3338,63],{"className":3339},[86],[2255,3341,3342],{"align":2179},[35,3343,3345,3364],{"className":3344},[38],[35,3346,3348],{"className":3347},[42],[44,3349,3350],{"xmlns":46},[48,3351,3352,3361],{},[51,3353,3354],{},[54,3355,3356,3358],{},[61,3357,529],{},[61,3359,3360],{},"9",[65,3362,3363],{"encoding":67},"\\frac{5}{9}",[35,3365,3367],{"className":3366,"ariaHidden":73},[72],[35,3368,3370,3373],{"className":3369},[77],[35,3371],{"className":3372,"style":293},[81],[35,3374,3376,3379,3438],{"className":3375},[86],[35,3377],{"className":3378},[90,91],[35,3380,3382],{"className":3381},[54],[35,3383,3385,3430],{"className":3384},[98,99],[35,3386,3388,3427],{"className":3387},[103],[35,3389,3391,3405,3413],{"className":3390,"style":312},[107],[35,3392,3393,3396],{"style":111},[35,3394],{"className":3395,"style":116},[115],[35,3397,3399],{"className":3398},[120,121,122,123],[35,3400,3402],{"className":3401},[86,123],[35,3403,3360],{"className":3404},[86,123],[35,3406,3407,3410],{"style":132},[35,3408],{"className":3409,"style":116},[115],[35,3411],{"className":3412,"style":140},[139],[35,3414,3415,3418],{"style":143},[35,3416],{"className":3417,"style":116},[115],[35,3419,3421],{"className":3420},[120,121,122,123],[35,3422,3424],{"className":3423},[86,123],[35,3425,529],{"className":3426},[86,123],[35,3428,162],{"className":3429},[161],[35,3431,3433],{"className":3432},[103],[35,3434,3436],{"className":3435,"style":169},[107],[35,3437],{},[35,3439],{"className":3440},[175,91],[2255,3442,3443],{"align":2179},"14",[2255,3445,3446],{"align":2179},[35,3447,3449,3468],{"className":3448},[38],[35,3450,3452],{"className":3451},[42],[44,3453,3454],{"xmlns":46},[48,3455,3456,3465],{},[51,3457,3458],{},[54,3459,3460,3462],{},[61,3461,272],{},[61,3463,3464],{},"512",[65,3466,3467],{"encoding":67},"\\frac{1}{512}",[35,3469,3471],{"className":3470,"ariaHidden":73},[72],[35,3472,3474,3477],{"className":3473},[77],[35,3475],{"className":3476,"style":293},[81],[35,3478,3480,3483,3542],{"className":3479},[86],[35,3481],{"className":3482},[90,91],[35,3484,3486],{"className":3485},[54],[35,3487,3489,3534],{"className":3488},[98,99],[35,3490,3492,3531],{"className":3491},[103],[35,3493,3495,3509,3517],{"className":3494,"style":312},[107],[35,3496,3497,3500],{"style":111},[35,3498],{"className":3499,"style":116},[115],[35,3501,3503],{"className":3502},[120,121,122,123],[35,3504,3506],{"className":3505},[86,123],[35,3507,3464],{"className":3508},[86,123],[35,3510,3511,3514],{"style":132},[35,3512],{"className":3513,"style":116},[115],[35,3515],{"className":3516,"style":140},[139],[35,3518,3519,3522],{"style":143},[35,3520],{"className":3521,"style":116},[115],[35,3523,3525],{"className":3524},[120,121,122,123],[35,3526,3528],{"className":3527},[86,123],[35,3529,272],{"className":3530},[86,123],[35,3532,162],{"className":3533},[161],[35,3535,3537],{"className":3536},[103],[35,3538,3540],{"className":3539,"style":169},[107],[35,3541],{},[35,3543],{"className":3544},[175,91],[17,3546,3547],{},"Observa los patrones:",[3549,3550,3551,3610,3742],"ul",{},[232,3552,3553,3554,176],{},"La longitud siempre es impar: ",[35,3555,3557,3577],{"className":3556},[38],[35,3558,3560],{"className":3559},[42],[44,3561,3562],{"xmlns":46},[48,3563,3564,3574],{},[51,3565,3566,3568,3570,3572],{},[61,3567,404],{},[57,3569,2197],{},[276,3571,1316],{},[61,3573,272],{},[65,3575,3576],{"encoding":67},"2n + 1",[35,3578,3580,3601],{"className":3579,"ariaHidden":73},[72],[35,3581,3583,3586,3589,3592,3595,3598],{"className":3582},[77],[35,3584],{"className":3585,"style":1331},[81],[35,3587,404],{"className":3588},[86],[35,3590,2197],{"className":3591},[86,156],[35,3593],{"className":3594,"style":2223},[366],[35,3596,1316],{"className":3597},[2227],[35,3599],{"className":3600,"style":2223},[366],[35,3602,3604,3607],{"className":3603},[77],[35,3605],{"className":3606,"style":381},[81],[35,3608,272],{"className":3609},[86],[232,3611,3612,3613,176],{},"La proporción de caras es siempre ",[35,3614,3616,3648],{"className":3615},[38],[35,3617,3619],{"className":3618},[42],[44,3620,3621],{"xmlns":46},[48,3622,3623,3645],{},[51,3624,3625],{},[54,3626,3627,3635],{},[51,3628,3629,3631,3633],{},[57,3630,2197],{},[276,3632,1316],{},[61,3634,272],{},[51,3636,3637,3639,3641,3643],{},[61,3638,404],{},[57,3640,2197],{},[276,3642,1316],{},[61,3644,272],{},[65,3646,3647],{"encoding":67},"\\frac{n+1}{2n+1}",[35,3649,3651],{"className":3650,"ariaHidden":73},[72],[35,3652,3654,3658],{"className":3653},[77],[35,3655],{"className":3656,"style":3657},[81],"height:1.2484em;vertical-align:-0.4033em;",[35,3659,3661,3664,3739],{"className":3660},[86],[35,3662],{"className":3663},[90,91],[35,3665,3667],{"className":3666},[54],[35,3668,3670,3730],{"className":3669},[98,99],[35,3671,3673,3727],{"className":3672},[103],[35,3674,3676,3699,3707],{"className":3675,"style":312},[107],[35,3677,3678,3681],{"style":111},[35,3679],{"className":3680,"style":116},[115],[35,3682,3684],{"className":3683},[120,121,122,123],[35,3685,3687,3690,3693,3696],{"className":3686},[86,123],[35,3688,404],{"className":3689},[86,123],[35,3691,2197],{"className":3692},[86,156,123],[35,3694,1316],{"className":3695},[2227,123],[35,3697,272],{"className":3698},[86,123],[35,3700,3701,3704],{"style":132},[35,3702],{"className":3703,"style":116},[115],[35,3705],{"className":3706,"style":140},[139],[35,3708,3709,3712],{"style":143},[35,3710],{"className":3711,"style":116},[115],[35,3713,3715],{"className":3714},[120,121,122,123],[35,3716,3718,3721,3724],{"className":3717},[86,123],[35,3719,2197],{"className":3720},[86,156,123],[35,3722,1316],{"className":3723},[2227,123],[35,3725,272],{"className":3726},[86,123],[35,3728,162],{"className":3729},[161],[35,3731,3733],{"className":3732},[103],[35,3734,3737],{"className":3735,"style":3736},[107],"height:0.4033em;",[35,3738],{},[35,3740],{"className":3741},[175,91],[232,3743,3744,3745,3914,3915,3970],{},"La probabilidad de cada secuencia individual es ",[35,3746,3748,3784],{"className":3747},[38],[35,3749,3751],{"className":3750},[42],[44,3752,3753],{"xmlns":46},[48,3754,3755,3781],{},[51,3756,3757],{},[1022,3758,3759,3771],{},[51,3760,3761,3763,3769],{},[276,3762,1028],{"fence":73},[54,3764,3765,3767],{},[61,3766,272],{},[61,3768,404],{},[276,3770,1037],{"fence":73},[51,3772,3773,3775,3777,3779],{},[61,3774,404],{},[57,3776,2197],{},[276,3778,1316],{},[61,3780,272],{},[65,3782,3783],{"encoding":67},"\\left(\\frac{1}{2}\\right)^{2n+1}",[35,3785,3787],{"className":3786,"ariaHidden":73},[72],[35,3788,3790,3793],{"className":3789},[77],[35,3791],{"className":3792,"style":1061},[81],[35,3794,3796,3879],{"className":3795},[1065],[35,3797,3799,3805,3873],{"className":3798},[1065],[35,3800,3802],{"className":3801,"style":1073},[90,1072],[35,3803,1028],{"className":3804},[1077,1078],[35,3806,3808,3811,3870],{"className":3807},[86],[35,3809],{"className":3810},[90,91],[35,3812,3814],{"className":3813},[54],[35,3815,3817,3862],{"className":3816},[98,99],[35,3818,3820,3859],{"className":3819},[103],[35,3821,3823,3837,3845],{"className":3822,"style":312},[107],[35,3824,3825,3828],{"style":111},[35,3826],{"className":3827,"style":116},[115],[35,3829,3831],{"className":3830},[120,121,122,123],[35,3832,3834],{"className":3833},[86,123],[35,3835,404],{"className":3836},[86,123],[35,3838,3839,3842],{"style":132},[35,3840],{"className":3841,"style":116},[115],[35,3843],{"className":3844,"style":140},[139],[35,3846,3847,3850],{"style":143},[35,3848],{"className":3849,"style":116},[115],[35,3851,3853],{"className":3852},[120,121,122,123],[35,3854,3856],{"className":3855},[86,123],[35,3857,272],{"className":3858},[86,123],[35,3860,162],{"className":3861},[161],[35,3863,3865],{"className":3864},[103],[35,3866,3868],{"className":3867,"style":169},[107],[35,3869],{},[35,3871],{"className":3872},[175,91],[35,3874,3876],{"className":3875,"style":1073},[175,1072],[35,3877,1037],{"className":3878},[1077,1078],[35,3880,3882],{"className":3881},[1156],[35,3883,3885],{"className":3884},[98],[35,3886,3888],{"className":3887},[103],[35,3889,3891],{"className":3890,"style":1166},[107],[35,3892,3893,3896],{"style":1169},[35,3894],{"className":3895,"style":1173},[115],[35,3897,3899],{"className":3898},[120,121,122,123],[35,3900,3902,3905,3908,3911],{"className":3901},[86,123],[35,3903,404],{"className":3904},[86,123],[35,3906,2197],{"className":3907},[86,156,123],[35,3909,1316],{"className":3910},[2227,123],[35,3912,272],{"className":3913},[86,123]," — cada uno de los ",[35,3916,3918,3937],{"className":3917},[38],[35,3919,3921],{"className":3920},[42],[44,3922,3923],{"xmlns":46},[48,3924,3925,3935],{},[51,3926,3927,3929,3931,3933],{},[61,3928,404],{},[57,3930,2197],{},[276,3932,1316],{},[61,3934,272],{},[65,3936,2204],{"encoding":67},[35,3938,3940,3961],{"className":3939,"ariaHidden":73},[72],[35,3941,3943,3946,3949,3952,3955,3958],{"className":3942},[77],[35,3944],{"className":3945,"style":1331},[81],[35,3947,404],{"className":3948},[86],[35,3950,2197],{"className":3951},[86,156],[35,3953],{"className":3954,"style":2223},[366],[35,3956,1316],{"className":3957},[2227],[35,3959],{"className":3960,"style":2223},[366],[35,3962,3964,3967],{"className":3963},[77],[35,3965],{"className":3966,"style":381},[81],[35,3968,272],{"className":3969},[86]," lanzamientos tiene una probabilidad de un medio.",[17,3972,3973,3974],{},"Pero la columna que realmente importa es la de las secuencias válidas: ",[35,3975,3977,4012],{"className":3976},[38],[35,3978,3980],{"className":3979},[42],[44,3981,3982],{"xmlns":46},[48,3983,3984,4009],{},[51,3985,3986,3988,3990,3992,3994,3996,3998,4000,4002,4004,4006],{},[61,3987,272],{},[276,3989,525],{"separator":73},[61,3991,272],{},[276,3993,525],{"separator":73},[61,3995,404],{},[276,3997,525],{"separator":73},[61,3999,529],{},[276,4001,525],{"separator":73},[61,4003,3443],{},[276,4005,525],{"separator":73},[276,4007,4008],{},"…",[65,4010,4011],{"encoding":67},"1, 1, 2, 5, 14, \\ldots",[35,4013,4015],{"className":4014,"ariaHidden":73},[72],[35,4016,4018,4021,4024,4027,4030,4033,4036,4039,4042,4045,4048,4051,4054,4057,4060,4063,4066],{"className":4017},[77],[35,4019],{"className":4020,"style":513},[81],[35,4022,272],{"className":4023},[86],[35,4025,525],{"className":4026},[524],[35,4028],{"className":4029,"style":1452},[366],[35,4031,272],{"className":4032},[86],[35,4034,525],{"className":4035},[524],[35,4037],{"className":4038,"style":1452},[366],[35,4040,404],{"className":4041},[86],[35,4043,525],{"className":4044},[524],[35,4046],{"className":4047,"style":1452},[366],[35,4049,529],{"className":4050},[86],[35,4052,525],{"className":4053},[524],[35,4055],{"className":4056,"style":1452},[366],[35,4058,3443],{"className":4059},[86],[35,4061,525],{"className":4062},[524],[35,4064],{"className":4065,"style":1452},[366],[35,4067,4008],{"className":4068},[1065],[17,4070,4071],{},"¿Te suena?",[561,4073,4075],{"id":4074},"del-conteo-manual-a-los-números-de-catalan","Del conteo manual a los números de Catalan",[17,4077,4078,4079,4201,4202,4209,4210,4213],{},"Esos números — ",[35,4080,4082,4126],{"className":4081},[38],[35,4083,4085],{"className":4084},[42],[44,4086,4087],{"xmlns":46},[48,4088,4089,4123],{},[51,4090,4091,4093,4095,4097,4099,4101,4103,4105,4107,4109,4111,4114,4116,4119,4121],{},[61,4092,272],{},[276,4094,525],{"separator":73},[61,4096,272],{},[276,4098,525],{"separator":73},[61,4100,404],{},[276,4102,525],{"separator":73},[61,4104,529],{},[276,4106,525],{"separator":73},[61,4108,3443],{},[276,4110,525],{"separator":73},[61,4112,4113],{},"42",[276,4115,525],{"separator":73},[61,4117,4118],{},"132",[276,4120,525],{"separator":73},[276,4122,4008],{},[65,4124,4125],{"encoding":67},"1, 1, 2, 5, 14, 42, 132, \\ldots",[35,4127,4129],{"className":4128,"ariaHidden":73},[72],[35,4130,4132,4135,4138,4141,4144,4147,4150,4153,4156,4159,4162,4165,4168,4171,4174,4177,4180,4183,4186,4189,4192,4195,4198],{"className":4131},[77],[35,4133],{"className":4134,"style":513},[81],[35,4136,272],{"className":4137},[86],[35,4139,525],{"className":4140},[524],[35,4142],{"className":4143,"style":1452},[366],[35,4145,272],{"className":4146},[86],[35,4148,525],{"className":4149},[524],[35,4151],{"className":4152,"style":1452},[366],[35,4154,404],{"className":4155},[86],[35,4157,525],{"className":4158},[524],[35,4160],{"className":4161,"style":1452},[366],[35,4163,529],{"className":4164},[86],[35,4166,525],{"className":4167},[524],[35,4169],{"className":4170,"style":1452},[366],[35,4172,3443],{"className":4173},[86],[35,4175,525],{"className":4176},[524],[35,4178],{"className":4179,"style":1452},[366],[35,4181,4113],{"className":4182},[86],[35,4184,525],{"className":4185},[524],[35,4187],{"className":4188,"style":1452},[366],[35,4190,4118],{"className":4191},[86],[35,4193,525],{"className":4194},[524],[35,4196],{"className":4197,"style":1452},[366],[35,4199,4008],{"className":4200},[1065]," — tienen nombre propio. Son los ",[21,4203,4206],{"href":4204,"rel":4205},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FN%C3%BAmeros_de_Catalan",[25],[239,4207,4208],{},"números de Catalan",", una de las secuencias más ubicuas de toda la combinatoria. (Y no, no son de origen ",[212,4211,4212],{},"Catalán",".)",[17,4215,4216,4217,4224,4225,4228,4229,4232,4233,4238,4239,4242],{},"Su historia es en sí misma una historia de conexiones inesperadas. El matemático mongol ",[21,4218,4221],{"href":4219,"rel":4220},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FMinggatu",[25],[239,4222,4223],{},"Mingantu"," (c. 1692–1763), trabajando en China, fue la primera persona conocida en utilizarlos — alrededor de la década de 1730, en series trigonométricas incluidas en su libro ",[212,4226,4227],{},"Ge Yuan Mi Lu Jie Fa",". Dos décadas más tarde, en 1751, ",[239,4230,4231],{},"Leonhard Euler"," ",[21,4234,4237],{"href":4235,"rel":4236},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FN%C3%BAmeros_de_Catalan#Historia",[25],"descubrió la misma secuencia de forma independiente"," en una carta a Goldbach, esta vez contando las formas de dividir un polígono en triángulos. Y casi un siglo después de Mingantu, el matemático franco-belga ",[239,4240,4241],{},"Eugène Charles Catalan"," (1814–1894) los encontró de nuevo en 1838, al contar las formas de colocar paréntesis en un producto de factores. La secuencia lleva su nombre — acuñado por John Riordan en el siglo XX — a pesar de que los números habían estado apareciendo, bajo diferentes formas, durante más de cien años antes que él.",[17,4244,4245],{},"La misma secuencia, descubierta de forma independiente en tres partes del mundo y a lo largo de tres siglos, cada vez en un contexto completamente distinto. Si ese patrón te suena familiar... sigue leyendo.",[17,4247,4248,4249,4277],{},"El ",[35,4250,4252,4265],{"className":4251},[38],[35,4253,4255],{"className":4254},[42],[44,4256,4257],{"xmlns":46},[48,4258,4259,4263],{},[51,4260,4261],{},[57,4262,2197],{},[65,4264,2197],{"encoding":67},[35,4266,4268],{"className":4267,"ariaHidden":73},[72],[35,4269,4271,4274],{"className":4270},[77],[35,4272],{"className":4273,"style":206},[81],[35,4275,2197],{"className":4276},[86,156],"-ésimo número de Catalan se define como:",[17,4279,4281],{"className":4280},[592],[35,4282,4284,4335],{"className":4283},[38],[35,4285,4287],{"className":4286},[42],[44,4288,4289],{"xmlns":46},[48,4290,4291,4332],{},[51,4292,4293,4301,4303,4315],{},[4294,4295,4296,4299],"msub",{},[57,4297,4298],{},"C",[57,4300,2197],{},[276,4302,278],{},[54,4304,4305,4307],{},[61,4306,272],{},[51,4308,4309,4311,4313],{},[57,4310,2197],{},[276,4312,1316],{},[61,4314,272],{},[51,4316,4317,4319,4330],{},[276,4318,1028],{"fence":73},[54,4320,4322,4328],{"linethickness":4321},"0px",[51,4323,4324,4326],{},[61,4325,404],{},[57,4327,2197],{},[57,4329,2197],{},[276,4331,1037],{"fence":73},[65,4333,4334],{"encoding":67},"C_n = \\frac{1}{n+1}\\binom{2n}{n}",[35,4336,4338,4398],{"className":4337,"ariaHidden":73},[72],[35,4339,4341,4345,4389,4392,4395],{"className":4340},[77],[35,4342],{"className":4343,"style":4344},[81],"height:0.8333em;vertical-align:-0.15em;",[35,4346,4348,4352],{"className":4347},[86],[35,4349,4298],{"className":4350,"style":4351},[86,156],"margin-right:0.0715em;",[35,4353,4355],{"className":4354},[1156],[35,4356,4358,4380],{"className":4357},[98,99],[35,4359,4361,4377],{"className":4360},[103],[35,4362,4365],{"className":4363,"style":4364},[107],"height:0.1514em;",[35,4366,4368,4371],{"style":4367},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[35,4369],{"className":4370,"style":1173},[115],[35,4372,4374],{"className":4373},[120,121,122,123],[35,4375,2197],{"className":4376},[86,156,123],[35,4378,162],{"className":4379},[161],[35,4381,4383],{"className":4382},[103],[35,4384,4387],{"className":4385,"style":4386},[107],"height:0.15em;",[35,4388],{},[35,4390],{"className":4391,"style":367},[366],[35,4393,278],{"className":4394},[371],[35,4396],{"className":4397,"style":367},[366],[35,4399,4401,4405,4479],{"className":4400},[77],[35,4402],{"className":4403,"style":4404},[81],"height:1.2984em;vertical-align:-0.4033em;",[35,4406,4408,4411,4476],{"className":4407},[86],[35,4409],{"className":4410},[90,91],[35,4412,4414],{"className":4413},[54],[35,4415,4417,4468],{"className":4416},[98,99],[35,4418,4420,4465],{"className":4419},[103],[35,4421,4423,4443,4451],{"className":4422,"style":312},[107],[35,4424,4425,4428],{"style":111},[35,4426],{"className":4427,"style":116},[115],[35,4429,4431],{"className":4430},[120,121,122,123],[35,4432,4434,4437,4440],{"className":4433},[86,123],[35,4435,2197],{"className":4436},[86,156,123],[35,4438,1316],{"className":4439},[2227,123],[35,4441,272],{"className":4442},[86,123],[35,4444,4445,4448],{"style":132},[35,4446],{"className":4447,"style":116},[115],[35,4449],{"className":4450,"style":140},[139],[35,4452,4453,4456],{"style":143},[35,4454],{"className":4455,"style":116},[115],[35,4457,4459],{"className":4458},[120,121,122,123],[35,4460,4462],{"className":4461},[86,123],[35,4463,272],{"className":4464},[86,123],[35,4466,162],{"className":4467},[161],[35,4469,4471],{"className":4470},[103],[35,4472,4474],{"className":4473,"style":3736},[107],[35,4475],{},[35,4477],{"className":4478},[175,91],[35,4480,4482,4488,4545],{"className":4481},[86],[35,4483,4485],{"className":4484,"style":1073},[90,1072],[35,4486,1028],{"className":4487},[1077,1078],[35,4489,4491],{"className":4490},[54],[35,4492,4494,4537],{"className":4493},[98,99],[35,4495,4497,4534],{"className":4496},[103],[35,4498,4501,4516],{"className":4499,"style":4500},[107],"height:0.8951em;",[35,4502,4504,4507],{"style":4503},"top:-2.355em;",[35,4505],{"className":4506,"style":1173},[115],[35,4508,4510],{"className":4509},[120,121,122,123],[35,4511,4513],{"className":4512},[86,123],[35,4514,2197],{"className":4515},[86,156,123],[35,4517,4519,4522],{"style":4518},"top:-3.144em;",[35,4520],{"className":4521,"style":1173},[115],[35,4523,4525],{"className":4524},[120,121,122,123],[35,4526,4528,4531],{"className":4527},[86,123],[35,4529,404],{"className":4530},[86,123],[35,4532,2197],{"className":4533},[86,156,123],[35,4535,162],{"className":4536},[161],[35,4538,4540],{"className":4539},[103],[35,4541,4543],{"className":4542,"style":169},[107],[35,4544],{},[35,4546,4548],{"className":4547,"style":1073},[175,1072],[35,4549,1037],{"className":4550},[1077,1078],[17,4552,4553,4554,4662,4663,4697,4698,4726,4727,4730,4731,4759,4760,4793,4794,4822,4823,4856,4857,176],{},"donde ",[35,4555,4557,4583],{"className":4556},[38],[35,4558,4560],{"className":4559},[42],[44,4561,4562],{"xmlns":46},[48,4563,4564,4580],{},[51,4565,4566,4568,4578],{},[276,4567,1028],{"fence":73},[54,4569,4570,4576],{"linethickness":4321},[51,4571,4572,4574],{},[61,4573,404],{},[57,4575,2197],{},[57,4577,2197],{},[276,4579,1037],{"fence":73},[65,4581,4582],{"encoding":67},"\\binom{2n}{n}",[35,4584,4586],{"className":4585,"ariaHidden":73},[72],[35,4587,4589,4593],{"className":4588},[77],[35,4590],{"className":4591,"style":4592},[81],"height:1.2451em;vertical-align:-0.35em;",[35,4594,4596,4602,4656],{"className":4595},[86],[35,4597,4599],{"className":4598,"style":1073},[90,1072],[35,4600,1028],{"className":4601},[1077,1078],[35,4603,4605],{"className":4604},[54],[35,4606,4608,4648],{"className":4607},[98,99],[35,4609,4611,4645],{"className":4610},[103],[35,4612,4614,4628],{"className":4613,"style":4500},[107],[35,4615,4616,4619],{"style":4503},[35,4617],{"className":4618,"style":1173},[115],[35,4620,4622],{"className":4621},[120,121,122,123],[35,4623,4625],{"className":4624},[86,123],[35,4626,2197],{"className":4627},[86,156,123],[35,4629,4630,4633],{"style":4518},[35,4631],{"className":4632,"style":1173},[115],[35,4634,4636],{"className":4635},[120,121,122,123],[35,4637,4639,4642],{"className":4638},[86,123],[35,4640,404],{"className":4641},[86,123],[35,4643,2197],{"className":4644},[86,156,123],[35,4646,162],{"className":4647},[161],[35,4649,4651],{"className":4650},[103],[35,4652,4654],{"className":4653,"style":169},[107],[35,4655],{},[35,4657,4659],{"className":4658,"style":1073},[175,1072],[35,4660,1037],{"className":4661},[1077,1078]," — se lee «",[35,4664,4666,4682],{"className":4665},[38],[35,4667,4669],{"className":4668},[42],[44,4670,4671],{"xmlns":46},[48,4672,4673,4679],{},[51,4674,4675,4677],{},[61,4676,404],{},[57,4678,2197],{},[65,4680,4681],{"encoding":67},"2n",[35,4683,4685],{"className":4684,"ariaHidden":73},[72],[35,4686,4688,4691,4694],{"className":4687},[77],[35,4689],{"className":4690,"style":381},[81],[35,4692,404],{"className":4693},[86],[35,4695,2197],{"className":4696},[86,156]," sobre ",[35,4699,4701,4714],{"className":4700},[38],[35,4702,4704],{"className":4703},[42],[44,4705,4706],{"xmlns":46},[48,4707,4708,4712],{},[51,4709,4710],{},[57,4711,2197],{},[65,4713,2197],{"encoding":67},[35,4715,4717],{"className":4716,"ariaHidden":73},[72],[35,4718,4720,4723],{"className":4719},[77],[35,4721],{"className":4722,"style":206},[81],[35,4724,2197],{"className":4725},[86,156],"» — es el ",[239,4728,4729],{},"coeficiente binomial",": el número de formas de elegir ",[35,4732,4734,4747],{"className":4733},[38],[35,4735,4737],{"className":4736},[42],[44,4738,4739],{"xmlns":46},[48,4740,4741,4745],{},[51,4742,4743],{},[57,4744,2197],{},[65,4746,2197],{"encoding":67},[35,4748,4750],{"className":4749,"ariaHidden":73},[72],[35,4751,4753,4756],{"className":4752},[77],[35,4754],{"className":4755,"style":206},[81],[35,4757,2197],{"className":4758},[86,156]," elementos de un conjunto de ",[35,4761,4763,4778],{"className":4762},[38],[35,4764,4766],{"className":4765},[42],[44,4767,4768],{"xmlns":46},[48,4769,4770,4776],{},[51,4771,4772,4774],{},[61,4773,404],{},[57,4775,2197],{},[65,4777,4681],{"encoding":67},[35,4779,4781],{"className":4780,"ariaHidden":73},[72],[35,4782,4784,4787,4790],{"className":4783},[77],[35,4785],{"className":4786,"style":381},[81],[35,4788,404],{"className":4789},[86],[35,4791,2197],{"className":4792},[86,156],". En la práctica, responde a preguntas como «¿de cuántas formas puedo distribuir ",[35,4795,4797,4810],{"className":4796},[38],[35,4798,4800],{"className":4799},[42],[44,4801,4802],{"xmlns":46},[48,4803,4804,4808],{},[51,4805,4806],{},[57,4807,2197],{},[65,4809,2197],{"encoding":67},[35,4811,4813],{"className":4812,"ariaHidden":73},[72],[35,4814,4816,4819],{"className":4815},[77],[35,4817],{"className":4818,"style":206},[81],[35,4820,2197],{"className":4821},[86,156]," caras en ",[35,4824,4826,4841],{"className":4825},[38],[35,4827,4829],{"className":4828},[42],[44,4830,4831],{"xmlns":46},[48,4832,4833,4839],{},[51,4834,4835,4837],{},[61,4836,404],{},[57,4838,2197],{},[65,4840,4681],{"encoding":67},[35,4842,4844],{"className":4843,"ariaHidden":73},[72],[35,4845,4847,4850,4853],{"className":4846},[77],[35,4848],{"className":4849,"style":381},[81],[35,4851,404],{"className":4852},[86],[35,4854,2197],{"className":4855},[86,156]," lanzamientos?» Exploramos esta notación en detalle en el ",[21,4858,4860],{"href":221,"rel":4859},[25],"artículo sobre el triángulo de Tartaglia",[17,4862,4863],{},"Comprobemos con nuestros datos:",[17,4865,4867],{"className":4866},[592],[35,4868,4870,4945],{"className":4869},[38],[35,4871,4873],{"className":4872},[42],[44,4874,4875],{"xmlns":46},[48,4876,4877,4942],{},[51,4878,4879,4885,4887,4893,4905,4907,4909,4912,4918,4920,4926,4938,4940],{},[4294,4880,4881,4883],{},[57,4882,4298],{},[61,4884,517],{},[276,4886,278],{},[54,4888,4889,4891],{},[61,4890,272],{},[61,4892,272],{},[51,4894,4895,4897,4903],{},[276,4896,1028],{"fence":73},[54,4898,4899,4901],{"linethickness":4321},[61,4900,517],{},[61,4902,517],{},[276,4904,1037],{"fence":73},[276,4906,278],{},[61,4908,272],{},[366,4910],{"width":4911},"2em",[4294,4913,4914,4916],{},[57,4915,4298],{},[61,4917,272],{},[276,4919,278],{},[54,4921,4922,4924],{},[61,4923,272],{},[61,4925,404],{},[51,4927,4928,4930,4936],{},[276,4929,1028],{"fence":73},[54,4931,4932,4934],{"linethickness":4321},[61,4933,404],{},[61,4935,272],{},[276,4937,1037],{"fence":73},[276,4939,278],{},[61,4941,272],{},[65,4943,4944],{"encoding":67},"C_0 = \\frac{1}{1}\\binom{0}{0} = 1 \\qquad C_1 = \\frac{1}{2}\\binom{2}{1} = 1",[35,4946,4948,5004,5153,5215,5364],{"className":4947,"ariaHidden":73},[72],[35,4949,4951,4954,4995,4998,5001],{"className":4950},[77],[35,4952],{"className":4953,"style":4344},[81],[35,4955,4957,4960],{"className":4956},[86],[35,4958,4298],{"className":4959,"style":4351},[86,156],[35,4961,4963],{"className":4962},[1156],[35,4964,4966,4987],{"className":4965},[98,99],[35,4967,4969,4984],{"className":4968},[103],[35,4970,4973],{"className":4971,"style":4972},[107],"height:0.3011em;",[35,4974,4975,4978],{"style":4367},[35,4976],{"className":4977,"style":1173},[115],[35,4979,4981],{"className":4980},[120,121,122,123],[35,4982,517],{"className":4983},[86,123],[35,4985,162],{"className":4986},[161],[35,4988,4990],{"className":4989},[103],[35,4991,4993],{"className":4992,"style":4386},[107],[35,4994],{},[35,4996],{"className":4997,"style":367},[366],[35,4999,278],{"className":5000},[371],[35,5002],{"className":5003,"style":367},[366],[35,5005,5007,5010,5078,5144,5147,5150],{"className":5006},[77],[35,5008],{"className":5009,"style":4592},[81],[35,5011,5013,5016,5075],{"className":5012},[86],[35,5014],{"className":5015},[90,91],[35,5017,5019],{"className":5018},[54],[35,5020,5022,5067],{"className":5021},[98,99],[35,5023,5025,5064],{"className":5024},[103],[35,5026,5028,5042,5050],{"className":5027,"style":312},[107],[35,5029,5030,5033],{"style":111},[35,5031],{"className":5032,"style":116},[115],[35,5034,5036],{"className":5035},[120,121,122,123],[35,5037,5039],{"className":5038},[86,123],[35,5040,272],{"className":5041},[86,123],[35,5043,5044,5047],{"style":132},[35,5045],{"className":5046,"style":116},[115],[35,5048],{"className":5049,"style":140},[139],[35,5051,5052,5055],{"style":143},[35,5053],{"className":5054,"style":116},[115],[35,5056,5058],{"className":5057},[120,121,122,123],[35,5059,5061],{"className":5060},[86,123],[35,5062,272],{"className":5063},[86,123],[35,5065,162],{"className":5066},[161],[35,5068,5070],{"className":5069},[103],[35,5071,5073],{"className":5072,"style":169},[107],[35,5074],{},[35,5076],{"className":5077},[175,91],[35,5079,5081,5087,5138],{"className":5080},[86],[35,5082,5084],{"className":5083,"style":1073},[90,1072],[35,5085,1028],{"className":5086},[1077,1078],[35,5088,5090],{"className":5089},[54],[35,5091,5093,5130],{"className":5092},[98,99],[35,5094,5096,5127],{"className":5095},[103],[35,5097,5099,5113],{"className":5098,"style":4500},[107],[35,5100,5101,5104],{"style":4503},[35,5102],{"className":5103,"style":1173},[115],[35,5105,5107],{"className":5106},[120,121,122,123],[35,5108,5110],{"className":5109},[86,123],[35,5111,517],{"className":5112},[86,123],[35,5114,5115,5118],{"style":4518},[35,5116],{"className":5117,"style":1173},[115],[35,5119,5121],{"className":5120},[120,121,122,123],[35,5122,5124],{"className":5123},[86,123],[35,5125,517],{"className":5126},[86,123],[35,5128,162],{"className":5129},[161],[35,5131,5133],{"className":5132},[103],[35,5134,5136],{"className":5135,"style":169},[107],[35,5137],{},[35,5139,5141],{"className":5140,"style":1073},[175,1072],[35,5142,1037],{"className":5143},[1077,1078],[35,5145],{"className":5146,"style":367},[366],[35,5148,278],{"className":5149},[371],[35,5151],{"className":5152,"style":367},[366],[35,5154,5156,5159,5162,5166,5206,5209,5212],{"className":5155},[77],[35,5157],{"className":5158,"style":4344},[81],[35,5160,272],{"className":5161},[86],[35,5163],{"className":5164,"style":5165},[366],"margin-right:2em;",[35,5167,5169,5172],{"className":5168},[86],[35,5170,4298],{"className":5171,"style":4351},[86,156],[35,5173,5175],{"className":5174},[1156],[35,5176,5178,5198],{"className":5177},[98,99],[35,5179,5181,5195],{"className":5180},[103],[35,5182,5184],{"className":5183,"style":4972},[107],[35,5185,5186,5189],{"style":4367},[35,5187],{"className":5188,"style":1173},[115],[35,5190,5192],{"className":5191},[120,121,122,123],[35,5193,272],{"className":5194},[86,123],[35,5196,162],{"className":5197},[161],[35,5199,5201],{"className":5200},[103],[35,5202,5204],{"className":5203,"style":4386},[107],[35,5205],{},[35,5207],{"className":5208,"style":367},[366],[35,5210,278],{"className":5211},[371],[35,5213],{"className":5214,"style":367},[366],[35,5216,5218,5221,5289,5355,5358,5361],{"className":5217},[77],[35,5219],{"className":5220,"style":4592},[81],[35,5222,5224,5227,5286],{"className":5223},[86],[35,5225],{"className":5226},[90,91],[35,5228,5230],{"className":5229},[54],[35,5231,5233,5278],{"className":5232},[98,99],[35,5234,5236,5275],{"className":5235},[103],[35,5237,5239,5253,5261],{"className":5238,"style":312},[107],[35,5240,5241,5244],{"style":111},[35,5242],{"className":5243,"style":116},[115],[35,5245,5247],{"className":5246},[120,121,122,123],[35,5248,5250],{"className":5249},[86,123],[35,5251,404],{"className":5252},[86,123],[35,5254,5255,5258],{"style":132},[35,5256],{"className":5257,"style":116},[115],[35,5259],{"className":5260,"style":140},[139],[35,5262,5263,5266],{"style":143},[35,5264],{"className":5265,"style":116},[115],[35,5267,5269],{"className":5268},[120,121,122,123],[35,5270,5272],{"className":5271},[86,123],[35,5273,272],{"className":5274},[86,123],[35,5276,162],{"className":5277},[161],[35,5279,5281],{"className":5280},[103],[35,5282,5284],{"className":5283,"style":169},[107],[35,5285],{},[35,5287],{"className":5288},[175,91],[35,5290,5292,5298,5349],{"className":5291},[86],[35,5293,5295],{"className":5294,"style":1073},[90,1072],[35,5296,1028],{"className":5297},[1077,1078],[35,5299,5301],{"className":5300},[54],[35,5302,5304,5341],{"className":5303},[98,99],[35,5305,5307,5338],{"className":5306},[103],[35,5308,5310,5324],{"className":5309,"style":4500},[107],[35,5311,5312,5315],{"style":4503},[35,5313],{"className":5314,"style":1173},[115],[35,5316,5318],{"className":5317},[120,121,122,123],[35,5319,5321],{"className":5320},[86,123],[35,5322,272],{"className":5323},[86,123],[35,5325,5326,5329],{"style":4518},[35,5327],{"className":5328,"style":1173},[115],[35,5330,5332],{"className":5331},[120,121,122,123],[35,5333,5335],{"className":5334},[86,123],[35,5336,404],{"className":5337},[86,123],[35,5339,162],{"className":5340},[161],[35,5342,5344],{"className":5343},[103],[35,5345,5347],{"className":5346,"style":169},[107],[35,5348],{},[35,5350,5352],{"className":5351,"style":1073},[175,1072],[35,5353,1037],{"className":5354},[1077,1078],[35,5356],{"className":5357,"style":367},[366],[35,5359,278],{"className":5360},[371],[35,5362],{"className":5363,"style":367},[366],[35,5365,5367,5370],{"className":5366},[77],[35,5368],{"className":5369,"style":381},[81],[35,5371,272],{"className":5372},[86],[17,5374,5376],{"className":5375},[592],[35,5377,5379,5454],{"className":5378},[38],[35,5380,5382],{"className":5381},[42],[44,5383,5384],{"xmlns":46},[48,5385,5386,5451],{},[51,5387,5388,5394,5396,5402,5414,5416,5418,5420,5426,5428,5434,5447,5449],{},[4294,5389,5390,5392],{},[57,5391,4298],{},[61,5393,404],{},[276,5395,278],{},[54,5397,5398,5400],{},[61,5399,272],{},[61,5401,927],{},[51,5403,5404,5406,5412],{},[276,5405,1028],{"fence":73},[54,5407,5408,5410],{"linethickness":4321},[61,5409,63],{},[61,5411,404],{},[276,5413,1037],{"fence":73},[276,5415,278],{},[61,5417,404],{},[366,5419],{"width":4911},[4294,5421,5422,5424],{},[57,5423,4298],{},[61,5425,927],{},[276,5427,278],{},[54,5429,5430,5432],{},[61,5431,272],{},[61,5433,63],{},[51,5435,5436,5438,5445],{},[276,5437,1028],{"fence":73},[54,5439,5440,5443],{"linethickness":4321},[61,5441,5442],{},"6",[61,5444,927],{},[276,5446,1037],{"fence":73},[276,5448,278],{},[61,5450,529],{},[65,5452,5453],{"encoding":67},"C_2 = \\frac{1}{3}\\binom{4}{2} = 2 \\qquad C_3 = \\frac{1}{4}\\binom{6}{3} = 5 ",[35,5455,5457,5512,5661,5722,5871],{"className":5456,"ariaHidden":73},[72],[35,5458,5460,5463,5503,5506,5509],{"className":5459},[77],[35,5461],{"className":5462,"style":4344},[81],[35,5464,5466,5469],{"className":5465},[86],[35,5467,4298],{"className":5468,"style":4351},[86,156],[35,5470,5472],{"className":5471},[1156],[35,5473,5475,5495],{"className":5474},[98,99],[35,5476,5478,5492],{"className":5477},[103],[35,5479,5481],{"className":5480,"style":4972},[107],[35,5482,5483,5486],{"style":4367},[35,5484],{"className":5485,"style":1173},[115],[35,5487,5489],{"className":5488},[120,121,122,123],[35,5490,404],{"className":5491},[86,123],[35,5493,162],{"className":5494},[161],[35,5496,5498],{"className":5497},[103],[35,5499,5501],{"className":5500,"style":4386},[107],[35,5502],{},[35,5504],{"className":5505,"style":367},[366],[35,5507,278],{"className":5508},[371],[35,5510],{"className":5511,"style":367},[366],[35,5513,5515,5518,5586,5652,5655,5658],{"className":5514},[77],[35,5516],{"className":5517,"style":4592},[81],[35,5519,5521,5524,5583],{"className":5520},[86],[35,5522],{"className":5523},[90,91],[35,5525,5527],{"className":5526},[54],[35,5528,5530,5575],{"className":5529},[98,99],[35,5531,5533,5572],{"className":5532},[103],[35,5534,5536,5550,5558],{"className":5535,"style":312},[107],[35,5537,5538,5541],{"style":111},[35,5539],{"className":5540,"style":116},[115],[35,5542,5544],{"className":5543},[120,121,122,123],[35,5545,5547],{"className":5546},[86,123],[35,5548,927],{"className":5549},[86,123],[35,5551,5552,5555],{"style":132},[35,5553],{"className":5554,"style":116},[115],[35,5556],{"className":5557,"style":140},[139],[35,5559,5560,5563],{"style":143},[35,5561],{"className":5562,"style":116},[115],[35,5564,5566],{"className":5565},[120,121,122,123],[35,5567,5569],{"className":5568},[86,123],[35,5570,272],{"className":5571},[86,123],[35,5573,162],{"className":5574},[161],[35,5576,5578],{"className":5577},[103],[35,5579,5581],{"className":5580,"style":169},[107],[35,5582],{},[35,5584],{"className":5585},[175,91],[35,5587,5589,5595,5646],{"className":5588},[86],[35,5590,5592],{"className":5591,"style":1073},[90,1072],[35,5593,1028],{"className":5594},[1077,1078],[35,5596,5598],{"className":5597},[54],[35,5599,5601,5638],{"className":5600},[98,99],[35,5602,5604,5635],{"className":5603},[103],[35,5605,5607,5621],{"className":5606,"style":4500},[107],[35,5608,5609,5612],{"style":4503},[35,5610],{"className":5611,"style":1173},[115],[35,5613,5615],{"className":5614},[120,121,122,123],[35,5616,5618],{"className":5617},[86,123],[35,5619,404],{"className":5620},[86,123],[35,5622,5623,5626],{"style":4518},[35,5624],{"className":5625,"style":1173},[115],[35,5627,5629],{"className":5628},[120,121,122,123],[35,5630,5632],{"className":5631},[86,123],[35,5633,63],{"className":5634},[86,123],[35,5636,162],{"className":5637},[161],[35,5639,5641],{"className":5640},[103],[35,5642,5644],{"className":5643,"style":169},[107],[35,5645],{},[35,5647,5649],{"className":5648,"style":1073},[175,1072],[35,5650,1037],{"className":5651},[1077,1078],[35,5653],{"className":5654,"style":367},[366],[35,5656,278],{"className":5657},[371],[35,5659],{"className":5660,"style":367},[366],[35,5662,5664,5667,5670,5673,5713,5716,5719],{"className":5663},[77],[35,5665],{"className":5666,"style":4344},[81],[35,5668,404],{"className":5669},[86],[35,5671],{"className":5672,"style":5165},[366],[35,5674,5676,5679],{"className":5675},[86],[35,5677,4298],{"className":5678,"style":4351},[86,156],[35,5680,5682],{"className":5681},[1156],[35,5683,5685,5705],{"className":5684},[98,99],[35,5686,5688,5702],{"className":5687},[103],[35,5689,5691],{"className":5690,"style":4972},[107],[35,5692,5693,5696],{"style":4367},[35,5694],{"className":5695,"style":1173},[115],[35,5697,5699],{"className":5698},[120,121,122,123],[35,5700,927],{"className":5701},[86,123],[35,5703,162],{"className":5704},[161],[35,5706,5708],{"className":5707},[103],[35,5709,5711],{"className":5710,"style":4386},[107],[35,5712],{},[35,5714],{"className":5715,"style":367},[366],[35,5717,278],{"className":5718},[371],[35,5720],{"className":5721,"style":367},[366],[35,5723,5725,5728,5796,5862,5865,5868],{"className":5724},[77],[35,5726],{"className":5727,"style":4592},[81],[35,5729,5731,5734,5793],{"className":5730},[86],[35,5732],{"className":5733},[90,91],[35,5735,5737],{"className":5736},[54],[35,5738,5740,5785],{"className":5739},[98,99],[35,5741,5743,5782],{"className":5742},[103],[35,5744,5746,5760,5768],{"className":5745,"style":312},[107],[35,5747,5748,5751],{"style":111},[35,5749],{"className":5750,"style":116},[115],[35,5752,5754],{"className":5753},[120,121,122,123],[35,5755,5757],{"className":5756},[86,123],[35,5758,63],{"className":5759},[86,123],[35,5761,5762,5765],{"style":132},[35,5763],{"className":5764,"style":116},[115],[35,5766],{"className":5767,"style":140},[139],[35,5769,5770,5773],{"style":143},[35,5771],{"className":5772,"style":116},[115],[35,5774,5776],{"className":5775},[120,121,122,123],[35,5777,5779],{"className":5778},[86,123],[35,5780,272],{"className":5781},[86,123],[35,5783,162],{"className":5784},[161],[35,5786,5788],{"className":5787},[103],[35,5789,5791],{"className":5790,"style":169},[107],[35,5792],{},[35,5794],{"className":5795},[175,91],[35,5797,5799,5805,5856],{"className":5798},[86],[35,5800,5802],{"className":5801,"style":1073},[90,1072],[35,5803,1028],{"className":5804},[1077,1078],[35,5806,5808],{"className":5807},[54],[35,5809,5811,5848],{"className":5810},[98,99],[35,5812,5814,5845],{"className":5813},[103],[35,5815,5817,5831],{"className":5816,"style":4500},[107],[35,5818,5819,5822],{"style":4503},[35,5820],{"className":5821,"style":1173},[115],[35,5823,5825],{"className":5824},[120,121,122,123],[35,5826,5828],{"className":5827},[86,123],[35,5829,927],{"className":5830},[86,123],[35,5832,5833,5836],{"style":4518},[35,5834],{"className":5835,"style":1173},[115],[35,5837,5839],{"className":5838},[120,121,122,123],[35,5840,5842],{"className":5841},[86,123],[35,5843,5442],{"className":5844},[86,123],[35,5846,162],{"className":5847},[161],[35,5849,5851],{"className":5850},[103],[35,5852,5854],{"className":5853,"style":169},[107],[35,5855],{},[35,5857,5859],{"className":5858,"style":1073},[175,1072],[35,5860,1037],{"className":5861},[1077,1078],[35,5863],{"className":5864,"style":367},[366],[35,5866,278],{"className":5867},[371],[35,5869],{"className":5870,"style":367},[366],[35,5872,5874,5877],{"className":5873},[77],[35,5875],{"className":5876,"style":381},[81],[35,5878,529],{"className":5879},[86],[17,5881,5883],{"className":5882},[592],[35,5884,5886,5928],{"className":5885},[38],[35,5887,5889],{"className":5888},[42],[44,5890,5891],{"xmlns":46},[48,5892,5893,5925],{},[51,5894,5895,5901,5903,5909,5921,5923],{},[4294,5896,5897,5899],{},[57,5898,4298],{},[61,5900,63],{},[276,5902,278],{},[54,5904,5905,5907],{},[61,5906,272],{},[61,5908,529],{},[51,5910,5911,5913,5919],{},[276,5912,1028],{"fence":73},[54,5914,5915,5917],{"linethickness":4321},[61,5916,1048],{},[61,5918,63],{},[276,5920,1037],{"fence":73},[276,5922,278],{},[61,5924,3443],{},[65,5926,5927],{"encoding":67},"C_4 = \\frac{1}{5}\\binom{8}{4} = 14",[35,5929,5931,5986,6135],{"className":5930,"ariaHidden":73},[72],[35,5932,5934,5937,5977,5980,5983],{"className":5933},[77],[35,5935],{"className":5936,"style":4344},[81],[35,5938,5940,5943],{"className":5939},[86],[35,5941,4298],{"className":5942,"style":4351},[86,156],[35,5944,5946],{"className":5945},[1156],[35,5947,5949,5969],{"className":5948},[98,99],[35,5950,5952,5966],{"className":5951},[103],[35,5953,5955],{"className":5954,"style":4972},[107],[35,5956,5957,5960],{"style":4367},[35,5958],{"className":5959,"style":1173},[115],[35,5961,5963],{"className":5962},[120,121,122,123],[35,5964,63],{"className":5965},[86,123],[35,5967,162],{"className":5968},[161],[35,5970,5972],{"className":5971},[103],[35,5973,5975],{"className":5974,"style":4386},[107],[35,5976],{},[35,5978],{"className":5979,"style":367},[366],[35,5981,278],{"className":5982},[371],[35,5984],{"className":5985,"style":367},[366],[35,5987,5989,5992,6060,6126,6129,6132],{"className":5988},[77],[35,5990],{"className":5991,"style":4592},[81],[35,5993,5995,5998,6057],{"className":5994},[86],[35,5996],{"className":5997},[90,91],[35,5999,6001],{"className":6000},[54],[35,6002,6004,6049],{"className":6003},[98,99],[35,6005,6007,6046],{"className":6006},[103],[35,6008,6010,6024,6032],{"className":6009,"style":312},[107],[35,6011,6012,6015],{"style":111},[35,6013],{"className":6014,"style":116},[115],[35,6016,6018],{"className":6017},[120,121,122,123],[35,6019,6021],{"className":6020},[86,123],[35,6022,529],{"className":6023},[86,123],[35,6025,6026,6029],{"style":132},[35,6027],{"className":6028,"style":116},[115],[35,6030],{"className":6031,"style":140},[139],[35,6033,6034,6037],{"style":143},[35,6035],{"className":6036,"style":116},[115],[35,6038,6040],{"className":6039},[120,121,122,123],[35,6041,6043],{"className":6042},[86,123],[35,6044,272],{"className":6045},[86,123],[35,6047,162],{"className":6048},[161],[35,6050,6052],{"className":6051},[103],[35,6053,6055],{"className":6054,"style":169},[107],[35,6056],{},[35,6058],{"className":6059},[175,91],[35,6061,6063,6069,6120],{"className":6062},[86],[35,6064,6066],{"className":6065,"style":1073},[90,1072],[35,6067,1028],{"className":6068},[1077,1078],[35,6070,6072],{"className":6071},[54],[35,6073,6075,6112],{"className":6074},[98,99],[35,6076,6078,6109],{"className":6077},[103],[35,6079,6081,6095],{"className":6080,"style":4500},[107],[35,6082,6083,6086],{"style":4503},[35,6084],{"className":6085,"style":1173},[115],[35,6087,6089],{"className":6088},[120,121,122,123],[35,6090,6092],{"className":6091},[86,123],[35,6093,63],{"className":6094},[86,123],[35,6096,6097,6100],{"style":4518},[35,6098],{"className":6099,"style":1173},[115],[35,6101,6103],{"className":6102},[120,121,122,123],[35,6104,6106],{"className":6105},[86,123],[35,6107,1048],{"className":6108},[86,123],[35,6110,162],{"className":6111},[161],[35,6113,6115],{"className":6114},[103],[35,6116,6118],{"className":6117,"style":169},[107],[35,6119],{},[35,6121,6123],{"className":6122,"style":1073},[175,1072],[35,6124,1037],{"className":6125},[1077,1078],[35,6127],{"className":6128,"style":367},[366],[35,6130,278],{"className":6131},[371],[35,6133],{"className":6134,"style":367},[366],[35,6136,6138,6141],{"className":6137},[77],[35,6139],{"className":6140,"style":381},[81],[35,6142,3443],{"className":6143},[86],[17,6145,6146,6147,6150],{},"Exacto. Los números de Catalan cuentan precisamente cuántas secuencias de lanzamientos terminan ",[239,6148,6149],{},"por primera vez"," con más caras que cruces en cada longitud posible.",[561,6152,6154],{"id":6153},"la-conexión-con-el-triángulo-de-tartaglia","La conexión con el triángulo de Tartaglia",[17,6156,6157,6158,6161,6162,6217,6218,6347,6348,6352,6353,176],{},"Hay otra forma de ver por qué aparecen los números de Catalan aquí. Pensemos en cada secuencia de lanzamientos como un ",[239,6159,6160],{},"camino"," en una retícula: cara sube un paso, cruz baja un paso. El número total de caminos de longitud ",[35,6163,6165,6184],{"className":6164},[38],[35,6166,6168],{"className":6167},[42],[44,6169,6170],{"xmlns":46},[48,6171,6172,6182],{},[51,6173,6174,6176,6178,6180],{},[61,6175,404],{},[57,6177,2197],{},[276,6179,1316],{},[61,6181,272],{},[65,6183,2204],{"encoding":67},[35,6185,6187,6208],{"className":6186,"ariaHidden":73},[72],[35,6188,6190,6193,6196,6199,6202,6205],{"className":6189},[77],[35,6191],{"className":6192,"style":1331},[81],[35,6194,404],{"className":6195},[86],[35,6197,2197],{"className":6198},[86,156],[35,6200],{"className":6201,"style":2223},[366],[35,6203,1316],{"className":6204},[2227],[35,6206],{"className":6207,"style":2223},[366],[35,6209,6211,6214],{"className":6210},[77],[35,6212],{"className":6213,"style":381},[81],[35,6215,272],{"className":6216},[86]," que terminan un paso por encima de cero es el coeficiente binomial ",[35,6219,6221,6257],{"className":6220},[38],[35,6222,6224],{"className":6223},[42],[44,6225,6226],{"xmlns":46},[48,6227,6228,6254],{},[51,6229,6230,6232,6252],{},[276,6231,1028],{"fence":73},[54,6233,6234,6244],{"linethickness":4321},[51,6235,6236,6238,6240,6242],{},[61,6237,404],{},[57,6239,2197],{},[276,6241,1316],{},[61,6243,272],{},[51,6245,6246,6248,6250],{},[57,6247,2197],{},[276,6249,1316],{},[61,6251,272],{},[276,6253,1037],{"fence":73},[65,6255,6256],{"encoding":67},"\\binom{2n+1}{n+1}",[35,6258,6260],{"className":6259,"ariaHidden":73},[72],[35,6261,6263,6266],{"className":6262},[77],[35,6264],{"className":6265,"style":4404},[81],[35,6267,6269,6275,6341],{"className":6268},[86],[35,6270,6272],{"className":6271,"style":1073},[90,1072],[35,6273,1028],{"className":6274},[1077,1078],[35,6276,6278],{"className":6277},[54],[35,6279,6281,6333],{"className":6280},[98,99],[35,6282,6284,6330],{"className":6283},[103],[35,6285,6287,6307],{"className":6286,"style":4500},[107],[35,6288,6289,6292],{"style":4503},[35,6290],{"className":6291,"style":1173},[115],[35,6293,6295],{"className":6294},[120,121,122,123],[35,6296,6298,6301,6304],{"className":6297},[86,123],[35,6299,2197],{"className":6300},[86,156,123],[35,6302,1316],{"className":6303},[2227,123],[35,6305,272],{"className":6306},[86,123],[35,6308,6309,6312],{"style":4518},[35,6310],{"className":6311,"style":1173},[115],[35,6313,6315],{"className":6314},[120,121,122,123],[35,6316,6318,6321,6324,6327],{"className":6317},[86,123],[35,6319,404],{"className":6320},[86,123],[35,6322,2197],{"className":6323},[86,156,123],[35,6325,1316],{"className":6326},[2227,123],[35,6328,272],{"className":6329},[86,123],[35,6331,162],{"className":6332},[161],[35,6334,6336],{"className":6335},[103],[35,6337,6339],{"className":6338,"style":3736},[107],[35,6340],{},[35,6342,6344],{"className":6343,"style":1073},[175,1072],[35,6345,1037],{"className":6346},[1077,1078]," — una entrada del triángulo de Tartaglia. Los números de Catalan cuentan sólo los caminos que nunca cruzan por encima de cero antes del paso final. Son, literalmente, una selección filtrada dentro del triángulo — la misma herramienta que usamos en el ",[21,6349,6351],{"href":221,"rel":6350},[25],"artículo anterior"," para contar combinaciones de monedas, ahora con una restricción adicional. Si quieres explorar esta conexión en más detalle, Matt Parker la explica de forma brillante en ",[21,6354,6356],{"href":30,"rel":6355},[25],"su vídeo",[561,6358,6360],{"id":6359},"la-serie-infinita","La serie infinita",[17,6362,6363,6364,6419],{},"Volvamos a nuestra tabla. La contribución de las secuencias de longitud ",[35,6365,6367,6386],{"className":6366},[38],[35,6368,6370],{"className":6369},[42],[44,6371,6372],{"xmlns":46},[48,6373,6374,6384],{},[51,6375,6376,6378,6380,6382],{},[61,6377,404],{},[57,6379,2197],{},[276,6381,1316],{},[61,6383,272],{},[65,6385,2204],{"encoding":67},[35,6387,6389,6410],{"className":6388,"ariaHidden":73},[72],[35,6390,6392,6395,6398,6401,6404,6407],{"className":6391},[77],[35,6393],{"className":6394,"style":1331},[81],[35,6396,404],{"className":6397},[86],[35,6399,2197],{"className":6400},[86,156],[35,6402],{"className":6403,"style":2223},[366],[35,6405,1316],{"className":6406},[2227],[35,6408],{"className":6409,"style":2223},[366],[35,6411,6413,6416],{"className":6412},[77],[35,6414],{"className":6415,"style":381},[81],[35,6417,272],{"className":6418},[86]," al promedio total es:",[17,6421,6423],{"className":6422},[592],[35,6424,6426,6536],{"className":6425},[38],[35,6427,6429],{"className":6428},[42],[44,6430,6431],{"xmlns":46},[48,6432,6433,6533],{},[51,6434,6435,6470,6473,6488,6490],{},[6436,6437,6438,6467],"munder",{},[6436,6439,6440,6464],{},[1022,6441,6442,6454],{},[51,6443,6444,6446,6452],{},[276,6445,1028],{"fence":73},[54,6447,6448,6450],{},[61,6449,272],{},[61,6451,404],{},[276,6453,1037],{"fence":73},[51,6455,6456,6458,6460,6462],{},[61,6457,404],{},[57,6459,2197],{},[276,6461,1316],{},[61,6463,272],{},[276,6465,6466],{"stretchy":73},"⏟",[606,6468,6469],{},"probabilidad",[276,6471,6472],{},"×",[6436,6474,6475,6485],{},[6436,6476,6477,6483],{},[4294,6478,6479,6481],{},[57,6480,4298],{},[57,6482,2197],{},[276,6484,6466],{"stretchy":73},[606,6486,6487],{},"n.º de secuencias",[276,6489,6472],{},[6436,6491,6492,6516],{},[6436,6493,6494,6514],{},[54,6495,6496,6504],{},[51,6497,6498,6500,6502],{},[57,6499,2197],{},[276,6501,1316],{},[61,6503,272],{},[51,6505,6506,6508,6510,6512],{},[61,6507,404],{},[57,6509,2197],{},[276,6511,1316],{},[61,6513,272],{},[276,6515,6466],{"stretchy":73},[51,6517,6518,6521,6530],{},[606,6519,6520],{},"proporci",[6522,6523,6524,6527],"mover",{"accent":73},[606,6525,6526],{},"o",[276,6528,6529],{},"ˊ",[606,6531,6532],{},"n de caras",[65,6534,6535],{"encoding":67},"\\underbrace{\\left(\\frac{1}{2}\\right)^{2n+1}}_{\\text{probabilidad}} \\times \\underbrace{C_n}_{\\text{n.º de secuencias}} \\times \\underbrace{\\frac{n+1}{2n+1}}_{\\text{proporción de caras}}",[35,6537,6539,6807,6975],{"className":6538,"ariaHidden":73},[72],[35,6540,6542,6546,6798,6801,6804],{"className":6541},[77],[35,6543],{"className":6544,"style":6545},[81],"height:4.0743em;vertical-align:-2.4202em;",[35,6547,6549],{"className":6548},[1065,6436],[35,6550,6552,6789],{"className":6551},[98,99],[35,6553,6555,6786],{"className":6554},[103],[35,6556,6559,6578],{"className":6557,"style":6558},[107],"height:1.654em;",[35,6560,6562,6566],{"style":6561},"top:-1.3699em;",[35,6563],{"className":6564,"style":6565},[115],"height:3.654em;",[35,6567,6569],{"className":6568},[120,121,122,123],[35,6570,6572],{"className":6571},[86,123],[35,6573,6575],{"className":6574},[86,625,123],[35,6576,6469],{"className":6577},[86,123],[35,6579,6581,6584],{"style":6580},"top:-3.654em;",[35,6582],{"className":6583,"style":6565},[115],[35,6585,6587],{"className":6586},[1065,6436],[35,6588,6590,6777],{"className":6589},[98,99],[35,6591,6593,6774],{"className":6592},[103],[35,6594,6596,6646],{"className":6595,"style":6558},[107],[35,6597,6601,6604],{"className":6598,"style":6600},[6599],"svg-align","top:-2.056em;",[35,6602],{"className":6603,"style":6565},[115],[35,6605,6609,6626,6636],{"className":6606,"style":6608},[6607],"stretchy","height:0.548em;min-width:1.6em;",[35,6610,6614],{"className":6611,"style":6613},[6612],"brace-left","height:0.548em;",[6615,6616,6622],"svg",{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6621},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","0.548em","0 0 400000 548","xMinYMin slice",[6623,6624],"path",{"d":6625},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[35,6627,6630],{"className":6628,"style":6613},[6629],"brace-center",[6615,6631,6633],{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6632},"xMidYMin slice",[6623,6634],{"d":6635},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[35,6637,6640],{"className":6638,"style":6613},[6639],"brace-right",[6615,6641,6643],{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6642},"xMaxYMin slice",[6623,6644],{"d":6645},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[35,6647,6648,6651],{"style":6580},[35,6649],{"className":6650,"style":6565},[115],[35,6652,6654],{"className":6653},[86],[35,6655,6657,6738],{"className":6656},[1065],[35,6658,6660,6666,6732],{"className":6659},[1065],[35,6661,6663],{"className":6662,"style":1073},[90,1072],[35,6664,1028],{"className":6665},[1077,122],[35,6667,6669,6672,6729],{"className":6668},[86],[35,6670],{"className":6671},[90,91],[35,6673,6675],{"className":6674},[54],[35,6676,6678,6720],{"className":6677},[98,99],[35,6679,6681,6717],{"className":6680},[103],[35,6682,6685,6697,6705],{"className":6683,"style":6684},[107],"height:1.3214em;",[35,6686,6688,6691],{"style":6687},"top:-2.314em;",[35,6689],{"className":6690,"style":116},[115],[35,6692,6694],{"className":6693},[86],[35,6695,404],{"className":6696},[86],[35,6698,6699,6702],{"style":132},[35,6700],{"className":6701,"style":116},[115],[35,6703],{"className":6704,"style":140},[139],[35,6706,6708,6711],{"style":6707},"top:-3.677em;",[35,6709],{"className":6710,"style":116},[115],[35,6712,6714],{"className":6713},[86],[35,6715,272],{"className":6716},[86],[35,6718,162],{"className":6719},[161],[35,6721,6723],{"className":6722},[103],[35,6724,6727],{"className":6725,"style":6726},[107],"height:0.686em;",[35,6728],{},[35,6730],{"className":6731},[175,91],[35,6733,6735],{"className":6734,"style":1073},[175,1072],[35,6736,1037],{"className":6737},[1077,122],[35,6739,6741],{"className":6740},[1156],[35,6742,6744],{"className":6743},[98],[35,6745,6747],{"className":6746},[103],[35,6748,6750],{"className":6749,"style":6558},[107],[35,6751,6753,6756],{"style":6752},"top:-3.9029em;margin-right:0.05em;",[35,6754],{"className":6755,"style":1173},[115],[35,6757,6759],{"className":6758},[120,121,122,123],[35,6760,6762,6765,6768,6771],{"className":6761},[86,123],[35,6763,404],{"className":6764},[86,123],[35,6766,2197],{"className":6767},[86,156,123],[35,6769,1316],{"className":6770},[2227,123],[35,6772,272],{"className":6773},[86,123],[35,6775,162],{"className":6776},[161],[35,6778,6780],{"className":6779},[103],[35,6781,6784],{"className":6782,"style":6783},[107],"height:1.598em;",[35,6785],{},[35,6787,162],{"className":6788},[161],[35,6790,6792],{"className":6791},[103],[35,6793,6796],{"className":6794,"style":6795},[107],"height:2.4202em;",[35,6797],{},[35,6799],{"className":6800,"style":2223},[366],[35,6802,6472],{"className":6803},[2227],[35,6805],{"className":6806,"style":2223},[366],[35,6808,6810,6814,6966,6969,6972],{"className":6809},[77],[35,6811],{"className":6812,"style":6813},[81],"height:2.1674em;vertical-align:-1.4841em;",[35,6815,6817],{"className":6816},[1065,6436],[35,6818,6820,6957],{"className":6819},[98,99],[35,6821,6823,6954],{"className":6822},[103],[35,6824,6827,6845],{"className":6825,"style":6826},[107],"height:0.6833em;",[35,6828,6830,6833],{"style":6829},"top:-1.5159em;",[35,6831],{"className":6832,"style":116},[115],[35,6834,6836],{"className":6835},[120,121,122,123],[35,6837,6839],{"className":6838},[86,123],[35,6840,6842],{"className":6841},[86,625,123],[35,6843,6487],{"className":6844},[86,123],[35,6846,6848,6851],{"style":6847},"top:-3em;",[35,6849],{"className":6850,"style":116},[115],[35,6852,6854],{"className":6853},[1065,6436],[35,6855,6857,6945],{"className":6856},[98,99],[35,6858,6860,6942],{"className":6859},[103],[35,6861,6863,6894],{"className":6862,"style":6826},[107],[35,6864,6867,6870],{"className":6865,"style":6866},[6599],"top:-2.202em;",[35,6868],{"className":6869,"style":116},[115],[35,6871,6873,6880,6887],{"className":6872,"style":6608},[6607],[35,6874,6876],{"className":6875,"style":6613},[6612],[6615,6877,6878],{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6621},[6623,6879],{"d":6625},[35,6881,6883],{"className":6882,"style":6613},[6629],[6615,6884,6885],{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6632},[6623,6886],{"d":6635},[35,6888,6890],{"className":6889,"style":6613},[6639],[6615,6891,6892],{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6642},[6623,6893],{"d":6645},[35,6895,6896,6899],{"style":6847},[35,6897],{"className":6898,"style":116},[115],[35,6900,6902],{"className":6901},[86],[35,6903,6905,6908],{"className":6904},[86],[35,6906,4298],{"className":6907,"style":4351},[86,156],[35,6909,6911],{"className":6910},[1156],[35,6912,6914,6934],{"className":6913},[98,99],[35,6915,6917,6931],{"className":6916},[103],[35,6918,6920],{"className":6919,"style":4364},[107],[35,6921,6922,6925],{"style":4367},[35,6923],{"className":6924,"style":1173},[115],[35,6926,6928],{"className":6927},[120,121,122,123],[35,6929,2197],{"className":6930},[86,156,123],[35,6932,162],{"className":6933},[161],[35,6935,6937],{"className":6936},[103],[35,6938,6940],{"className":6939,"style":4386},[107],[35,6941],{},[35,6943,162],{"className":6944},[161],[35,6946,6948],{"className":6947},[103],[35,6949,6952],{"className":6950,"style":6951},[107],"height:0.798em;",[35,6953],{},[35,6955,162],{"className":6956},[161],[35,6958,6960],{"className":6959},[103],[35,6961,6964],{"className":6962,"style":6963},[107],"height:1.4841em;",[35,6965],{},[35,6967],{"className":6968,"style":2223},[366],[35,6970,6472],{"className":6971},[2227],[35,6973],{"className":6974,"style":2223},[366],[35,6976,6978,6982],{"className":6977},[77],[35,6979],{"className":6980,"style":6981},[81],"height:3.561em;vertical-align:-2.2395em;",[35,6983,6985],{"className":6984},[1065,6436],[35,6986,6988,7214],{"className":6987},[98,99],[35,6989,6991,7211],{"className":6990},[103],[35,6992,6994,7052],{"className":6993,"style":6684},[107],[35,6995,6997,7001],{"style":6996},"top:-1.218em;",[35,6998],{"className":6999,"style":7000},[115],"height:3.3214em;",[35,7002,7004],{"className":7003},[120,121,122,123],[35,7005,7007],{"className":7006},[86,123],[35,7008,7010,7013,7049],{"className":7009},[86,625,123],[35,7011,6520],{"className":7012},[86,123],[35,7014,7017],{"className":7015},[86,7016,123],"accent",[35,7018,7020],{"className":7019},[98],[35,7021,7023],{"className":7022},[103],[35,7024,7027,7036],{"className":7025,"style":7026},[107],"height:0.6944em;",[35,7028,7030,7033],{"style":7029},"top:-2.7em;",[35,7031],{"className":7032,"style":1173},[115],[35,7034,6526],{"className":7035},[86,123],[35,7037,7038,7041],{"style":7029},[35,7039],{"className":7040,"style":1173},[115],[35,7042,7046],{"className":7043,"style":7045},[7044],"accent-body","left:-0.25em;",[35,7047,6529],{"className":7048},[86,123],[35,7050,6532],{"className":7051},[86,123],[35,7053,7055,7058],{"style":7054},"top:-3.3214em;",[35,7056],{"className":7057,"style":7000},[115],[35,7059,7061],{"className":7060},[1065,6436],[35,7062,7064,7202],{"className":7063},[98,99],[35,7065,7067,7199],{"className":7066},[103],[35,7068,7070,7101],{"className":7069,"style":6684},[107],[35,7071,7074,7077],{"className":7072,"style":7073},[6599],"top:-1.9041em;",[35,7075],{"className":7076,"style":7000},[115],[35,7078,7080,7087,7094],{"className":7079,"style":6608},[6607],[35,7081,7083],{"className":7082,"style":6613},[6612],[6615,7084,7085],{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6621},[6623,7086],{"d":6625},[35,7088,7090],{"className":7089,"style":6613},[6629],[6615,7091,7092],{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6632},[6623,7093],{"d":6635},[35,7095,7097],{"className":7096,"style":6613},[6639],[6615,7098,7099],{"xmlns":6617,"width":6618,"height":6619,"viewBox":6620,"preserveAspectRatio":6642},[6623,7100],{"d":6645},[35,7102,7103,7106],{"style":7054},[35,7104],{"className":7105,"style":7000},[115],[35,7107,7109],{"className":7108},[86],[35,7110,7112,7115,7196],{"className":7111},[86],[35,7113],{"className":7114},[90,91],[35,7116,7118],{"className":7117},[54],[35,7119,7121,7187],{"className":7120},[98,99],[35,7122,7124,7184],{"className":7123},[103],[35,7125,7127,7153,7161],{"className":7126,"style":6684},[107],[35,7128,7129,7132],{"style":6687},[35,7130],{"className":7131,"style":116},[115],[35,7133,7135,7138,7141,7144,7147,7150],{"className":7134},[86],[35,7136,404],{"className":7137},[86],[35,7139,2197],{"className":7140},[86,156],[35,7142],{"className":7143,"style":2223},[366],[35,7145,1316],{"className":7146},[2227],[35,7148],{"className":7149,"style":2223},[366],[35,7151,272],{"className":7152},[86],[35,7154,7155,7158],{"style":132},[35,7156],{"className":7157,"style":116},[115],[35,7159],{"className":7160,"style":140},[139],[35,7162,7163,7166],{"style":6707},[35,7164],{"className":7165,"style":116},[115],[35,7167,7169,7172,7175,7178,7181],{"className":7168},[86],[35,7170,2197],{"className":7171},[86,156],[35,7173],{"className":7174,"style":2223},[366],[35,7176,1316],{"className":7177},[2227],[35,7179],{"className":7180,"style":2223},[366],[35,7182,272],{"className":7183},[86],[35,7185,162],{"className":7186},[161],[35,7188,7190],{"className":7189},[103],[35,7191,7194],{"className":7192,"style":7193},[107],"height:0.7693em;",[35,7195],{},[35,7197],{"className":7198},[175,91],[35,7200,162],{"className":7201},[161],[35,7203,7205],{"className":7204},[103],[35,7206,7209],{"className":7207,"style":7208},[107],"height:1.4173em;",[35,7210],{},[35,7212,162],{"className":7213},[161],[35,7215,7217],{"className":7216},[103],[35,7218,7221],{"className":7219,"style":7220},[107],"height:2.2395em;",[35,7222],{},[17,7224,7225,7226,7229,7230,7281],{},"Y la proporción media global es la suma de ",[239,7227,7228],{},"todas"," estas contribuciones, desde ",[35,7231,7233,7251],{"className":7232},[38],[35,7234,7236],{"className":7235},[42],[44,7237,7238],{"xmlns":46},[48,7239,7240,7248],{},[51,7241,7242,7244,7246],{},[57,7243,2197],{},[276,7245,278],{},[61,7247,517],{},[65,7249,7250],{"encoding":67},"n = 0",[35,7252,7254,7272],{"className":7253,"ariaHidden":73},[72],[35,7255,7257,7260,7263,7266,7269],{"className":7256},[77],[35,7258],{"className":7259,"style":206},[81],[35,7261,2197],{"className":7262},[86,156],[35,7264],{"className":7265,"style":367},[366],[35,7267,278],{"className":7268},[371],[35,7270],{"className":7271,"style":367},[366],[35,7273,7275,7278],{"className":7274},[77],[35,7276],{"className":7277,"style":381},[81],[35,7279,517],{"className":7280},[86]," hasta el infinito:",[17,7283,7285],{"className":7284},[592],[35,7286,7288,7368],{"className":7287},[38],[35,7289,7291],{"className":7290},[42],[44,7292,7293],{"xmlns":46},[48,7294,7295,7365],{},[51,7296,7297,7300,7302,7320,7342,7345],{},[606,7298,7299],{},"Media",[276,7301,278],{},[7303,7304,7305,7308,7316],"msubsup",{},[276,7306,7307],{},"∑",[51,7309,7310,7312,7314],{},[57,7311,2197],{},[276,7313,278],{},[61,7315,517],{},[57,7317,7319],{"mathvariant":7318},"normal","∞",[54,7321,7322,7328],{},[4294,7323,7324,7326],{},[57,7325,4298],{},[57,7327,2197],{},[1022,7329,7330,7332],{},[61,7331,404],{},[51,7333,7334,7336,7338,7340],{},[61,7335,404],{},[57,7337,2197],{},[276,7339,1316],{},[61,7341,272],{},[276,7343,7344],{},"⋅",[54,7346,7347,7355],{},[51,7348,7349,7351,7353],{},[57,7350,2197],{},[276,7352,1316],{},[61,7354,272],{},[51,7356,7357,7359,7361,7363],{},[61,7358,404],{},[57,7360,2197],{},[276,7362,1316],{},[61,7364,272],{},[65,7366,7367],{"encoding":67},"\\text{Media} = \\sum_{n=0}^{\\infty} \\frac{C_n}{2^{2n+1}} \\cdot \\frac{n+1}{2n+1}",[35,7369,7371,7392,7634],{"className":7370,"ariaHidden":73},[72],[35,7372,7374,7377,7383,7386,7389],{"className":7373},[77],[35,7375],{"className":7376,"style":7026},[81],[35,7378,7380],{"className":7379},[86,625],[35,7381,7299],{"className":7382},[86],[35,7384],{"className":7385,"style":367},[366],[35,7387,278],{"className":7388},[371],[35,7390],{"className":7391,"style":367},[366],[35,7393,7395,7399,7470,7473,7625,7628,7631],{"className":7394},[77],[35,7396],{"className":7397,"style":7398},[81],"height:1.2334em;vertical-align:-0.345em;",[35,7400,7403,7409],{"className":7401},[7402],"mop",[35,7404,7307],{"className":7405,"style":7408},[7402,7406,7407],"op-symbol","small-op","position:relative;top:0em;",[35,7410,7412],{"className":7411},[1156],[35,7413,7415,7461],{"className":7414},[98,99],[35,7416,7418,7458],{"className":7417},[103],[35,7419,7422,7443],{"className":7420,"style":7421},[107],"height:0.8043em;",[35,7423,7425,7428],{"style":7424},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[35,7426],{"className":7427,"style":1173},[115],[35,7429,7431],{"className":7430},[120,121,122,123],[35,7432,7434,7437,7440],{"className":7433},[86,123],[35,7435,2197],{"className":7436},[86,156,123],[35,7438,278],{"className":7439},[371,123],[35,7441,517],{"className":7442},[86,123],[35,7444,7446,7449],{"style":7445},"top:-3.2029em;margin-right:0.05em;",[35,7447],{"className":7448,"style":1173},[115],[35,7450,7452],{"className":7451},[120,121,122,123],[35,7453,7455],{"className":7454},[86,123],[35,7456,7319],{"className":7457},[86,123],[35,7459,162],{"className":7460},[161],[35,7462,7464],{"className":7463},[103],[35,7465,7468],{"className":7466,"style":7467},[107],"height:0.2997em;",[35,7469],{},[35,7471],{"className":7472,"style":1452},[366],[35,7474,7476,7479,7622],{"className":7475},[86],[35,7477],{"className":7478},[90,91],[35,7480,7482],{"className":7481},[54],[35,7483,7485,7614],{"className":7484},[98,99],[35,7486,7488,7611],{"className":7487},[103],[35,7489,7492,7548,7556],{"className":7490,"style":7491},[107],"height:0.8884em;",[35,7493,7494,7497],{"style":111},[35,7495],{"className":7496,"style":116},[115],[35,7498,7500],{"className":7499},[120,121,122,123],[35,7501,7503],{"className":7502},[86,123],[35,7504,7506,7509],{"className":7505},[86,123],[35,7507,404],{"className":7508},[86,123],[35,7510,7512],{"className":7511},[1156],[35,7513,7515],{"className":7514},[98],[35,7516,7518],{"className":7517},[103],[35,7519,7522],{"className":7520,"style":7521},[107],"height:0.7463em;",[35,7523,7525,7529],{"style":7524},"top:-2.786em;margin-right:0.0714em;",[35,7526],{"className":7527,"style":7528},[115],"height:2.5em;",[35,7530,7533],{"className":7531},[120,7532,1078,123],"reset-size3",[35,7534,7536,7539,7542,7545],{"className":7535},[86,123],[35,7537,404],{"className":7538},[86,123],[35,7540,2197],{"className":7541},[86,156,123],[35,7543,1316],{"className":7544},[2227,123],[35,7546,272],{"className":7547},[86,123],[35,7549,7550,7553],{"style":132},[35,7551],{"className":7552,"style":116},[115],[35,7554],{"className":7555,"style":140},[139],[35,7557,7559,7562],{"style":7558},"top:-3.4101em;",[35,7560],{"className":7561,"style":116},[115],[35,7563,7565],{"className":7564},[120,121,122,123],[35,7566,7568],{"className":7567},[86,123],[35,7569,7571,7574],{"className":7570},[86,123],[35,7572,4298],{"className":7573,"style":4351},[86,156,123],[35,7575,7577],{"className":7576},[1156],[35,7578,7580,7602],{"className":7579},[98,99],[35,7581,7583,7599],{"className":7582},[103],[35,7584,7587],{"className":7585,"style":7586},[107],"height:0.1645em;",[35,7588,7590,7593],{"style":7589},"top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;",[35,7591],{"className":7592,"style":7528},[115],[35,7594,7596],{"className":7595},[120,7532,1078,123],[35,7597,2197],{"className":7598},[86,156,123],[35,7600,162],{"className":7601},[161],[35,7603,7605],{"className":7604},[103],[35,7606,7609],{"className":7607,"style":7608},[107],"height:0.143em;",[35,7610],{},[35,7612,162],{"className":7613},[161],[35,7615,7617],{"className":7616},[103],[35,7618,7620],{"className":7619,"style":169},[107],[35,7621],{},[35,7623],{"className":7624},[175,91],[35,7626],{"className":7627,"style":2223},[366],[35,7629,7344],{"className":7630},[2227],[35,7632],{"className":7633,"style":2223},[366],[35,7635,7637,7640],{"className":7636},[77],[35,7638],{"className":7639,"style":3657},[81],[35,7641,7643,7646,7720],{"className":7642},[86],[35,7644],{"className":7645},[90,91],[35,7647,7649],{"className":7648},[54],[35,7650,7652,7712],{"className":7651},[98,99],[35,7653,7655,7709],{"className":7654},[103],[35,7656,7658,7681,7689],{"className":7657,"style":312},[107],[35,7659,7660,7663],{"style":111},[35,7661],{"className":7662,"style":116},[115],[35,7664,7666],{"className":7665},[120,121,122,123],[35,7667,7669,7672,7675,7678],{"className":7668},[86,123],[35,7670,404],{"className":7671},[86,123],[35,7673,2197],{"className":7674},[86,156,123],[35,7676,1316],{"className":7677},[2227,123],[35,7679,272],{"className":7680},[86,123],[35,7682,7683,7686],{"style":132},[35,7684],{"className":7685,"style":116},[115],[35,7687],{"className":7688,"style":140},[139],[35,7690,7691,7694],{"style":143},[35,7692],{"className":7693,"style":116},[115],[35,7695,7697],{"className":7696},[120,121,122,123],[35,7698,7700,7703,7706],{"className":7699},[86,123],[35,7701,2197],{"className":7702},[86,156,123],[35,7704,1316],{"className":7705},[2227,123],[35,7707,272],{"className":7708},[86,123],[35,7710,162],{"className":7711},[161],[35,7713,7715],{"className":7714},[103],[35,7716,7718],{"className":7717,"style":3736},[107],[35,7719],{},[35,7721],{"className":7722},[175,91],[17,7724,7725],{},"Calculemos los primeros términos:",[2168,7727,7728,7769],{},[2171,7729,7730],{},[2174,7731,7732,7762,7766],{},[2177,7733,7734],{"align":2179},[35,7735,7737,7750],{"className":7736},[38],[35,7738,7740],{"className":7739},[42],[44,7741,7742],{"xmlns":46},[48,7743,7744,7748],{},[51,7745,7746],{},[57,7747,2197],{},[65,7749,2197],{"encoding":67},[35,7751,7753],{"className":7752,"ariaHidden":73},[72],[35,7754,7756,7759],{"className":7755},[77],[35,7757],{"className":7758,"style":206},[81],[35,7760,2197],{"className":7761},[86,156],[2177,7763,7765],{"align":7764},"left","Contribución",[2177,7767,7768],{"align":7764},"Acumulado",[2250,7770,7771,7992,8285,8577,8869],{},[2174,7772,7773,7775,7950],{},[2255,7774,517],{"align":2179},[2255,7776,7777],{"align":7764},[35,7778,7780,7810],{"className":7779},[38],[35,7781,7783],{"className":7782},[42],[44,7784,7785],{"xmlns":46},[48,7786,7787,7807],{},[51,7788,7789,7795,7797,7799,7801,7803,7805],{},[54,7790,7791,7793],{},[61,7792,272],{},[61,7794,404],{},[276,7796,6472],{},[61,7798,272],{},[276,7800,6472],{},[61,7802,272],{},[276,7804,278],{},[61,7806,500],{},[65,7808,7809],{"encoding":67},"\\frac{1}{2} \\times 1 \\times 1 = 0{,}5",[35,7811,7813,7896,7914,7932],{"className":7812,"ariaHidden":73},[72],[35,7814,7816,7819,7887,7890,7893],{"className":7815},[77],[35,7817],{"className":7818,"style":293},[81],[35,7820,7822,7825,7884],{"className":7821},[86],[35,7823],{"className":7824},[90,91],[35,7826,7828],{"className":7827},[54],[35,7829,7831,7876],{"className":7830},[98,99],[35,7832,7834,7873],{"className":7833},[103],[35,7835,7837,7851,7859],{"className":7836,"style":312},[107],[35,7838,7839,7842],{"style":111},[35,7840],{"className":7841,"style":116},[115],[35,7843,7845],{"className":7844},[120,121,122,123],[35,7846,7848],{"className":7847},[86,123],[35,7849,404],{"className":7850},[86,123],[35,7852,7853,7856],{"style":132},[35,7854],{"className":7855,"style":116},[115],[35,7857],{"className":7858,"style":140},[139],[35,7860,7861,7864],{"style":143},[35,7862],{"className":7863,"style":116},[115],[35,7865,7867],{"className":7866},[120,121,122,123],[35,7868,7870],{"className":7869},[86,123],[35,7871,272],{"className":7872},[86,123],[35,7874,162],{"className":7875},[161],[35,7877,7879],{"className":7878},[103],[35,7880,7882],{"className":7881,"style":169},[107],[35,7883],{},[35,7885],{"className":7886},[175,91],[35,7888],{"className":7889,"style":2223},[366],[35,7891,6472],{"className":7892},[2227],[35,7894],{"className":7895,"style":2223},[366],[35,7897,7899,7902,7905,7908,7911],{"className":7898},[77],[35,7900],{"className":7901,"style":1331},[81],[35,7903,272],{"className":7904},[86],[35,7906],{"className":7907,"style":2223},[366],[35,7909,6472],{"className":7910},[2227],[35,7912],{"className":7913,"style":2223},[366],[35,7915,7917,7920,7923,7926,7929],{"className":7916},[77],[35,7918],{"className":7919,"style":381},[81],[35,7921,272],{"className":7922},[86],[35,7924],{"className":7925,"style":367},[366],[35,7927,278],{"className":7928},[371],[35,7930],{"className":7931,"style":367},[366],[35,7933,7935,7938,7941,7947],{"className":7934},[77],[35,7936],{"className":7937,"style":513},[81],[35,7939,517],{"className":7940},[86],[35,7942,7944],{"className":7943},[86],[35,7945,525],{"className":7946},[524],[35,7948,529],{"className":7949},[86],[2255,7951,7952],{"align":7764},[35,7953,7955,7970],{"className":7954},[38],[35,7956,7958],{"className":7957},[42],[44,7959,7960],{"xmlns":46},[48,7961,7962,7967],{},[51,7963,7964],{},[61,7965,7966],{},"0,5000",[65,7968,7969],{"encoding":67},"0{,}5000",[35,7971,7973],{"className":7972,"ariaHidden":73},[72],[35,7974,7976,7979,7982,7988],{"className":7975},[77],[35,7977],{"className":7978,"style":513},[81],[35,7980,517],{"className":7981},[86],[35,7983,7985],{"className":7984},[86],[35,7986,525],{"className":7987},[524],[35,7989,7991],{"className":7990},[86],"5000",[2174,7993,7994,7996,8243],{},[2255,7995,272],{"align":2179},[2255,7997,7998],{"align":7764},[35,7999,8001,8037],{"className":8000},[38],[35,8002,8004],{"className":8003},[42],[44,8005,8006],{"xmlns":46},[48,8007,8008,8034],{},[51,8009,8010,8016,8018,8020,8022,8028,8031],{},[54,8011,8012,8014],{},[61,8013,272],{},[61,8015,1048],{},[276,8017,6472],{},[61,8019,272],{},[276,8021,6472],{},[54,8023,8024,8026],{},[61,8025,404],{},[61,8027,927],{},[276,8029,8030],{},"≈",[61,8032,8033],{},"0,0833",[65,8035,8036],{"encoding":67},"\\frac{1}{8} \\times 1 \\times \\frac{2}{3} \\approx 0{,}0833",[35,8038,8040,8123,8141,8224],{"className":8039,"ariaHidden":73},[72],[35,8041,8043,8046,8114,8117,8120],{"className":8042},[77],[35,8044],{"className":8045,"style":293},[81],[35,8047,8049,8052,8111],{"className":8048},[86],[35,8050],{"className":8051},[90,91],[35,8053,8055],{"className":8054},[54],[35,8056,8058,8103],{"className":8057},[98,99],[35,8059,8061,8100],{"className":8060},[103],[35,8062,8064,8078,8086],{"className":8063,"style":312},[107],[35,8065,8066,8069],{"style":111},[35,8067],{"className":8068,"style":116},[115],[35,8070,8072],{"className":8071},[120,121,122,123],[35,8073,8075],{"className":8074},[86,123],[35,8076,1048],{"className":8077},[86,123],[35,8079,8080,8083],{"style":132},[35,8081],{"className":8082,"style":116},[115],[35,8084],{"className":8085,"style":140},[139],[35,8087,8088,8091],{"style":143},[35,8089],{"className":8090,"style":116},[115],[35,8092,8094],{"className":8093},[120,121,122,123],[35,8095,8097],{"className":8096},[86,123],[35,8098,272],{"className":8099},[86,123],[35,8101,162],{"className":8102},[161],[35,8104,8106],{"className":8105},[103],[35,8107,8109],{"className":8108,"style":169},[107],[35,8110],{},[35,8112],{"className":8113},[175,91],[35,8115],{"className":8116,"style":2223},[366],[35,8118,6472],{"className":8119},[2227],[35,8121],{"className":8122,"style":2223},[366],[35,8124,8126,8129,8132,8135,8138],{"className":8125},[77],[35,8127],{"className":8128,"style":1331},[81],[35,8130,272],{"className":8131},[86],[35,8133],{"className":8134,"style":2223},[366],[35,8136,6472],{"className":8137},[2227],[35,8139],{"className":8140,"style":2223},[366],[35,8142,8144,8147,8215,8218,8221],{"className":8143},[77],[35,8145],{"className":8146,"style":293},[81],[35,8148,8150,8153,8212],{"className":8149},[86],[35,8151],{"className":8152},[90,91],[35,8154,8156],{"className":8155},[54],[35,8157,8159,8204],{"className":8158},[98,99],[35,8160,8162,8201],{"className":8161},[103],[35,8163,8165,8179,8187],{"className":8164,"style":312},[107],[35,8166,8167,8170],{"style":111},[35,8168],{"className":8169,"style":116},[115],[35,8171,8173],{"className":8172},[120,121,122,123],[35,8174,8176],{"className":8175},[86,123],[35,8177,927],{"className":8178},[86,123],[35,8180,8181,8184],{"style":132},[35,8182],{"className":8183,"style":116},[115],[35,8185],{"className":8186,"style":140},[139],[35,8188,8189,8192],{"style":143},[35,8190],{"className":8191,"style":116},[115],[35,8193,8195],{"className":8194},[120,121,122,123],[35,8196,8198],{"className":8197},[86,123],[35,8199,404],{"className":8200},[86,123],[35,8202,162],{"className":8203},[161],[35,8205,8207],{"className":8206},[103],[35,8208,8210],{"className":8209,"style":169},[107],[35,8211],{},[35,8213],{"className":8214},[175,91],[35,8216],{"className":8217,"style":367},[366],[35,8219,8030],{"className":8220},[371],[35,8222],{"className":8223,"style":367},[366],[35,8225,8227,8230,8233,8239],{"className":8226},[77],[35,8228],{"className":8229,"style":513},[81],[35,8231,517],{"className":8232},[86],[35,8234,8236],{"className":8235},[86],[35,8237,525],{"className":8238},[524],[35,8240,8242],{"className":8241},[86],"0833",[2255,8244,8245],{"align":7764},[35,8246,8248,8263],{"className":8247},[38],[35,8249,8251],{"className":8250},[42],[44,8252,8253],{"xmlns":46},[48,8254,8255,8260],{},[51,8256,8257],{},[61,8258,8259],{},"0,5833",[65,8261,8262],{"encoding":67},"0{,}5833",[35,8264,8266],{"className":8265,"ariaHidden":73},[72],[35,8267,8269,8272,8275,8281],{"className":8268},[77],[35,8270],{"className":8271,"style":513},[81],[35,8273,517],{"className":8274},[86],[35,8276,8278],{"className":8277},[86],[35,8279,525],{"className":8280},[524],[35,8282,8284],{"className":8283},[86],"5833",[2174,8286,8287,8289,8535],{},[2255,8288,404],{"align":2179},[2255,8290,8291],{"align":7764},[35,8292,8294,8329],{"className":8293},[38],[35,8295,8297],{"className":8296},[42],[44,8298,8299],{"xmlns":46},[48,8300,8301,8326],{},[51,8302,8303,8309,8311,8313,8315,8321,8323],{},[54,8304,8305,8307],{},[61,8306,272],{},[61,8308,1847],{},[276,8310,6472],{},[61,8312,404],{},[276,8314,6472],{},[54,8316,8317,8319],{},[61,8318,927],{},[61,8320,529],{},[276,8322,278],{},[61,8324,8325],{},"0,0375",[65,8327,8328],{"encoding":67},"\\frac{1}{32} \\times 2 \\times \\frac{3}{5} = 0{,}0375",[35,8330,8332,8415,8433,8516],{"className":8331,"ariaHidden":73},[72],[35,8333,8335,8338,8406,8409,8412],{"className":8334},[77],[35,8336],{"className":8337,"style":293},[81],[35,8339,8341,8344,8403],{"className":8340},[86],[35,8342],{"className":8343},[90,91],[35,8345,8347],{"className":8346},[54],[35,8348,8350,8395],{"className":8349},[98,99],[35,8351,8353,8392],{"className":8352},[103],[35,8354,8356,8370,8378],{"className":8355,"style":312},[107],[35,8357,8358,8361],{"style":111},[35,8359],{"className":8360,"style":116},[115],[35,8362,8364],{"className":8363},[120,121,122,123],[35,8365,8367],{"className":8366},[86,123],[35,8368,1847],{"className":8369},[86,123],[35,8371,8372,8375],{"style":132},[35,8373],{"className":8374,"style":116},[115],[35,8376],{"className":8377,"style":140},[139],[35,8379,8380,8383],{"style":143},[35,8381],{"className":8382,"style":116},[115],[35,8384,8386],{"className":8385},[120,121,122,123],[35,8387,8389],{"className":8388},[86,123],[35,8390,272],{"className":8391},[86,123],[35,8393,162],{"className":8394},[161],[35,8396,8398],{"className":8397},[103],[35,8399,8401],{"className":8400,"style":169},[107],[35,8402],{},[35,8404],{"className":8405},[175,91],[35,8407],{"className":8408,"style":2223},[366],[35,8410,6472],{"className":8411},[2227],[35,8413],{"className":8414,"style":2223},[366],[35,8416,8418,8421,8424,8427,8430],{"className":8417},[77],[35,8419],{"className":8420,"style":1331},[81],[35,8422,404],{"className":8423},[86],[35,8425],{"className":8426,"style":2223},[366],[35,8428,6472],{"className":8429},[2227],[35,8431],{"className":8432,"style":2223},[366],[35,8434,8436,8439,8507,8510,8513],{"className":8435},[77],[35,8437],{"className":8438,"style":293},[81],[35,8440,8442,8445,8504],{"className":8441},[86],[35,8443],{"className":8444},[90,91],[35,8446,8448],{"className":8447},[54],[35,8449,8451,8496],{"className":8450},[98,99],[35,8452,8454,8493],{"className":8453},[103],[35,8455,8457,8471,8479],{"className":8456,"style":312},[107],[35,8458,8459,8462],{"style":111},[35,8460],{"className":8461,"style":116},[115],[35,8463,8465],{"className":8464},[120,121,122,123],[35,8466,8468],{"className":8467},[86,123],[35,8469,529],{"className":8470},[86,123],[35,8472,8473,8476],{"style":132},[35,8474],{"className":8475,"style":116},[115],[35,8477],{"className":8478,"style":140},[139],[35,8480,8481,8484],{"style":143},[35,8482],{"className":8483,"style":116},[115],[35,8485,8487],{"className":8486},[120,121,122,123],[35,8488,8490],{"className":8489},[86,123],[35,8491,927],{"className":8492},[86,123],[35,8494,162],{"className":8495},[161],[35,8497,8499],{"className":8498},[103],[35,8500,8502],{"className":8501,"style":169},[107],[35,8503],{},[35,8505],{"className":8506},[175,91],[35,8508],{"className":8509,"style":367},[366],[35,8511,278],{"className":8512},[371],[35,8514],{"className":8515,"style":367},[366],[35,8517,8519,8522,8525,8531],{"className":8518},[77],[35,8520],{"className":8521,"style":513},[81],[35,8523,517],{"className":8524},[86],[35,8526,8528],{"className":8527},[86],[35,8529,525],{"className":8530},[524],[35,8532,8534],{"className":8533},[86],"0375",[2255,8536,8537],{"align":7764},[35,8538,8540,8555],{"className":8539},[38],[35,8541,8543],{"className":8542},[42],[44,8544,8545],{"xmlns":46},[48,8546,8547,8552],{},[51,8548,8549],{},[61,8550,8551],{},"0,6208",[65,8553,8554],{"encoding":67},"0{,}6208",[35,8556,8558],{"className":8557,"ariaHidden":73},[72],[35,8559,8561,8564,8567,8573],{"className":8560},[77],[35,8562],{"className":8563,"style":513},[81],[35,8565,517],{"className":8566},[86],[35,8568,8570],{"className":8569},[86],[35,8571,525],{"className":8572},[524],[35,8574,8576],{"className":8575},[86],"6208",[2174,8578,8579,8581,8827],{},[2255,8580,927],{"align":2179},[2255,8582,8583],{"align":7764},[35,8584,8586,8621],{"className":8585},[38],[35,8587,8589],{"className":8588},[42],[44,8590,8591],{"xmlns":46},[48,8592,8593,8618],{},[51,8594,8595,8601,8603,8605,8607,8613,8615],{},[54,8596,8597,8599],{},[61,8598,272],{},[61,8600,3203],{},[276,8602,6472],{},[61,8604,529],{},[276,8606,6472],{},[54,8608,8609,8611],{},[61,8610,63],{},[61,8612,3100],{},[276,8614,8030],{},[61,8616,8617],{},"0,0223",[65,8619,8620],{"encoding":67},"\\frac{1}{128} \\times 5 \\times \\frac{4}{7} \\approx 0{,}0223",[35,8622,8624,8707,8725,8808],{"className":8623,"ariaHidden":73},[72],[35,8625,8627,8630,8698,8701,8704],{"className":8626},[77],[35,8628],{"className":8629,"style":293},[81],[35,8631,8633,8636,8695],{"className":8632},[86],[35,8634],{"className":8635},[90,91],[35,8637,8639],{"className":8638},[54],[35,8640,8642,8687],{"className":8641},[98,99],[35,8643,8645,8684],{"className":8644},[103],[35,8646,8648,8662,8670],{"className":8647,"style":312},[107],[35,8649,8650,8653],{"style":111},[35,8651],{"className":8652,"style":116},[115],[35,8654,8656],{"className":8655},[120,121,122,123],[35,8657,8659],{"className":8658},[86,123],[35,8660,3203],{"className":8661},[86,123],[35,8663,8664,8667],{"style":132},[35,8665],{"className":8666,"style":116},[115],[35,8668],{"className":8669,"style":140},[139],[35,8671,8672,8675],{"style":143},[35,8673],{"className":8674,"style":116},[115],[35,8676,8678],{"className":8677},[120,121,122,123],[35,8679,8681],{"className":8680},[86,123],[35,8682,272],{"className":8683},[86,123],[35,8685,162],{"className":8686},[161],[35,8688,8690],{"className":8689},[103],[35,8691,8693],{"className":8692,"style":169},[107],[35,8694],{},[35,8696],{"className":8697},[175,91],[35,8699],{"className":8700,"style":2223},[366],[35,8702,6472],{"className":8703},[2227],[35,8705],{"className":8706,"style":2223},[366],[35,8708,8710,8713,8716,8719,8722],{"className":8709},[77],[35,8711],{"className":8712,"style":1331},[81],[35,8714,529],{"className":8715},[86],[35,8717],{"className":8718,"style":2223},[366],[35,8720,6472],{"className":8721},[2227],[35,8723],{"className":8724,"style":2223},[366],[35,8726,8728,8731,8799,8802,8805],{"className":8727},[77],[35,8729],{"className":8730,"style":293},[81],[35,8732,8734,8737,8796],{"className":8733},[86],[35,8735],{"className":8736},[90,91],[35,8738,8740],{"className":8739},[54],[35,8741,8743,8788],{"className":8742},[98,99],[35,8744,8746,8785],{"className":8745},[103],[35,8747,8749,8763,8771],{"className":8748,"style":312},[107],[35,8750,8751,8754],{"style":111},[35,8752],{"className":8753,"style":116},[115],[35,8755,8757],{"className":8756},[120,121,122,123],[35,8758,8760],{"className":8759},[86,123],[35,8761,3100],{"className":8762},[86,123],[35,8764,8765,8768],{"style":132},[35,8766],{"className":8767,"style":116},[115],[35,8769],{"className":8770,"style":140},[139],[35,8772,8773,8776],{"style":143},[35,8774],{"className":8775,"style":116},[115],[35,8777,8779],{"className":8778},[120,121,122,123],[35,8780,8782],{"className":8781},[86,123],[35,8783,63],{"className":8784},[86,123],[35,8786,162],{"className":8787},[161],[35,8789,8791],{"className":8790},[103],[35,8792,8794],{"className":8793,"style":169},[107],[35,8795],{},[35,8797],{"className":8798},[175,91],[35,8800],{"className":8801,"style":367},[366],[35,8803,8030],{"className":8804},[371],[35,8806],{"className":8807,"style":367},[366],[35,8809,8811,8814,8817,8823],{"className":8810},[77],[35,8812],{"className":8813,"style":513},[81],[35,8815,517],{"className":8816},[86],[35,8818,8820],{"className":8819},[86],[35,8821,525],{"className":8822},[524],[35,8824,8826],{"className":8825},[86],"0223",[2255,8828,8829],{"align":7764},[35,8830,8832,8847],{"className":8831},[38],[35,8833,8835],{"className":8834},[42],[44,8836,8837],{"xmlns":46},[48,8838,8839,8844],{},[51,8840,8841],{},[61,8842,8843],{},"0,6431",[65,8845,8846],{"encoding":67},"0{,}6431",[35,8848,8850],{"className":8849,"ariaHidden":73},[72],[35,8851,8853,8856,8859,8865],{"className":8852},[77],[35,8854],{"className":8855,"style":513},[81],[35,8857,517],{"className":8858},[86],[35,8860,8862],{"className":8861},[86],[35,8863,525],{"className":8864},[524],[35,8866,8868],{"className":8867},[86],"6431",[2174,8870,8871,8873,9119],{},[2255,8872,63],{"align":2179},[2255,8874,8875],{"align":7764},[35,8876,8878,8913],{"className":8877},[38],[35,8879,8881],{"className":8880},[42],[44,8882,8883],{"xmlns":46},[48,8884,8885,8910],{},[51,8886,8887,8893,8895,8897,8899,8905,8907],{},[54,8888,8889,8891],{},[61,8890,272],{},[61,8892,3464],{},[276,8894,6472],{},[61,8896,3443],{},[276,8898,6472],{},[54,8900,8901,8903],{},[61,8902,529],{},[61,8904,3360],{},[276,8906,8030],{},[61,8908,8909],{},"0,0152",[65,8911,8912],{"encoding":67},"\\frac{1}{512} \\times 14 \\times \\frac{5}{9} \\approx 0{,}0152",[35,8914,8916,8999,9017,9100],{"className":8915,"ariaHidden":73},[72],[35,8917,8919,8922,8990,8993,8996],{"className":8918},[77],[35,8920],{"className":8921,"style":293},[81],[35,8923,8925,8928,8987],{"className":8924},[86],[35,8926],{"className":8927},[90,91],[35,8929,8931],{"className":8930},[54],[35,8932,8934,8979],{"className":8933},[98,99],[35,8935,8937,8976],{"className":8936},[103],[35,8938,8940,8954,8962],{"className":8939,"style":312},[107],[35,8941,8942,8945],{"style":111},[35,8943],{"className":8944,"style":116},[115],[35,8946,8948],{"className":8947},[120,121,122,123],[35,8949,8951],{"className":8950},[86,123],[35,8952,3464],{"className":8953},[86,123],[35,8955,8956,8959],{"style":132},[35,8957],{"className":8958,"style":116},[115],[35,8960],{"className":8961,"style":140},[139],[35,8963,8964,8967],{"style":143},[35,8965],{"className":8966,"style":116},[115],[35,8968,8970],{"className":8969},[120,121,122,123],[35,8971,8973],{"className":8972},[86,123],[35,8974,272],{"className":8975},[86,123],[35,8977,162],{"className":8978},[161],[35,8980,8982],{"className":8981},[103],[35,8983,8985],{"className":8984,"style":169},[107],[35,8986],{},[35,8988],{"className":8989},[175,91],[35,8991],{"className":8992,"style":2223},[366],[35,8994,6472],{"className":8995},[2227],[35,8997],{"className":8998,"style":2223},[366],[35,9000,9002,9005,9008,9011,9014],{"className":9001},[77],[35,9003],{"className":9004,"style":1331},[81],[35,9006,3443],{"className":9007},[86],[35,9009],{"className":9010,"style":2223},[366],[35,9012,6472],{"className":9013},[2227],[35,9015],{"className":9016,"style":2223},[366],[35,9018,9020,9023,9091,9094,9097],{"className":9019},[77],[35,9021],{"className":9022,"style":293},[81],[35,9024,9026,9029,9088],{"className":9025},[86],[35,9027],{"className":9028},[90,91],[35,9030,9032],{"className":9031},[54],[35,9033,9035,9080],{"className":9034},[98,99],[35,9036,9038,9077],{"className":9037},[103],[35,9039,9041,9055,9063],{"className":9040,"style":312},[107],[35,9042,9043,9046],{"style":111},[35,9044],{"className":9045,"style":116},[115],[35,9047,9049],{"className":9048},[120,121,122,123],[35,9050,9052],{"className":9051},[86,123],[35,9053,3360],{"className":9054},[86,123],[35,9056,9057,9060],{"style":132},[35,9058],{"className":9059,"style":116},[115],[35,9061],{"className":9062,"style":140},[139],[35,9064,9065,9068],{"style":143},[35,9066],{"className":9067,"style":116},[115],[35,9069,9071],{"className":9070},[120,121,122,123],[35,9072,9074],{"className":9073},[86,123],[35,9075,529],{"className":9076},[86,123],[35,9078,162],{"className":9079},[161],[35,9081,9083],{"className":9082},[103],[35,9084,9086],{"className":9085,"style":169},[107],[35,9087],{},[35,9089],{"className":9090},[175,91],[35,9092],{"className":9093,"style":367},[366],[35,9095,8030],{"className":9096},[371],[35,9098],{"className":9099,"style":367},[366],[35,9101,9103,9106,9109,9115],{"className":9102},[77],[35,9104],{"className":9105,"style":513},[81],[35,9107,517],{"className":9108},[86],[35,9110,9112],{"className":9111},[86],[35,9113,525],{"className":9114},[524],[35,9116,9118],{"className":9117},[86],"0152",[2255,9120,9121],{"align":7764},[35,9122,9124,9139],{"className":9123},[38],[35,9125,9127],{"className":9126},[42],[44,9128,9129],{"xmlns":46},[48,9130,9131,9136],{},[51,9132,9133],{},[61,9134,9135],{},"0,6583",[65,9137,9138],{"encoding":67},"0{,}6583",[35,9140,9142],{"className":9141,"ariaHidden":73},[72],[35,9143,9145,9148,9151,9157],{"className":9144},[77],[35,9146],{"className":9147,"style":513},[81],[35,9149,517],{"className":9150},[86],[35,9152,9154],{"className":9153},[86],[35,9155,525],{"className":9156},[524],[35,9158,9160],{"className":9159},[86],"6583",[17,9162,9163,9164,9201,9202,9333],{},"Con sólo cinco términos ya estamos en ",[35,9165,9167,9180],{"className":9166},[38],[35,9168,9170],{"className":9169},[42],[44,9171,9172],{"xmlns":46},[48,9173,9174,9178],{},[51,9175,9176],{},[61,9177,9135],{},[65,9179,9138],{"encoding":67},[35,9181,9183],{"className":9182,"ariaHidden":73},[72],[35,9184,9186,9189,9192,9198],{"className":9185},[77],[35,9187],{"className":9188,"style":513},[81],[35,9190,517],{"className":9191},[86],[35,9193,9195],{"className":9194},[86],[35,9196,525],{"className":9197},[524],[35,9199,9160],{"className":9200},[86],", acercándonos lentamente a ",[35,9203,9205,9228],{"className":9204},[38],[35,9206,9208],{"className":9207},[42],[44,9209,9210],{"xmlns":46},[48,9211,9212,9225],{},[51,9213,9214,9220,9222],{},[54,9215,9216,9218],{},[57,9217,59],{},[61,9219,63],{},[276,9221,8030],{},[61,9223,9224],{},"0,7854",[65,9226,9227],{"encoding":67},"\\frac{\\pi}{4} \\approx 0{,}7854",[35,9229,9231,9314],{"className":9230,"ariaHidden":73},[72],[35,9232,9234,9237,9305,9308,9311],{"className":9233},[77],[35,9235],{"className":9236,"style":82},[81],[35,9238,9240,9243,9302],{"className":9239},[86],[35,9241],{"className":9242},[90,91],[35,9244,9246],{"className":9245},[54],[35,9247,9249,9294],{"className":9248},[98,99],[35,9250,9252,9291],{"className":9251},[103],[35,9253,9255,9269,9277],{"className":9254,"style":108},[107],[35,9256,9257,9260],{"style":111},[35,9258],{"className":9259,"style":116},[115],[35,9261,9263],{"className":9262},[120,121,122,123],[35,9264,9266],{"className":9265},[86,123],[35,9267,63],{"className":9268},[86,123],[35,9270,9271,9274],{"style":132},[35,9272],{"className":9273,"style":116},[115],[35,9275],{"className":9276,"style":140},[139],[35,9278,9279,9282],{"style":143},[35,9280],{"className":9281,"style":116},[115],[35,9283,9285],{"className":9284},[120,121,122,123],[35,9286,9288],{"className":9287},[86,123],[35,9289,59],{"className":9290,"style":157},[86,156,123],[35,9292,162],{"className":9293},[161],[35,9295,9297],{"className":9296},[103],[35,9298,9300],{"className":9299,"style":169},[107],[35,9301],{},[35,9303],{"className":9304},[175,91],[35,9306],{"className":9307,"style":367},[366],[35,9309,8030],{"className":9310},[371],[35,9312],{"className":9313,"style":367},[366],[35,9315,9317,9320,9323,9329],{"className":9316},[77],[35,9318],{"className":9319,"style":513},[81],[35,9321,517],{"className":9322},[86],[35,9324,9326],{"className":9325},[86],[35,9327,525],{"className":9328},[524],[35,9330,9332],{"className":9331},[86],"7854",". La convergencia es lenta, pero la dirección es inequívoca cuando el número de lanzamientos es suficientemente grande.",[561,9335,9337],{"id":9336},"la-aparición-del-arcoseno","La aparición del arcoseno",[17,9339,9340],{},"Aquí es donde ocurre la magia. Sustituyamos la fórmula de los números de Catalan en nuestra serie:",[17,9342,9344],{"className":9343},[592],[35,9345,9347,9394],{"className":9346},[38],[35,9348,9350],{"className":9349},[42],[44,9351,9352],{"xmlns":46},[48,9353,9354,9392],{},[51,9355,9356,9362,9364,9376],{},[4294,9357,9358,9360],{},[57,9359,4298],{},[57,9361,2197],{},[276,9363,278],{},[54,9365,9366,9368],{},[61,9367,272],{},[51,9369,9370,9372,9374],{},[57,9371,2197],{},[276,9373,1316],{},[61,9375,272],{},[51,9377,9378,9380,9390],{},[276,9379,1028],{"fence":73},[54,9381,9382,9388],{"linethickness":4321},[51,9383,9384,9386],{},[61,9385,404],{},[57,9387,2197],{},[57,9389,2197],{},[276,9391,1037],{"fence":73},[65,9393,4334],{"encoding":67},[35,9395,9397,9452],{"className":9396,"ariaHidden":73},[72],[35,9398,9400,9403,9443,9446,9449],{"className":9399},[77],[35,9401],{"className":9402,"style":4344},[81],[35,9404,9406,9409],{"className":9405},[86],[35,9407,4298],{"className":9408,"style":4351},[86,156],[35,9410,9412],{"className":9411},[1156],[35,9413,9415,9435],{"className":9414},[98,99],[35,9416,9418,9432],{"className":9417},[103],[35,9419,9421],{"className":9420,"style":4364},[107],[35,9422,9423,9426],{"style":4367},[35,9424],{"className":9425,"style":1173},[115],[35,9427,9429],{"className":9428},[120,121,122,123],[35,9430,2197],{"className":9431},[86,156,123],[35,9433,162],{"className":9434},[161],[35,9436,9438],{"className":9437},[103],[35,9439,9441],{"className":9440,"style":4386},[107],[35,9442],{},[35,9444],{"className":9445,"style":367},[366],[35,9447,278],{"className":9448},[371],[35,9450],{"className":9451,"style":367},[366],[35,9453,9455,9458,9532],{"className":9454},[77],[35,9456],{"className":9457,"style":4404},[81],[35,9459,9461,9464,9529],{"className":9460},[86],[35,9462],{"className":9463},[90,91],[35,9465,9467],{"className":9466},[54],[35,9468,9470,9521],{"className":9469},[98,99],[35,9471,9473,9518],{"className":9472},[103],[35,9474,9476,9496,9504],{"className":9475,"style":312},[107],[35,9477,9478,9481],{"style":111},[35,9479],{"className":9480,"style":116},[115],[35,9482,9484],{"className":9483},[120,121,122,123],[35,9485,9487,9490,9493],{"className":9486},[86,123],[35,9488,2197],{"className":9489},[86,156,123],[35,9491,1316],{"className":9492},[2227,123],[35,9494,272],{"className":9495},[86,123],[35,9497,9498,9501],{"style":132},[35,9499],{"className":9500,"style":116},[115],[35,9502],{"className":9503,"style":140},[139],[35,9505,9506,9509],{"style":143},[35,9507],{"className":9508,"style":116},[115],[35,9510,9512],{"className":9511},[120,121,122,123],[35,9513,9515],{"className":9514},[86,123],[35,9516,272],{"className":9517},[86,123],[35,9519,162],{"className":9520},[161],[35,9522,9524],{"className":9523},[103],[35,9525,9527],{"className":9526,"style":3736},[107],[35,9528],{},[35,9530],{"className":9531},[175,91],[35,9533,9535,9541,9595],{"className":9534},[86],[35,9536,9538],{"className":9537,"style":1073},[90,1072],[35,9539,1028],{"className":9540},[1077,1078],[35,9542,9544],{"className":9543},[54],[35,9545,9547,9587],{"className":9546},[98,99],[35,9548,9550,9584],{"className":9549},[103],[35,9551,9553,9567],{"className":9552,"style":4500},[107],[35,9554,9555,9558],{"style":4503},[35,9556],{"className":9557,"style":1173},[115],[35,9559,9561],{"className":9560},[120,121,122,123],[35,9562,9564],{"className":9563},[86,123],[35,9565,2197],{"className":9566},[86,156,123],[35,9568,9569,9572],{"style":4518},[35,9570],{"className":9571,"style":1173},[115],[35,9573,9575],{"className":9574},[120,121,122,123],[35,9576,9578,9581],{"className":9577},[86,123],[35,9579,404],{"className":9580},[86,123],[35,9582,2197],{"className":9583},[86,156,123],[35,9585,162],{"className":9586},[161],[35,9588,9590],{"className":9589},[103],[35,9591,9593],{"className":9592,"style":169},[107],[35,9594],{},[35,9596,9598],{"className":9597,"style":1073},[175,1072],[35,9599,1037],{"className":9600},[1077,1078],[17,9602,9603,9604,9674,9675,9801,9802,9865],{},"Al multiplicar ",[35,9605,9607,9625],{"className":9606},[38],[35,9608,9610],{"className":9609},[42],[44,9611,9612],{"xmlns":46},[48,9613,9614,9622],{},[51,9615,9616],{},[4294,9617,9618,9620],{},[57,9619,4298],{},[57,9621,2197],{},[65,9623,9624],{"encoding":67},"C_n",[35,9626,9628],{"className":9627,"ariaHidden":73},[72],[35,9629,9631,9634],{"className":9630},[77],[35,9632],{"className":9633,"style":4344},[81],[35,9635,9637,9640],{"className":9636},[86],[35,9638,4298],{"className":9639,"style":4351},[86,156],[35,9641,9643],{"className":9642},[1156],[35,9644,9646,9666],{"className":9645},[98,99],[35,9647,9649,9663],{"className":9648},[103],[35,9650,9652],{"className":9651,"style":4364},[107],[35,9653,9654,9657],{"style":4367},[35,9655],{"className":9656,"style":1173},[115],[35,9658,9660],{"className":9659},[120,121,122,123],[35,9661,2197],{"className":9662},[86,156,123],[35,9664,162],{"className":9665},[161],[35,9667,9669],{"className":9668},[103],[35,9670,9672],{"className":9671,"style":4386},[107],[35,9673],{}," por ",[35,9676,9678,9709],{"className":9677},[38],[35,9679,9681],{"className":9680},[42],[44,9682,9683],{"xmlns":46},[48,9684,9685,9707],{},[51,9686,9687],{},[54,9688,9689,9697],{},[51,9690,9691,9693,9695],{},[57,9692,2197],{},[276,9694,1316],{},[61,9696,272],{},[51,9698,9699,9701,9703,9705],{},[61,9700,404],{},[57,9702,2197],{},[276,9704,1316],{},[61,9706,272],{},[65,9708,3647],{"encoding":67},[35,9710,9712],{"className":9711,"ariaHidden":73},[72],[35,9713,9715,9718],{"className":9714},[77],[35,9716],{"className":9717,"style":3657},[81],[35,9719,9721,9724,9798],{"className":9720},[86],[35,9722],{"className":9723},[90,91],[35,9725,9727],{"className":9726},[54],[35,9728,9730,9790],{"className":9729},[98,99],[35,9731,9733,9787],{"className":9732},[103],[35,9734,9736,9759,9767],{"className":9735,"style":312},[107],[35,9737,9738,9741],{"style":111},[35,9739],{"className":9740,"style":116},[115],[35,9742,9744],{"className":9743},[120,121,122,123],[35,9745,9747,9750,9753,9756],{"className":9746},[86,123],[35,9748,404],{"className":9749},[86,123],[35,9751,2197],{"className":9752},[86,156,123],[35,9754,1316],{"className":9755},[2227,123],[35,9757,272],{"className":9758},[86,123],[35,9760,9761,9764],{"style":132},[35,9762],{"className":9763,"style":116},[115],[35,9765],{"className":9766,"style":140},[139],[35,9768,9769,9772],{"style":143},[35,9770],{"className":9771,"style":116},[115],[35,9773,9775],{"className":9774},[120,121,122,123],[35,9776,9778,9781,9784],{"className":9777},[86,123],[35,9779,2197],{"className":9780},[86,156,123],[35,9782,1316],{"className":9783},[2227,123],[35,9785,272],{"className":9786},[86,123],[35,9788,162],{"className":9789},[161],[35,9791,9793],{"className":9792},[103],[35,9794,9796],{"className":9795,"style":3736},[107],[35,9797],{},[35,9799],{"className":9800},[175,91],", el factor ",[35,9803,9805,9828],{"className":9804},[38],[35,9806,9808],{"className":9807},[42],[44,9809,9810],{"xmlns":46},[48,9811,9812,9825],{},[51,9813,9814,9817,9819,9821,9823],{},[276,9815,1028],{"stretchy":9816},"false",[57,9818,2197],{},[276,9820,1316],{},[61,9822,272],{},[276,9824,1037],{"stretchy":9816},[65,9826,9827],{"encoding":67},"(n+1)",[35,9829,9831,9853],{"className":9830,"ariaHidden":73},[72],[35,9832,9834,9838,9841,9844,9847,9850],{"className":9833},[77],[35,9835],{"className":9836,"style":9837},[81],"height:1em;vertical-align:-0.25em;",[35,9839,1028],{"className":9840},[90],[35,9842,2197],{"className":9843},[86,156],[35,9845],{"className":9846,"style":2223},[366],[35,9848,1316],{"className":9849},[2227],[35,9851],{"className":9852,"style":2223},[366],[35,9854,9856,9859,9862],{"className":9855},[77],[35,9857],{"className":9858,"style":9837},[81],[35,9860,272],{"className":9861},[86],[35,9863,1037],{"className":9864},[175]," se cancela limpiamente:",[17,9867,9869],{"className":9868},[592],[35,9870,9872,9964],{"className":9871},[38],[35,9873,9875],{"className":9874},[42],[44,9876,9877],{"xmlns":46},[48,9878,9879,9961],{},[51,9880,9881,9893,9909,9911,9931,9933],{},[54,9882,9883,9885],{},[61,9884,272],{},[51,9886,9887,9889,9891],{},[57,9888,2197],{},[276,9890,1316],{},[61,9892,272],{},[51,9894,9895,9897,9907],{},[276,9896,1028],{"fence":73},[54,9898,9899,9905],{"linethickness":4321},[51,9900,9901,9903],{},[61,9902,404],{},[57,9904,2197],{},[57,9906,2197],{},[276,9908,1037],{"fence":73},[276,9910,7344],{},[54,9912,9913,9921],{},[51,9914,9915,9917,9919],{},[57,9916,2197],{},[276,9918,1316],{},[61,9920,272],{},[51,9922,9923,9925,9927,9929],{},[61,9924,404],{},[57,9926,2197],{},[276,9928,1316],{},[61,9930,272],{},[276,9932,278],{},[54,9934,9935,9951],{},[51,9936,9937,9939,9949],{},[276,9938,1028],{"fence":73},[54,9940,9941,9947],{"linethickness":4321},[51,9942,9943,9945],{},[61,9944,404],{},[57,9946,2197],{},[57,9948,2197],{},[276,9950,1037],{"fence":73},[51,9952,9953,9955,9957,9959],{},[61,9954,404],{},[57,9956,2197],{},[276,9958,1316],{},[61,9960,272],{},[65,9962,9963],{"encoding":67},"\\frac{1}{n+1}\\binom{2n}{n} \\cdot \\frac{n+1}{2n+1} = \\frac{\\binom{2n}{n}}{2n+1}",[35,9965,9967,10125,10223],{"className":9966,"ariaHidden":73},[72],[35,9968,9970,9973,10047,10116,10119,10122],{"className":9969},[77],[35,9971],{"className":9972,"style":4404},[81],[35,9974,9976,9979,10044],{"className":9975},[86],[35,9977],{"className":9978},[90,91],[35,9980,9982],{"className":9981},[54],[35,9983,9985,10036],{"className":9984},[98,99],[35,9986,9988,10033],{"className":9987},[103],[35,9989,9991,10011,10019],{"className":9990,"style":312},[107],[35,9992,9993,9996],{"style":111},[35,9994],{"className":9995,"style":116},[115],[35,9997,9999],{"className":9998},[120,121,122,123],[35,10000,10002,10005,10008],{"className":10001},[86,123],[35,10003,2197],{"className":10004},[86,156,123],[35,10006,1316],{"className":10007},[2227,123],[35,10009,272],{"className":10010},[86,123],[35,10012,10013,10016],{"style":132},[35,10014],{"className":10015,"style":116},[115],[35,10017],{"className":10018,"style":140},[139],[35,10020,10021,10024],{"style":143},[35,10022],{"className":10023,"style":116},[115],[35,10025,10027],{"className":10026},[120,121,122,123],[35,10028,10030],{"className":10029},[86,123],[35,10031,272],{"className":10032},[86,123],[35,10034,162],{"className":10035},[161],[35,10037,10039],{"className":10038},[103],[35,10040,10042],{"className":10041,"style":3736},[107],[35,10043],{},[35,10045],{"className":10046},[175,91],[35,10048,10050,10056,10110],{"className":10049},[86],[35,10051,10053],{"className":10052,"style":1073},[90,1072],[35,10054,1028],{"className":10055},[1077,1078],[35,10057,10059],{"className":10058},[54],[35,10060,10062,10102],{"className":10061},[98,99],[35,10063,10065,10099],{"className":10064},[103],[35,10066,10068,10082],{"className":10067,"style":4500},[107],[35,10069,10070,10073],{"style":4503},[35,10071],{"className":10072,"style":1173},[115],[35,10074,10076],{"className":10075},[120,121,122,123],[35,10077,10079],{"className":10078},[86,123],[35,10080,2197],{"className":10081},[86,156,123],[35,10083,10084,10087],{"style":4518},[35,10085],{"className":10086,"style":1173},[115],[35,10088,10090],{"className":10089},[120,121,122,123],[35,10091,10093,10096],{"className":10092},[86,123],[35,10094,404],{"className":10095},[86,123],[35,10097,2197],{"className":10098},[86,156,123],[35,10100,162],{"className":10101},[161],[35,10103,10105],{"className":10104},[103],[35,10106,10108],{"className":10107,"style":169},[107],[35,10109],{},[35,10111,10113],{"className":10112,"style":1073},[175,1072],[35,10114,1037],{"className":10115},[1077,1078],[35,10117],{"className":10118,"style":2223},[366],[35,10120,7344],{"className":10121},[2227],[35,10123],{"className":10124,"style":2223},[366],[35,10126,10128,10131,10214,10217,10220],{"className":10127},[77],[35,10129],{"className":10130,"style":3657},[81],[35,10132,10134,10137,10211],{"className":10133},[86],[35,10135],{"className":10136},[90,91],[35,10138,10140],{"className":10139},[54],[35,10141,10143,10203],{"className":10142},[98,99],[35,10144,10146,10200],{"className":10145},[103],[35,10147,10149,10172,10180],{"className":10148,"style":312},[107],[35,10150,10151,10154],{"style":111},[35,10152],{"className":10153,"style":116},[115],[35,10155,10157],{"className":10156},[120,121,122,123],[35,10158,10160,10163,10166,10169],{"className":10159},[86,123],[35,10161,404],{"className":10162},[86,123],[35,10164,2197],{"className":10165},[86,156,123],[35,10167,1316],{"className":10168},[2227,123],[35,10170,272],{"className":10171},[86,123],[35,10173,10174,10177],{"style":132},[35,10175],{"className":10176,"style":116},[115],[35,10178],{"className":10179,"style":140},[139],[35,10181,10182,10185],{"style":143},[35,10183],{"className":10184,"style":116},[115],[35,10186,10188],{"className":10187},[120,121,122,123],[35,10189,10191,10194,10197],{"className":10190},[86,123],[35,10192,2197],{"className":10193},[86,156,123],[35,10195,1316],{"className":10196},[2227,123],[35,10198,272],{"className":10199},[86,123],[35,10201,162],{"className":10202},[161],[35,10204,10206],{"className":10205},[103],[35,10207,10209],{"className":10208,"style":3736},[107],[35,10210],{},[35,10212],{"className":10213},[175,91],[35,10215],{"className":10216,"style":367},[366],[35,10218,278],{"className":10219},[371],[35,10221],{"className":10222,"style":367},[366],[35,10224,10226,10230],{"className":10225},[77],[35,10227],{"className":10228,"style":10229},[81],"height:1.9133em;vertical-align:-0.4033em;",[35,10231,10233,10236,10384],{"className":10232},[86],[35,10234],{"className":10235},[90,91],[35,10237,10239],{"className":10238},[54],[35,10240,10242,10376],{"className":10241},[98,99],[35,10243,10245,10373],{"className":10244},[103],[35,10246,10249,10272,10280],{"className":10247,"style":10248},[107],"height:1.51em;",[35,10250,10251,10254],{"style":111},[35,10252],{"className":10253,"style":116},[115],[35,10255,10257],{"className":10256},[120,121,122,123],[35,10258,10260,10263,10266,10269],{"className":10259},[86,123],[35,10261,404],{"className":10262},[86,123],[35,10264,2197],{"className":10265},[86,156,123],[35,10267,1316],{"className":10268},[2227,123],[35,10270,272],{"className":10271},[86,123],[35,10273,10274,10277],{"style":132},[35,10275],{"className":10276,"style":116},[115],[35,10278],{"className":10279,"style":140},[139],[35,10281,10283,10286],{"style":10282},"top:-3.7125em;",[35,10284],{"className":10285,"style":116},[115],[35,10287,10289],{"className":10288},[120,121,122,123],[35,10290,10292],{"className":10291},[86,123],[35,10293,10295,10306,10364],{"className":10294},[86,123],[35,10296,10300],{"className":10297,"style":10299},[90,120,7532,10298,123,1072],"size6","top:0.075em;",[35,10301,10303],{"className":10302},[1077,1078,123],[35,10304,1028],{"className":10305},[123],[35,10307,10309],{"className":10308},[54],[35,10310,10312,10355],{"className":10311},[98,99],[35,10313,10315,10352],{"className":10314},[103],[35,10316,10319,10334],{"className":10317,"style":10318},[107],"height:0.9313em;",[35,10320,10322,10325],{"style":10321},"top:-2.156em;",[35,10323],{"className":10324,"style":7528},[115],[35,10326,10328],{"className":10327},[120,7532,1078,123],[35,10329,10331],{"className":10330},[86,123],[35,10332,2197],{"className":10333},[86,156,123],[35,10335,10337,10340],{"style":10336},"top:-2.971em;",[35,10338],{"className":10339,"style":7528},[115],[35,10341,10343],{"className":10342},[120,7532,1078,123],[35,10344,10346,10349],{"className":10345},[86,123],[35,10347,404],{"className":10348},[86,123],[35,10350,2197],{"className":10351},[86,156,123],[35,10353,162],{"className":10354},[161],[35,10356,10358],{"className":10357},[103],[35,10359,10362],{"className":10360,"style":10361},[107],"height:0.344em;",[35,10363],{},[35,10365,10367],{"className":10366,"style":10299},[175,120,7532,10298,123,1072],[35,10368,10370],{"className":10369},[1077,1078,123],[35,10371,1037],{"className":10372},[123],[35,10374,162],{"className":10375},[161],[35,10377,10379],{"className":10378},[103],[35,10380,10382],{"className":10381,"style":3736},[107],[35,10383],{},[35,10385],{"className":10386},[175,91],[17,10388,10389],{},"Y la serie queda:",[17,10391,10393],{"className":10392},[592],[35,10394,10396,10532],{"className":10395},[38],[35,10397,10399],{"className":10398},[42],[44,10400,10401],{"xmlns":46},[48,10402,10403,10529],{},[51,10404,10405,10407,10409,10423,10469,10471,10477,10491],{},[606,10406,7299],{},[276,10408,278],{},[7303,10410,10411,10413,10421],{},[276,10412,7307],{},[51,10414,10415,10417,10419],{},[57,10416,2197],{},[276,10418,278],{},[61,10420,517],{},[57,10422,7319],{"mathvariant":7318},[54,10424,10425,10441],{},[51,10426,10427,10429,10439],{},[276,10428,1028],{"fence":73},[54,10430,10431,10437],{"linethickness":4321},[51,10432,10433,10435],{},[61,10434,404],{},[57,10436,2197],{},[57,10438,2197],{},[276,10440,1037],{"fence":73},[51,10442,10443,10457,10459,10461,10463,10465,10467],{},[1022,10444,10445,10447],{},[61,10446,404],{},[51,10448,10449,10451,10453,10455],{},[61,10450,404],{},[57,10452,2197],{},[276,10454,1316],{},[61,10456,272],{},[276,10458,1028],{"stretchy":9816},[61,10460,404],{},[57,10462,2197],{},[276,10464,1316],{},[61,10466,272],{},[276,10468,1037],{"stretchy":9816},[276,10470,278],{},[54,10472,10473,10475],{},[61,10474,272],{},[61,10476,404],{},[7303,10478,10479,10481,10489],{},[276,10480,7307],{},[51,10482,10483,10485,10487],{},[57,10484,2197],{},[276,10486,278],{},[61,10488,517],{},[57,10490,7319],{"mathvariant":7318},[54,10492,10493,10509],{},[51,10494,10495,10497,10507],{},[276,10496,1028],{"fence":73},[54,10498,10499,10505],{"linethickness":4321},[51,10500,10501,10503],{},[61,10502,404],{},[57,10504,2197],{},[57,10506,2197],{},[276,10508,1037],{"fence":73},[51,10510,10511,10517,10519,10521,10523,10525,10527],{},[1022,10512,10513,10515],{},[61,10514,63],{},[57,10516,2197],{},[276,10518,1028],{"stretchy":9816},[61,10520,404],{},[57,10522,2197],{},[276,10524,1316],{},[61,10526,272],{},[276,10528,1037],{"stretchy":9816},[65,10530,10531],{"encoding":67},"\\text{Media} = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{2^{2n+1}(2n+1)} = \\frac{1}{2} \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n(2n+1)}",[35,10533,10535,10556,10835],{"className":10534,"ariaHidden":73},[72],[35,10536,10538,10541,10547,10550,10553],{"className":10537},[77],[35,10539],{"className":10540,"style":7026},[81],[35,10542,10544],{"className":10543},[86,625],[35,10545,7299],{"className":10546},[86],[35,10548],{"className":10549,"style":367},[366],[35,10551,278],{"className":10552},[371],[35,10554],{"className":10555,"style":367},[366],[35,10557,10559,10563,10626,10629,10826,10829,10832],{"className":10558},[77],[35,10560],{"className":10561,"style":10562},[81],"height:2.03em;vertical-align:-0.52em;",[35,10564,10566,10569],{"className":10565},[7402],[35,10567,7307],{"className":10568,"style":7408},[7402,7406,7407],[35,10570,10572],{"className":10571},[1156],[35,10573,10575,10618],{"className":10574},[98,99],[35,10576,10578,10615],{"className":10577},[103],[35,10579,10581,10601],{"className":10580,"style":7421},[107],[35,10582,10583,10586],{"style":7424},[35,10584],{"className":10585,"style":1173},[115],[35,10587,10589],{"className":10588},[120,121,122,123],[35,10590,10592,10595,10598],{"className":10591},[86,123],[35,10593,2197],{"className":10594},[86,156,123],[35,10596,278],{"className":10597},[371,123],[35,10599,517],{"className":10600},[86,123],[35,10602,10603,10606],{"style":7445},[35,10604],{"className":10605,"style":1173},[115],[35,10607,10609],{"className":10608},[120,121,122,123],[35,10610,10612],{"className":10611},[86,123],[35,10613,7319],{"className":10614},[86,123],[35,10616,162],{"className":10617},[161],[35,10619,10621],{"className":10620},[103],[35,10622,10624],{"className":10623,"style":7467},[107],[35,10625],{},[35,10627],{"className":10628,"style":1452},[366],[35,10630,10632,10635,10823],{"className":10631},[86],[35,10633],{"className":10634},[90,91],[35,10636,10638],{"className":10637},[54],[35,10639,10641,10814],{"className":10640},[98,99],[35,10642,10644,10811],{"className":10643},[103],[35,10645,10647,10717,10725],{"className":10646,"style":10248},[107],[35,10648,10649,10652],{"style":111},[35,10650],{"className":10651,"style":116},[115],[35,10653,10655],{"className":10654},[120,121,122,123],[35,10656,10658,10699,10702,10705,10708,10711,10714],{"className":10657},[86,123],[35,10659,10661,10664],{"className":10660},[86,123],[35,10662,404],{"className":10663},[86,123],[35,10665,10667],{"className":10666},[1156],[35,10668,10670],{"className":10669},[98],[35,10671,10673],{"className":10672},[103],[35,10674,10676],{"className":10675,"style":7521},[107],[35,10677,10678,10681],{"style":7524},[35,10679],{"className":10680,"style":7528},[115],[35,10682,10684],{"className":10683},[120,7532,1078,123],[35,10685,10687,10690,10693,10696],{"className":10686},[86,123],[35,10688,404],{"className":10689},[86,123],[35,10691,2197],{"className":10692},[86,156,123],[35,10694,1316],{"className":10695},[2227,123],[35,10697,272],{"className":10698},[86,123],[35,10700,1028],{"className":10701},[90,123],[35,10703,404],{"className":10704},[86,123],[35,10706,2197],{"className":10707},[86,156,123],[35,10709,1316],{"className":10710},[2227,123],[35,10712,272],{"className":10713},[86,123],[35,10715,1037],{"className":10716},[175,123],[35,10718,10719,10722],{"style":132},[35,10720],{"className":10721,"style":116},[115],[35,10723],{"className":10724,"style":140},[139],[35,10726,10727,10730],{"style":10282},[35,10728],{"className":10729,"style":116},[115],[35,10731,10733],{"className":10732},[120,121,122,123],[35,10734,10736],{"className":10735},[86,123],[35,10737,10739,10748,10802],{"className":10738},[86,123],[35,10740,10742],{"className":10741,"style":10299},[90,120,7532,10298,123,1072],[35,10743,10745],{"className":10744},[1077,1078,123],[35,10746,1028],{"className":10747},[123],[35,10749,10751],{"className":10750},[54],[35,10752,10754,10794],{"className":10753},[98,99],[35,10755,10757,10791],{"className":10756},[103],[35,10758,10760,10774],{"className":10759,"style":10318},[107],[35,10761,10762,10765],{"style":10321},[35,10763],{"className":10764,"style":7528},[115],[35,10766,10768],{"className":10767},[120,7532,1078,123],[35,10769,10771],{"className":10770},[86,123],[35,10772,2197],{"className":10773},[86,156,123],[35,10775,10776,10779],{"style":10336},[35,10777],{"className":10778,"style":7528},[115],[35,10780,10782],{"className":10781},[120,7532,1078,123],[35,10783,10785,10788],{"className":10784},[86,123],[35,10786,404],{"className":10787},[86,123],[35,10789,2197],{"className":10790},[86,156,123],[35,10792,162],{"className":10793},[161],[35,10795,10797],{"className":10796},[103],[35,10798,10800],{"className":10799,"style":10361},[107],[35,10801],{},[35,10803,10805],{"className":10804,"style":10299},[175,120,7532,10298,123,1072],[35,10806,10808],{"className":10807},[1077,1078,123],[35,10809,1037],{"className":10810},[123],[35,10812,162],{"className":10813},[161],[35,10815,10817],{"className":10816},[103],[35,10818,10821],{"className":10819,"style":10820},[107],"height:0.52em;",[35,10822],{},[35,10824],{"className":10825},[175,91],[35,10827],{"className":10828,"style":367},[366],[35,10830,278],{"className":10831},[371],[35,10833],{"className":10834,"style":367},[366],[35,10836,10838,10841,10909,10912,10975,10978],{"className":10837},[77],[35,10839],{"className":10840,"style":10562},[81],[35,10842,10844,10847,10906],{"className":10843},[86],[35,10845],{"className":10846},[90,91],[35,10848,10850],{"className":10849},[54],[35,10851,10853,10898],{"className":10852},[98,99],[35,10854,10856,10895],{"className":10855},[103],[35,10857,10859,10873,10881],{"className":10858,"style":312},[107],[35,10860,10861,10864],{"style":111},[35,10862],{"className":10863,"style":116},[115],[35,10865,10867],{"className":10866},[120,121,122,123],[35,10868,10870],{"className":10869},[86,123],[35,10871,404],{"className":10872},[86,123],[35,10874,10875,10878],{"style":132},[35,10876],{"className":10877,"style":116},[115],[35,10879],{"className":10880,"style":140},[139],[35,10882,10883,10886],{"style":143},[35,10884],{"className":10885,"style":116},[115],[35,10887,10889],{"className":10888},[120,121,122,123],[35,10890,10892],{"className":10891},[86,123],[35,10893,272],{"className":10894},[86,123],[35,10896,162],{"className":10897},[161],[35,10899,10901],{"className":10900},[103],[35,10902,10904],{"className":10903,"style":169},[107],[35,10905],{},[35,10907],{"className":10908},[175,91],[35,10910],{"className":10911,"style":1452},[366],[35,10913,10915,10918],{"className":10914},[7402],[35,10916,7307],{"className":10917,"style":7408},[7402,7406,7407],[35,10919,10921],{"className":10920},[1156],[35,10922,10924,10967],{"className":10923},[98,99],[35,10925,10927,10964],{"className":10926},[103],[35,10928,10930,10950],{"className":10929,"style":7421},[107],[35,10931,10932,10935],{"style":7424},[35,10933],{"className":10934,"style":1173},[115],[35,10936,10938],{"className":10937},[120,121,122,123],[35,10939,10941,10944,10947],{"className":10940},[86,123],[35,10942,2197],{"className":10943},[86,156,123],[35,10945,278],{"className":10946},[371,123],[35,10948,517],{"className":10949},[86,123],[35,10951,10952,10955],{"style":7445},[35,10953],{"className":10954,"style":1173},[115],[35,10956,10958],{"className":10957},[120,121,122,123],[35,10959,10961],{"className":10960},[86,123],[35,10962,7319],{"className":10963},[86,123],[35,10965,162],{"className":10966},[161],[35,10968,10970],{"className":10969},[103],[35,10971,10973],{"className":10972,"style":7467},[107],[35,10974],{},[35,10976],{"className":10977,"style":1452},[366],[35,10979,10981,10984,11160],{"className":10980},[86],[35,10982],{"className":10983},[90,91],[35,10985,10987],{"className":10986},[54],[35,10988,10990,11152],{"className":10989},[98,99],[35,10991,10993,11149],{"className":10992},[103],[35,10994,10996,11055,11063],{"className":10995,"style":10248},[107],[35,10997,10998,11001],{"style":111},[35,10999],{"className":11000,"style":116},[115],[35,11002,11004],{"className":11003},[120,121,122,123],[35,11005,11007,11037,11040,11043,11046,11049,11052],{"className":11006},[86,123],[35,11008,11010,11013],{"className":11009},[86,123],[35,11011,63],{"className":11012},[86,123],[35,11014,11016],{"className":11015},[1156],[35,11017,11019],{"className":11018},[98],[35,11020,11022],{"className":11021},[103],[35,11023,11026],{"className":11024,"style":11025},[107],"height:0.5935em;",[35,11027,11028,11031],{"style":7524},[35,11029],{"className":11030,"style":7528},[115],[35,11032,11034],{"className":11033},[120,7532,1078,123],[35,11035,2197],{"className":11036},[86,156,123],[35,11038,1028],{"className":11039},[90,123],[35,11041,404],{"className":11042},[86,123],[35,11044,2197],{"className":11045},[86,156,123],[35,11047,1316],{"className":11048},[2227,123],[35,11050,272],{"className":11051},[86,123],[35,11053,1037],{"className":11054},[175,123],[35,11056,11057,11060],{"style":132},[35,11058],{"className":11059,"style":116},[115],[35,11061],{"className":11062,"style":140},[139],[35,11064,11065,11068],{"style":10282},[35,11066],{"className":11067,"style":116},[115],[35,11069,11071],{"className":11070},[120,121,122,123],[35,11072,11074],{"className":11073},[86,123],[35,11075,11077,11086,11140],{"className":11076},[86,123],[35,11078,11080],{"className":11079,"style":10299},[90,120,7532,10298,123,1072],[35,11081,11083],{"className":11082},[1077,1078,123],[35,11084,1028],{"className":11085},[123],[35,11087,11089],{"className":11088},[54],[35,11090,11092,11132],{"className":11091},[98,99],[35,11093,11095,11129],{"className":11094},[103],[35,11096,11098,11112],{"className":11097,"style":10318},[107],[35,11099,11100,11103],{"style":10321},[35,11101],{"className":11102,"style":7528},[115],[35,11104,11106],{"className":11105},[120,7532,1078,123],[35,11107,11109],{"className":11108},[86,123],[35,11110,2197],{"className":11111},[86,156,123],[35,11113,11114,11117],{"style":10336},[35,11115],{"className":11116,"style":7528},[115],[35,11118,11120],{"className":11119},[120,7532,1078,123],[35,11121,11123,11126],{"className":11122},[86,123],[35,11124,404],{"className":11125},[86,123],[35,11127,2197],{"className":11128},[86,156,123],[35,11130,162],{"className":11131},[161],[35,11133,11135],{"className":11134},[103],[35,11136,11138],{"className":11137,"style":10361},[107],[35,11139],{},[35,11141,11143],{"className":11142,"style":10299},[175,120,7532,10298,123,1072],[35,11144,11146],{"className":11145},[1077,1078,123],[35,11147,1037],{"className":11148},[123],[35,11150,162],{"className":11151},[161],[35,11153,11155],{"className":11154},[103],[35,11156,11158],{"className":11157,"style":10820},[107],[35,11159],{},[35,11161],{"className":11162},[175,91],[17,11164,11165,11166,11169],{},"Ahora viene el momento que a Matt Parker le hace «enfadar de pura incredulidad». Resulta que existe una función clásica del análisis matemático llamada ",[239,11167,11168],{},"arcoseno"," — la función inversa del seno — cuya expansión en serie de potencias es:",[17,11171,11173],{"className":11172},[592],[35,11174,11176,11269],{"className":11175},[38],[35,11177,11179],{"className":11178},[42],[44,11180,11181],{"xmlns":46},[48,11182,11183,11266],{},[51,11184,11185,11188,11191,11193,11196,11198,11200,11214,11238,11240],{},[57,11186,11187],{},"arcsin",[276,11189,11190],{},"⁡",[276,11192,1028],{"stretchy":9816},[57,11194,11195],{},"x",[276,11197,1037],{"stretchy":9816},[276,11199,278],{},[7303,11201,11202,11204,11212],{},[276,11203,7307],{},[51,11205,11206,11208,11210],{},[57,11207,2197],{},[276,11209,278],{},[61,11211,517],{},[57,11213,7319],{"mathvariant":7318},[54,11215,11216,11232],{},[51,11217,11218,11220,11230],{},[276,11219,1028],{"fence":73},[54,11221,11222,11228],{"linethickness":4321},[51,11223,11224,11226],{},[61,11225,404],{},[57,11227,2197],{},[57,11229,2197],{},[276,11231,1037],{"fence":73},[1022,11233,11234,11236],{},[61,11235,63],{},[57,11237,2197],{},[276,11239,7344],{},[54,11241,11242,11256],{},[1022,11243,11244,11246],{},[57,11245,11195],{},[51,11247,11248,11250,11252,11254],{},[61,11249,404],{},[57,11251,2197],{},[276,11253,1316],{},[61,11255,272],{},[51,11257,11258,11260,11262,11264],{},[61,11259,404],{},[57,11261,2197],{},[276,11263,1316],{},[61,11265,272],{},[65,11267,11268],{"encoding":67},"\\arcsin(x) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n} \\cdot \\frac{x^{2n+1}}{2n+1}",[35,11270,11272,11299,11547],{"className":11271,"ariaHidden":73},[72],[35,11273,11275,11278,11281,11284,11287,11290,11293,11296],{"className":11274},[77],[35,11276],{"className":11277,"style":9837},[81],[35,11279,11187],{"className":11280},[7402],[35,11282,1028],{"className":11283},[90],[35,11285,11195],{"className":11286},[86,156],[35,11288,1037],{"className":11289},[175],[35,11291],{"className":11292,"style":367},[366],[35,11294,278],{"className":11295},[371],[35,11297],{"className":11298,"style":367},[366],[35,11300,11302,11306,11369,11372,11538,11541,11544],{"className":11301},[77],[35,11303],{"className":11304,"style":11305},[81],"height:1.855em;vertical-align:-0.345em;",[35,11307,11309,11312],{"className":11308},[7402],[35,11310,7307],{"className":11311,"style":7408},[7402,7406,7407],[35,11313,11315],{"className":11314},[1156],[35,11316,11318,11361],{"className":11317},[98,99],[35,11319,11321,11358],{"className":11320},[103],[35,11322,11324,11344],{"className":11323,"style":7421},[107],[35,11325,11326,11329],{"style":7424},[35,11327],{"className":11328,"style":1173},[115],[35,11330,11332],{"className":11331},[120,121,122,123],[35,11333,11335,11338,11341],{"className":11334},[86,123],[35,11336,2197],{"className":11337},[86,156,123],[35,11339,278],{"className":11340},[371,123],[35,11342,517],{"className":11343},[86,123],[35,11345,11346,11349],{"style":7445},[35,11347],{"className":11348,"style":1173},[115],[35,11350,11352],{"className":11351},[120,121,122,123],[35,11353,11355],{"className":11354},[86,123],[35,11356,7319],{"className":11357},[86,123],[35,11359,162],{"className":11360},[161],[35,11362,11364],{"className":11363},[103],[35,11365,11367],{"className":11366,"style":7467},[107],[35,11368],{},[35,11370],{"className":11371,"style":1452},[366],[35,11373,11375,11378,11535],{"className":11374},[86],[35,11376],{"className":11377},[90,91],[35,11379,11381],{"className":11380},[54],[35,11382,11384,11527],{"className":11383},[98,99],[35,11385,11387,11524],{"className":11386},[103],[35,11388,11390,11430,11438],{"className":11389,"style":10248},[107],[35,11391,11392,11395],{"style":111},[35,11393],{"className":11394,"style":116},[115],[35,11396,11398],{"className":11397},[120,121,122,123],[35,11399,11401],{"className":11400},[86,123],[35,11402,11404,11407],{"className":11403},[86,123],[35,11405,63],{"className":11406},[86,123],[35,11408,11410],{"className":11409},[1156],[35,11411,11413],{"className":11412},[98],[35,11414,11416],{"className":11415},[103],[35,11417,11419],{"className":11418,"style":11025},[107],[35,11420,11421,11424],{"style":7524},[35,11422],{"className":11423,"style":7528},[115],[35,11425,11427],{"className":11426},[120,7532,1078,123],[35,11428,2197],{"className":11429},[86,156,123],[35,11431,11432,11435],{"style":132},[35,11433],{"className":11434,"style":116},[115],[35,11436],{"className":11437,"style":140},[139],[35,11439,11440,11443],{"style":10282},[35,11441],{"className":11442,"style":116},[115],[35,11444,11446],{"className":11445},[120,121,122,123],[35,11447,11449],{"className":11448},[86,123],[35,11450,11452,11461,11515],{"className":11451},[86,123],[35,11453,11455],{"className":11454,"style":10299},[90,120,7532,10298,123,1072],[35,11456,11458],{"className":11457},[1077,1078,123],[35,11459,1028],{"className":11460},[123],[35,11462,11464],{"className":11463},[54],[35,11465,11467,11507],{"className":11466},[98,99],[35,11468,11470,11504],{"className":11469},[103],[35,11471,11473,11487],{"className":11472,"style":10318},[107],[35,11474,11475,11478],{"style":10321},[35,11476],{"className":11477,"style":7528},[115],[35,11479,11481],{"className":11480},[120,7532,1078,123],[35,11482,11484],{"className":11483},[86,123],[35,11485,2197],{"className":11486},[86,156,123],[35,11488,11489,11492],{"style":10336},[35,11490],{"className":11491,"style":7528},[115],[35,11493,11495],{"className":11494},[120,7532,1078,123],[35,11496,11498,11501],{"className":11497},[86,123],[35,11499,404],{"className":11500},[86,123],[35,11502,2197],{"className":11503},[86,156,123],[35,11505,162],{"className":11506},[161],[35,11508,11510],{"className":11509},[103],[35,11511,11513],{"className":11512,"style":10361},[107],[35,11514],{},[35,11516,11518],{"className":11517,"style":10299},[175,120,7532,10298,123,1072],[35,11519,11521],{"className":11520},[1077,1078,123],[35,11522,1037],{"className":11523},[123],[35,11525,162],{"className":11526},[161],[35,11528,11530],{"className":11529},[103],[35,11531,11533],{"className":11532,"style":169},[107],[35,11534],{},[35,11536],{"className":11537},[175,91],[35,11539],{"className":11540,"style":2223},[366],[35,11542,7344],{"className":11543},[2227],[35,11545],{"className":11546,"style":2223},[366],[35,11548,11550,11554],{"className":11549},[77],[35,11551],{"className":11552,"style":11553},[81],"height:1.4213em;vertical-align:-0.4033em;",[35,11555,11557,11560,11669],{"className":11556},[86],[35,11558],{"className":11559},[90,91],[35,11561,11563],{"className":11562},[54],[35,11564,11566,11661],{"className":11565},[98,99],[35,11567,11569,11658],{"className":11568},[103],[35,11570,11573,11596,11604],{"className":11571,"style":11572},[107],"height:1.0179em;",[35,11574,11575,11578],{"style":111},[35,11576],{"className":11577,"style":116},[115],[35,11579,11581],{"className":11580},[120,121,122,123],[35,11582,11584,11587,11590,11593],{"className":11583},[86,123],[35,11585,404],{"className":11586},[86,123],[35,11588,2197],{"className":11589},[86,156,123],[35,11591,1316],{"className":11592},[2227,123],[35,11594,272],{"className":11595},[86,123],[35,11597,11598,11601],{"style":132},[35,11599],{"className":11600,"style":116},[115],[35,11602],{"className":11603,"style":140},[139],[35,11605,11606,11609],{"style":143},[35,11607],{"className":11608,"style":116},[115],[35,11610,11612],{"className":11611},[120,121,122,123],[35,11613,11615],{"className":11614},[86,123],[35,11616,11618,11621],{"className":11617},[86,123],[35,11619,11195],{"className":11620},[86,156,123],[35,11622,11624],{"className":11623},[1156],[35,11625,11627],{"className":11626},[98],[35,11628,11630],{"className":11629},[103],[35,11631,11634],{"className":11632,"style":11633},[107],"height:0.8913em;",[35,11635,11637,11640],{"style":11636},"top:-2.931em;margin-right:0.0714em;",[35,11638],{"className":11639,"style":7528},[115],[35,11641,11643],{"className":11642},[120,7532,1078,123],[35,11644,11646,11649,11652,11655],{"className":11645},[86,123],[35,11647,404],{"className":11648},[86,123],[35,11650,2197],{"className":11651},[86,156,123],[35,11653,1316],{"className":11654},[2227,123],[35,11656,272],{"className":11657},[86,123],[35,11659,162],{"className":11660},[161],[35,11662,11664],{"className":11663},[103],[35,11665,11667],{"className":11666,"style":3736},[107],[35,11668],{},[35,11670],{"className":11671},[175,91],[17,11673,11674,11675,11703,11704,11732,11733,176],{},"¿Te parece familiar? Es exactamente nuestra serie, multiplicada por ",[35,11676,11678,11691],{"className":11677},[38],[35,11679,11681],{"className":11680},[42],[44,11682,11683],{"xmlns":46},[48,11684,11685,11689],{},[51,11686,11687],{},[61,11688,404],{},[65,11690,404],{"encoding":67},[35,11692,11694],{"className":11693,"ariaHidden":73},[72],[35,11695,11697,11700],{"className":11696},[77],[35,11698],{"className":11699,"style":381},[81],[35,11701,404],{"className":11702},[86]," y con ",[35,11705,11707,11720],{"className":11706},[38],[35,11708,11710],{"className":11709},[42],[44,11711,11712],{"xmlns":46},[48,11713,11714,11718],{},[51,11715,11716],{},[57,11717,11195],{},[65,11719,11195],{"encoding":67},[35,11721,11723],{"className":11722,"ariaHidden":73},[72],[35,11724,11726,11729],{"className":11725},[77],[35,11727],{"className":11728,"style":206},[81],[35,11730,11195],{"className":11731},[86,156]," en lugar de ",[35,11734,11736,11749],{"className":11735},[38],[35,11737,11739],{"className":11738},[42],[44,11740,11741],{"xmlns":46},[48,11742,11743,11747],{},[51,11744,11745],{},[61,11746,272],{},[65,11748,272],{"encoding":67},[35,11750,11752],{"className":11751,"ariaHidden":73},[72],[35,11753,11755,11758],{"className":11754},[77],[35,11756],{"className":11757,"style":381},[81],[35,11759,272],{"className":11760},[86],[17,11762,11763,11764,11815],{},"Si evaluamos en ",[35,11765,11767,11785],{"className":11766},[38],[35,11768,11770],{"className":11769},[42],[44,11771,11772],{"xmlns":46},[48,11773,11774,11782],{},[51,11775,11776,11778,11780],{},[57,11777,11195],{},[276,11779,278],{},[61,11781,272],{},[65,11783,11784],{"encoding":67},"x = 1",[35,11786,11788,11806],{"className":11787,"ariaHidden":73},[72],[35,11789,11791,11794,11797,11800,11803],{"className":11790},[77],[35,11792],{"className":11793,"style":206},[81],[35,11795,11195],{"className":11796},[86,156],[35,11798],{"className":11799,"style":367},[366],[35,11801,278],{"className":11802},[371],[35,11804],{"className":11805,"style":367},[366],[35,11807,11809,11812],{"className":11808},[77],[35,11810],{"className":11811,"style":381},[81],[35,11813,272],{"className":11814},[86],":",[17,11817,11819],{"className":11818},[592],[35,11820,11822,11898],{"className":11821},[38],[35,11823,11825],{"className":11824},[42],[44,11826,11827],{"xmlns":46},[48,11828,11829,11895],{},[51,11830,11831,11833,11835,11837,11839,11841,11843,11857],{},[57,11832,11187],{},[276,11834,11190],{},[276,11836,1028],{"stretchy":9816},[61,11838,272],{},[276,11840,1037],{"stretchy":9816},[276,11842,278],{},[7303,11844,11845,11847,11855],{},[276,11846,7307],{},[51,11848,11849,11851,11853],{},[57,11850,2197],{},[276,11852,278],{},[61,11854,517],{},[57,11856,7319],{"mathvariant":7318},[54,11858,11859,11875],{},[51,11860,11861,11863,11873],{},[276,11862,1028],{"fence":73},[54,11864,11865,11871],{"linethickness":4321},[51,11866,11867,11869],{},[61,11868,404],{},[57,11870,2197],{},[57,11872,2197],{},[276,11874,1037],{"fence":73},[51,11876,11877,11883,11885,11887,11889,11891,11893],{},[1022,11878,11879,11881],{},[61,11880,63],{},[57,11882,2197],{},[276,11884,1028],{"stretchy":9816},[61,11886,404],{},[57,11888,2197],{},[276,11890,1316],{},[61,11892,272],{},[276,11894,1037],{"stretchy":9816},[65,11896,11897],{"encoding":67},"\\arcsin(1) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n(2n+1)}",[35,11899,11901,11928],{"className":11900,"ariaHidden":73},[72],[35,11902,11904,11907,11910,11913,11916,11919,11922,11925],{"className":11903},[77],[35,11905],{"className":11906,"style":9837},[81],[35,11908,11187],{"className":11909},[7402],[35,11911,1028],{"className":11912},[90],[35,11914,272],{"className":11915},[86],[35,11917,1037],{"className":11918},[175],[35,11920],{"className":11921,"style":367},[366],[35,11923,278],{"className":11924},[371],[35,11926],{"className":11927,"style":367},[366],[35,11929,11931,11934,11997,12000],{"className":11930},[77],[35,11932],{"className":11933,"style":10562},[81],[35,11935,11937,11940],{"className":11936},[7402],[35,11938,7307],{"className":11939,"style":7408},[7402,7406,7407],[35,11941,11943],{"className":11942},[1156],[35,11944,11946,11989],{"className":11945},[98,99],[35,11947,11949,11986],{"className":11948},[103],[35,11950,11952,11972],{"className":11951,"style":7421},[107],[35,11953,11954,11957],{"style":7424},[35,11955],{"className":11956,"style":1173},[115],[35,11958,11960],{"className":11959},[120,121,122,123],[35,11961,11963,11966,11969],{"className":11962},[86,123],[35,11964,2197],{"className":11965},[86,156,123],[35,11967,278],{"className":11968},[371,123],[35,11970,517],{"className":11971},[86,123],[35,11973,11974,11977],{"style":7445},[35,11975],{"className":11976,"style":1173},[115],[35,11978,11980],{"className":11979},[120,121,122,123],[35,11981,11983],{"className":11982},[86,123],[35,11984,7319],{"className":11985},[86,123],[35,11987,162],{"className":11988},[161],[35,11990,11992],{"className":11991},[103],[35,11993,11995],{"className":11994,"style":7467},[107],[35,11996],{},[35,11998],{"className":11999,"style":1452},[366],[35,12001,12003,12006,12181],{"className":12002},[86],[35,12004],{"className":12005},[90,91],[35,12007,12009],{"className":12008},[54],[35,12010,12012,12173],{"className":12011},[98,99],[35,12013,12015,12170],{"className":12014},[103],[35,12016,12018,12076,12084],{"className":12017,"style":10248},[107],[35,12019,12020,12023],{"style":111},[35,12021],{"className":12022,"style":116},[115],[35,12024,12026],{"className":12025},[120,121,122,123],[35,12027,12029,12058,12061,12064,12067,12070,12073],{"className":12028},[86,123],[35,12030,12032,12035],{"className":12031},[86,123],[35,12033,63],{"className":12034},[86,123],[35,12036,12038],{"className":12037},[1156],[35,12039,12041],{"className":12040},[98],[35,12042,12044],{"className":12043},[103],[35,12045,12047],{"className":12046,"style":11025},[107],[35,12048,12049,12052],{"style":7524},[35,12050],{"className":12051,"style":7528},[115],[35,12053,12055],{"className":12054},[120,7532,1078,123],[35,12056,2197],{"className":12057},[86,156,123],[35,12059,1028],{"className":12060},[90,123],[35,12062,404],{"className":12063},[86,123],[35,12065,2197],{"className":12066},[86,156,123],[35,12068,1316],{"className":12069},[2227,123],[35,12071,272],{"className":12072},[86,123],[35,12074,1037],{"className":12075},[175,123],[35,12077,12078,12081],{"style":132},[35,12079],{"className":12080,"style":116},[115],[35,12082],{"className":12083,"style":140},[139],[35,12085,12086,12089],{"style":10282},[35,12087],{"className":12088,"style":116},[115],[35,12090,12092],{"className":12091},[120,121,122,123],[35,12093,12095],{"className":12094},[86,123],[35,12096,12098,12107,12161],{"className":12097},[86,123],[35,12099,12101],{"className":12100,"style":10299},[90,120,7532,10298,123,1072],[35,12102,12104],{"className":12103},[1077,1078,123],[35,12105,1028],{"className":12106},[123],[35,12108,12110],{"className":12109},[54],[35,12111,12113,12153],{"className":12112},[98,99],[35,12114,12116,12150],{"className":12115},[103],[35,12117,12119,12133],{"className":12118,"style":10318},[107],[35,12120,12121,12124],{"style":10321},[35,12122],{"className":12123,"style":7528},[115],[35,12125,12127],{"className":12126},[120,7532,1078,123],[35,12128,12130],{"className":12129},[86,123],[35,12131,2197],{"className":12132},[86,156,123],[35,12134,12135,12138],{"style":10336},[35,12136],{"className":12137,"style":7528},[115],[35,12139,12141],{"className":12140},[120,7532,1078,123],[35,12142,12144,12147],{"className":12143},[86,123],[35,12145,404],{"className":12146},[86,123],[35,12148,2197],{"className":12149},[86,156,123],[35,12151,162],{"className":12152},[161],[35,12154,12156],{"className":12155},[103],[35,12157,12159],{"className":12158,"style":10361},[107],[35,12160],{},[35,12162,12164],{"className":12163,"style":10299},[175,120,7532,10298,123,1072],[35,12165,12167],{"className":12166},[1077,1078,123],[35,12168,1037],{"className":12169},[123],[35,12171,162],{"className":12172},[161],[35,12174,12176],{"className":12175},[103],[35,12177,12179],{"className":12178,"style":10820},[107],[35,12180],{},[35,12182],{"className":12183},[175,91],[17,12185,12186,12187,12233,12234,12262,12263,11815],{},"¿Y cuánto vale ",[35,12188,12190,12212],{"className":12189},[38],[35,12191,12193],{"className":12192},[42],[44,12194,12195],{"xmlns":46},[48,12196,12197,12209],{},[51,12198,12199,12201,12203,12205,12207],{},[57,12200,11187],{},[276,12202,11190],{},[276,12204,1028],{"stretchy":9816},[61,12206,272],{},[276,12208,1037],{"stretchy":9816},[65,12210,12211],{"encoding":67},"\\arcsin(1)",[35,12213,12215],{"className":12214,"ariaHidden":73},[72],[35,12216,12218,12221,12224,12227,12230],{"className":12217},[77],[35,12219],{"className":12220,"style":9837},[81],[35,12222,11187],{"className":12223},[7402],[35,12225,1028],{"className":12226},[90],[35,12228,272],{"className":12229},[86],[35,12231,1037],{"className":12232},[175],"? Es el ángulo cuyo seno es ",[35,12235,12237,12250],{"className":12236},[38],[35,12238,12240],{"className":12239},[42],[44,12241,12242],{"xmlns":46},[48,12243,12244,12248],{},[51,12245,12246],{},[61,12247,272],{},[65,12249,272],{"encoding":67},[35,12251,12253],{"className":12252,"ariaHidden":73},[72],[35,12254,12256,12259],{"className":12255},[77],[35,12257],{"className":12258,"style":381},[81],[35,12260,272],{"className":12261},[86],", es decir, ",[35,12264,12266,12284],{"className":12265},[38],[35,12267,12269],{"className":12268},[42],[44,12270,12271],{"xmlns":46},[48,12272,12273,12281],{},[51,12274,12275,12278],{},[61,12276,12277],{},"90",[57,12279,12280],{"mathvariant":7318},"°",[65,12282,12283],{"encoding":67},"90°",[35,12285,12287],{"className":12286,"ariaHidden":73},[72],[35,12288,12290,12293],{"className":12289},[77],[35,12291],{"className":12292,"style":7026},[81],[35,12294,12283],{"className":12295},[86],[17,12297,12299],{"className":12298},[592],[35,12300,12302,12332],{"className":12301},[38],[35,12303,12305],{"className":12304},[42],[44,12306,12307],{"xmlns":46},[48,12308,12309,12329],{},[51,12310,12311,12313,12315,12317,12319,12321,12323],{},[57,12312,11187],{},[276,12314,11190],{},[276,12316,1028],{"stretchy":9816},[61,12318,272],{},[276,12320,1037],{"stretchy":9816},[276,12322,278],{},[54,12324,12325,12327],{},[57,12326,59],{},[61,12328,404],{},[65,12330,12331],{"encoding":67},"\\arcsin(1) = \\frac{\\pi}{2}",[35,12333,12335,12362],{"className":12334,"ariaHidden":73},[72],[35,12336,12338,12341,12344,12347,12350,12353,12356,12359],{"className":12337},[77],[35,12339],{"className":12340,"style":9837},[81],[35,12342,11187],{"className":12343},[7402],[35,12345,1028],{"className":12346},[90],[35,12348,272],{"className":12349},[86],[35,12351,1037],{"className":12352},[175],[35,12354],{"className":12355,"style":367},[366],[35,12357,278],{"className":12358},[371],[35,12360],{"className":12361,"style":367},[366],[35,12363,12365,12368],{"className":12364},[77],[35,12366],{"className":12367,"style":82},[81],[35,12369,12371,12374,12433],{"className":12370},[86],[35,12372],{"className":12373},[90,91],[35,12375,12377],{"className":12376},[54],[35,12378,12380,12425],{"className":12379},[98,99],[35,12381,12383,12422],{"className":12382},[103],[35,12384,12386,12400,12408],{"className":12385,"style":108},[107],[35,12387,12388,12391],{"style":111},[35,12389],{"className":12390,"style":116},[115],[35,12392,12394],{"className":12393},[120,121,122,123],[35,12395,12397],{"className":12396},[86,123],[35,12398,404],{"className":12399},[86,123],[35,12401,12402,12405],{"style":132},[35,12403],{"className":12404,"style":116},[115],[35,12406],{"className":12407,"style":140},[139],[35,12409,12410,12413],{"style":143},[35,12411],{"className":12412,"style":116},[115],[35,12414,12416],{"className":12415},[120,121,122,123],[35,12417,12419],{"className":12418},[86,123],[35,12420,59],{"className":12421,"style":157},[86,156,123],[35,12423,162],{"className":12424},[161],[35,12426,12428],{"className":12427},[103],[35,12429,12431],{"className":12430,"style":169},[107],[35,12432],{},[35,12434],{"className":12435},[175,91],[17,12437,12438],{},"Por lo tanto:",[17,12440,12442],{"className":12441},[592],[35,12443,12445,12483],{"className":12444},[38],[35,12446,12448],{"className":12447},[42],[44,12449,12450],{"xmlns":46},[48,12451,12452,12480],{},[51,12453,12454,12456,12458,12464,12466,12472,12474],{},[606,12455,7299],{},[276,12457,278],{},[54,12459,12460,12462],{},[61,12461,272],{},[61,12463,404],{},[276,12465,7344],{},[54,12467,12468,12470],{},[57,12469,59],{},[61,12471,404],{},[276,12473,278],{},[54,12475,12476,12478],{},[57,12477,59],{},[61,12479,63],{},[65,12481,12482],{"encoding":67},"\\text{Media} = \\frac{1}{2} \\cdot \\frac{\\pi}{2} = \\frac{\\pi}{4}",[35,12484,12486,12507,12590,12673],{"className":12485,"ariaHidden":73},[72],[35,12487,12489,12492,12498,12501,12504],{"className":12488},[77],[35,12490],{"className":12491,"style":7026},[81],[35,12493,12495],{"className":12494},[86,625],[35,12496,7299],{"className":12497},[86],[35,12499],{"className":12500,"style":367},[366],[35,12502,278],{"className":12503},[371],[35,12505],{"className":12506,"style":367},[366],[35,12508,12510,12513,12581,12584,12587],{"className":12509},[77],[35,12511],{"className":12512,"style":293},[81],[35,12514,12516,12519,12578],{"className":12515},[86],[35,12517],{"className":12518},[90,91],[35,12520,12522],{"className":12521},[54],[35,12523,12525,12570],{"className":12524},[98,99],[35,12526,12528,12567],{"className":12527},[103],[35,12529,12531,12545,12553],{"className":12530,"style":312},[107],[35,12532,12533,12536],{"style":111},[35,12534],{"className":12535,"style":116},[115],[35,12537,12539],{"className":12538},[120,121,122,123],[35,12540,12542],{"className":12541},[86,123],[35,12543,404],{"className":12544},[86,123],[35,12546,12547,12550],{"style":132},[35,12548],{"className":12549,"style":116},[115],[35,12551],{"className":12552,"style":140},[139],[35,12554,12555,12558],{"style":143},[35,12556],{"className":12557,"style":116},[115],[35,12559,12561],{"className":12560},[120,121,122,123],[35,12562,12564],{"className":12563},[86,123],[35,12565,272],{"className":12566},[86,123],[35,12568,162],{"className":12569},[161],[35,12571,12573],{"className":12572},[103],[35,12574,12576],{"className":12575,"style":169},[107],[35,12577],{},[35,12579],{"className":12580},[175,91],[35,12582],{"className":12583,"style":2223},[366],[35,12585,7344],{"className":12586},[2227],[35,12588],{"className":12589,"style":2223},[366],[35,12591,12593,12596,12664,12667,12670],{"className":12592},[77],[35,12594],{"className":12595,"style":82},[81],[35,12597,12599,12602,12661],{"className":12598},[86],[35,12600],{"className":12601},[90,91],[35,12603,12605],{"className":12604},[54],[35,12606,12608,12653],{"className":12607},[98,99],[35,12609,12611,12650],{"className":12610},[103],[35,12612,12614,12628,12636],{"className":12613,"style":108},[107],[35,12615,12616,12619],{"style":111},[35,12617],{"className":12618,"style":116},[115],[35,12620,12622],{"className":12621},[120,121,122,123],[35,12623,12625],{"className":12624},[86,123],[35,12626,404],{"className":12627},[86,123],[35,12629,12630,12633],{"style":132},[35,12631],{"className":12632,"style":116},[115],[35,12634],{"className":12635,"style":140},[139],[35,12637,12638,12641],{"style":143},[35,12639],{"className":12640,"style":116},[115],[35,12642,12644],{"className":12643},[120,121,122,123],[35,12645,12647],{"className":12646},[86,123],[35,12648,59],{"className":12649,"style":157},[86,156,123],[35,12651,162],{"className":12652},[161],[35,12654,12656],{"className":12655},[103],[35,12657,12659],{"className":12658,"style":169},[107],[35,12660],{},[35,12662],{"className":12663},[175,91],[35,12665],{"className":12666,"style":367},[366],[35,12668,278],{"className":12669},[371],[35,12671],{"className":12672,"style":367},[366],[35,12674,12676,12679],{"className":12675},[77],[35,12677],{"className":12678,"style":82},[81],[35,12680,12682,12685,12744],{"className":12681},[86],[35,12683],{"className":12684},[90,91],[35,12686,12688],{"className":12687},[54],[35,12689,12691,12736],{"className":12690},[98,99],[35,12692,12694,12733],{"className":12693},[103],[35,12695,12697,12711,12719],{"className":12696,"style":108},[107],[35,12698,12699,12702],{"style":111},[35,12700],{"className":12701,"style":116},[115],[35,12703,12705],{"className":12704},[120,121,122,123],[35,12706,12708],{"className":12707},[86,123],[35,12709,63],{"className":12710},[86,123],[35,12712,12713,12716],{"style":132},[35,12714],{"className":12715,"style":116},[115],[35,12717],{"className":12718,"style":140},[139],[35,12720,12721,12724],{"style":143},[35,12722],{"className":12723,"style":116},[115],[35,12725,12727],{"className":12726},[120,121,122,123],[35,12728,12730],{"className":12729},[86,123],[35,12731,59],{"className":12732,"style":157},[86,156,123],[35,12734,162],{"className":12735},[161],[35,12737,12739],{"className":12738},[103],[35,12740,12742],{"className":12741,"style":169},[107],[35,12743],{},[35,12745],{"className":12746},[175,91],[17,12748,12749,12750,12758,12759,12787,12788,12791],{},"Esta conexión fue ",[21,12751,12754,12755],{"href":12752,"rel":12753},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.14487",[25],"descubierta en 2025 por el matemático ",[239,12756,12757],{},"Jim Propp",". Y lo que la hace extraordinaria no es que ",[35,12760,12762,12775],{"className":12761},[38],[35,12763,12765],{"className":12764},[42],[44,12766,12767],{"xmlns":46},[48,12768,12769,12773],{},[51,12770,12771],{},[57,12772,59],{},[65,12774,196],{"encoding":67},[35,12776,12778],{"className":12777,"ariaHidden":73},[72],[35,12779,12781,12784],{"className":12780},[77],[35,12782],{"className":12783,"style":206},[81],[35,12785,59],{"className":12786,"style":157},[86,156]," aparezca en una serie infinita — eso es habitual. Lo extraordinario es que una serie nacida de un proceso tan elemental como lanzar monedas resulte ser ",[212,12789,12790],{},"exactamente"," la misma serie que define una función trigonométrica.",[17,12793,12794],{},"No es una aproximación ni una coincidencia numérica — es una identidad matemática exacta.",[17,12796,12797,12798,12826,12827,11815],{},"Eso sí, no intentes calcular ",[35,12799,12801,12814],{"className":12800},[38],[35,12802,12804],{"className":12803},[42],[44,12805,12806],{"xmlns":46},[48,12807,12808,12812],{},[51,12809,12810],{},[57,12811,59],{},[65,12813,196],{"encoding":67},[35,12815,12817],{"className":12816,"ariaHidden":73},[72],[35,12818,12820,12823],{"className":12819},[77],[35,12821],{"className":12822,"style":206},[81],[35,12824,59],{"className":12825,"style":157},[86,156]," de esta forma. Como el propio Propp ",[21,12828,12830],{"href":12752,"rel":12829},[25],"explicó en el propio documento de Arxiv",[12832,12833,12834],"blockquote",{},[17,12835,12836],{},"Para obtener pi con la precisión de 3,14, podrían hacer falta hasta un billón de lanzamientos de moneda. En parte porque las secuencias pueden alargarse mucho antes de que las caras superen a las cruces — tanto que el valor esperado de la longitud de una secuencia es ¡infinito! Además, no se pueden lanzar todas las monedas a la vez como se pueden soltar agujas: el orden de caras y cruces importa.",[561,12838,12840],{"id":12839},"reflexión-final","Reflexión final",[17,12842,12843],{},"En el artículo anterior, descubrimos que el triángulo de Tartaglia conectaba la combinatoria, el álgebra y la probabilidad. Tres lenguajes que contaban la misma historia.",[17,12845,12846,12847,176],{},"Hoy hemos añadido un cuarto personaje a la trama: el número ",[35,12848,12850,12863],{"className":12849},[38],[35,12851,12853],{"className":12852},[42],[44,12854,12855],{"xmlns":46},[48,12856,12857,12861],{},[51,12858,12859],{},[57,12860,59],{},[65,12862,196],{"encoding":67},[35,12864,12866],{"className":12865,"ariaHidden":73},[72],[35,12867,12869,12872],{"className":12868},[77],[35,12870],{"className":12871,"style":206},[81],[35,12873,59],{"className":12874,"style":157},[86,156],[17,12876,12877,12878,12906],{},"Lanzar una moneda parece un proceso puramente aleatorio. El triángulo de Tartaglia parece pura aritmética. Los números de Catalan parecen un capricho de la combinatoria. Y ",[35,12879,12881,12894],{"className":12880},[38],[35,12882,12884],{"className":12883},[42],[44,12885,12886],{"xmlns":46},[48,12887,12888,12892],{},[51,12889,12890],{},[57,12891,59],{},[65,12893,196],{"encoding":67},[35,12895,12897],{"className":12896,"ariaHidden":73},[72],[35,12898,12900,12903],{"className":12899},[77],[35,12901],{"className":12902,"style":206},[81],[35,12904,59],{"className":12905,"style":157},[86,156]," parece pertenecer exclusivamente a la geometría — a los círculos, a las curvas, al mundo continuo.",[17,12908,12909],{},"Y sin embargo, todos terminan conectados en una sola ecuación.",[17,12911,12912,12913,12941],{},"Hay algo profundamente satisfactorio en descubrir que un hecho nuevo sobre ",[35,12914,12916,12929],{"className":12915},[38],[35,12917,12919],{"className":12918},[42],[44,12920,12921],{"xmlns":46},[48,12922,12923,12927],{},[51,12924,12925],{},[57,12926,59],{},[65,12928,196],{"encoding":67},[35,12930,12932],{"className":12931,"ariaHidden":73},[72],[35,12933,12935,12938],{"className":12934},[77],[35,12936],{"className":12937,"style":206},[81],[35,12939,59],{"className":12940,"style":157},[86,156]," se escondía, desde siempre, detrás de algo tan mundano como lanzar una moneda. Y que nadie lo notó hasta 2025.",[17,12943,12944],{},"Las matemáticas siguen teniendo secretos. Y a veces, están más cerca de lo que pensamos.",[17,12946,12947,12948,12952],{},"Pon la teoría en práctica: ",[21,12949,12951],{"href":578,"rel":12950},[25],"lanza una moneda online"," y observa los patrones en acción.",{"title":1291,"searchDepth":12954,"depth":12954,"links":12955},2,[12956,12964,12965,12966,12967,12968],{"id":563,"depth":12954,"text":564,"children":12957},[12958,12960,12961,12962,12963],{"id":584,"depth":12959,"text":585},3,{"id":852,"depth":12959,"text":853},{"id":1266,"depth":12959,"text":1267},{"id":2154,"depth":12959,"text":2155},{"id":2165,"depth":12959,"text":2166},{"id":4074,"depth":12954,"text":4075},{"id":6153,"depth":12954,"text":6154},{"id":6359,"depth":12954,"text":6360},{"id":9336,"depth":12954,"text":9337},{"id":12839,"depth":12954,"text":12840},"2026-05-27T10:00:00Z","Descubre cómo lanzar una moneda hasta que haya más caras que cruces conduce, inesperadamente, al número π — a través de los números de Catalan y la serie del arcoseno.","md",{"excerpt":12973},{"type":10,"value":12974},[12975,12977,13082,13114],[13,12976,7],{"id":15},[17,12978,19,12979,27,12982,33,12985,176],{},[21,12980,26],{"href":23,"rel":12981},[25],[21,12983,32],{"href":30,"rel":12984},[25],[35,12986,12988,13005],{"className":12987},[38],[35,12989,12991],{"className":12990},[42],[44,12992,12993],{"xmlns":46},[48,12994,12995,13003],{},[51,12996,12997],{},[54,12998,12999,13001],{},[57,13000,59],{},[61,13002,63],{},[65,13004,68],{"encoding":67},[35,13006,13008],{"className":13007,"ariaHidden":73},[72],[35,13009,13011,13014],{"className":13010},[77],[35,13012],{"className":13013,"style":82},[81],[35,13015,13017,13020,13079],{"className":13016},[86],[35,13018],{"className":13019},[90,91],[35,13021,13023],{"className":13022},[54],[35,13024,13026,13071],{"className":13025},[98,99],[35,13027,13029,13068],{"className":13028},[103],[35,13030,13032,13046,13054],{"className":13031,"style":108},[107],[35,13033,13034,13037],{"style":111},[35,13035],{"className":13036,"style":116},[115],[35,13038,13040],{"className":13039},[120,121,122,123],[35,13041,13043],{"className":13042},[86,123],[35,13044,63],{"className":13045},[86,123],[35,13047,13048,13051],{"style":132},[35,13049],{"className":13050,"style":116},[115],[35,13052],{"className":13053,"style":140},[139],[35,13055,13056,13059],{"style":143},[35,13057],{"className":13058,"style":116},[115],[35,13060,13062],{"className":13061},[120,121,122,123],[35,13063,13065],{"className":13064},[86,123],[35,13066,59],{"className":13067,"style":157},[86,156,123],[35,13069,162],{"className":13070},[161],[35,13072,13074],{"className":13073},[103],[35,13075,13077],{"className":13076,"style":169},[107],[35,13078],{},[35,13080],{"className":13081},[175,91],[17,13083,179,13084,210,13112,215],{},[35,13085,13087,13100],{"className":13086},[38],[35,13088,13090],{"className":13089},[42],[44,13091,13092],{"xmlns":46},[48,13093,13094,13098],{},[51,13095,13096],{},[57,13097,59],{},[65,13099,196],{"encoding":67},[35,13101,13103],{"className":13102,"ariaHidden":73},[72],[35,13104,13106,13109],{"className":13105},[77],[35,13107],{"className":13108,"style":206},[81],[35,13110,59],{"className":13111,"style":157},[86,156],[212,13113,214],{},[17,13115,218,13116,224],{},[21,13117,223],{"href":221,"rel":13118},[25],true,"\u002Fblog\u002Fpi-coin-flips-catalan-numbers.es",{"title":7,"description":12970},"lanzando-una-moneda-hasta-obtener-pi","blog\u002Fpi-coin-flips-catalan-numbers.es","tftvlIus-Vzedr65fsds1GlLWOwi3lEU2RboB8VhTN4",{"id":13126,"title":13127,"_locale":8,"body":13128,"date":23664,"description":23665,"extension":12971,"meta":23666,"navigation":13119,"path":23673,"published":13119,"seo":23674,"slug":23675,"stem":23676,"__hash__":23677},"blog\u002Fblog\u002Fpascals-triangle-and-probability.es.md","¿Qué tienen en común el triángulo de Tartaglia y una moneda?",{"type":10,"value":13129,"toc":23649},[13130,13133,13136,13140,13155,13158,13164,13168,13171,13192,13195,13201,13211,13457,13461,13467,13574,13693,13849,13852,14098,14400,14843,14875,15274,15278,15288,15497,15503,15567,15627,15759,15762,15770,15774,15860,15866,15873,15926,16002,16220,16224,16229,16364,16396,16822,16825,17532,17718,17721,17992,17996,17999,18083,18167,18299,18305,18437,18440,19231,19234,19240,19243,19263,19271,19275,19282,19377,19380,19574,19606,19611,19841,19900,19905,20227,20297,20302,20808,20905,20915,21281,21517,21521,21585,21897,21900,22867,23242,23245,23511,23606,23608,23611,23628,23634,23637,23640,23643],[13,13131,13127],{"id":13132},"qué-tienen-en-común-el-triángulo-de-tartaglia-y-una-moneda",[17,13134,13135],{},"Existe un triángulo de números que lleva siglos fascinando a matemáticos y matemáticas de todo el mundo. Su construcción es tan sencilla que un niño o niña podría dibujarlo en un cuaderno. Y sin embargo, dentro de sus filas se esconde una conexión profunda entre tres ramas de las matemáticas que, a primera vista, no parecen tener nada que ver entre sí: la combinatoria, el álgebra y la probabilidad. Lo más sorprendente es que esa conexión se revela con algo tan cotidiano como lanzar una moneda.",[561,13137,13139],{"id":13138},"un-poco-de-historia","Un poco de historia",[17,13141,13142,13143,13146,13147,13150,13151,13154],{},"En muchos países de habla hispana este patrón se conoce como ",[239,13144,13145],{},"triángulo de Tartaglia",", en honor al matemático italiano Niccolò Fontana Tartaglia (1499–1557), que lo estudió en el contexto de la resolución de ecuaciones cúbicas. En el mundo anglosajón se le llama ",[239,13148,13149],{},"triángulo de Pascal",", porque Blaise Pascal publicó en 1654 un tratado — el ",[212,13152,13153],{},"Traité du triangle arithmétique"," — donde formalizó sus propiedades y, lo que es más importante, lo conectó por primera vez con los problemas de azar que le planteaba el caballero de Méré.",[17,13156,13157],{},"Pero la historia real de este triángulo es mucho más antigua. En la China del siglo XI, el matemático Jia Xian ya describía el mismo patrón para calcular coeficientes de potencias. Más tarde, en el siglo XIII, Yang Hui lo reprodujo en un diagrama que ha llegado hasta nuestros días. En Persia, Omar Jayyam trabajó con las mismas ideas en el siglo XII. Y en la India, Pingala exploró patrones combinatorios similares siglos antes de nuestra era.",[17,13159,13160,13161],{},"Que civilizaciones tan distantes en el tiempo y el espacio descubrieran el mismo triángulo de forma independiente nos dice algo importante: no se trata de una invención humana arbitraria, sino de una estructura que emerge de forma natural cada vez que alguien se pregunta ",[212,13162,13163],{},"¿de cuántas maneras puedo organizar esto?",[561,13165,13167],{"id":13166},"cómo-construir-el-triángulo","Cómo construir el triángulo",[17,13169,13170],{},"La regla de construcción es elegantemente simple:",[229,13172,13173,13179,13185],{},[232,13174,13175,13176,13178],{},"Empieza colocando un ",[239,13177,272],{}," en la cima.",[232,13180,13181,13182,13184],{},"En cada fila nueva, coloca un ",[239,13183,272],{}," en cada extremo.",[232,13186,13187,13188,13191],{},"Cada número interior es la ",[239,13189,13190],{},"suma de los dos números"," que tiene justo encima.",[17,13193,13194],{},"Las primeras filas quedan así:",[1286,13196,13199],{"className":13197,"code":13198,"language":625,"meta":1291},[1289],"            1\n          1   1\n        1   2   1\n      1   3   3   1\n    1   4   6   4   1\n  1   5  10  10   5   1\n",[1293,13200,13198],{"__ignoreMap":1291},[17,13202,13203,13204,13206,13207,13210],{},"La primera fila (el ",[239,13205,272],{}," solitario de la cima) se llama ",[239,13208,13209],{},"fila 0",". La siguiente es la fila 1, y así sucesivamente. Esto no es un capricho: numerar desde cero hará que todo encaje mejor cuando conectemos el triángulo con las fórmulas.",[17,13212,13213,13214,13295,13296,13324,13325,13295,13376,13404,13405,13456],{},"Fíjate en la fila 4: ",[35,13215,13217,13247],{"className":13216},[38],[35,13218,13220],{"className":13219},[42],[44,13221,13222],{"xmlns":46},[48,13223,13224,13244],{},[51,13225,13226,13228,13230,13232,13234,13236,13238,13240,13242],{},[61,13227,272],{},[276,13229,525],{"separator":73},[61,13231,63],{},[276,13233,525],{"separator":73},[61,13235,5442],{},[276,13237,525],{"separator":73},[61,13239,63],{},[276,13241,525],{"separator":73},[61,13243,272],{},[65,13245,13246],{"encoding":67},"1, 4, 6, 4, 1",[35,13248,13250],{"className":13249,"ariaHidden":73},[72],[35,13251,13253,13256,13259,13262,13265,13268,13271,13274,13277,13280,13283,13286,13289,13292],{"className":13252},[77],[35,13254],{"className":13255,"style":513},[81],[35,13257,272],{"className":13258},[86],[35,13260,525],{"className":13261},[524],[35,13263],{"className":13264,"style":1452},[366],[35,13266,63],{"className":13267},[86],[35,13269,525],{"className":13270},[524],[35,13272],{"className":13273,"style":1452},[366],[35,13275,5442],{"className":13276},[86],[35,13278,525],{"className":13279},[524],[35,13281],{"className":13282,"style":1452},[366],[35,13284,63],{"className":13285},[86],[35,13287,525],{"className":13288},[524],[35,13290],{"className":13291,"style":1452},[366],[35,13293,272],{"className":13294},[86],". El ",[35,13297,13299,13312],{"className":13298},[38],[35,13300,13302],{"className":13301},[42],[44,13303,13304],{"xmlns":46},[48,13305,13306,13310],{},[51,13307,13308],{},[61,13309,5442],{},[65,13311,5442],{"encoding":67},[35,13313,13315],{"className":13314,"ariaHidden":73},[72],[35,13316,13318,13321],{"className":13317},[77],[35,13319],{"className":13320,"style":381},[81],[35,13322,5442],{"className":13323},[86]," del centro viene de sumar los dos números que tiene encima: ",[35,13326,13328,13346],{"className":13327},[38],[35,13329,13331],{"className":13330},[42],[44,13332,13333],{"xmlns":46},[48,13334,13335,13343],{},[51,13336,13337,13339,13341],{},[61,13338,927],{},[276,13340,1316],{},[61,13342,927],{},[65,13344,13345],{"encoding":67},"3 + 3",[35,13347,13349,13367],{"className":13348,"ariaHidden":73},[72],[35,13350,13352,13355,13358,13361,13364],{"className":13351},[77],[35,13353],{"className":13354,"style":1331},[81],[35,13356,927],{"className":13357},[86],[35,13359],{"className":13360,"style":2223},[366],[35,13362,1316],{"className":13363},[2227],[35,13365],{"className":13366,"style":2223},[366],[35,13368,13370,13373],{"className":13369},[77],[35,13371],{"className":13372,"style":381},[81],[35,13374,927],{"className":13375},[86],[35,13377,13379,13392],{"className":13378},[38],[35,13380,13382],{"className":13381},[42],[44,13383,13384],{"xmlns":46},[48,13385,13386,13390],{},[51,13387,13388],{},[61,13389,63],{},[65,13391,63],{"encoding":67},[35,13393,13395],{"className":13394,"ariaHidden":73},[72],[35,13396,13398,13401],{"className":13397},[77],[35,13399],{"className":13400,"style":381},[81],[35,13402,63],{"className":13403},[86]," de la izquierda viene de ",[35,13406,13408,13426],{"className":13407},[38],[35,13409,13411],{"className":13410},[42],[44,13412,13413],{"xmlns":46},[48,13414,13415,13423],{},[51,13416,13417,13419,13421],{},[61,13418,272],{},[276,13420,1316],{},[61,13422,927],{},[65,13424,13425],{"encoding":67},"1 + 3",[35,13427,13429,13447],{"className":13428,"ariaHidden":73},[72],[35,13430,13432,13435,13438,13441,13444],{"className":13431},[77],[35,13433],{"className":13434,"style":1331},[81],[35,13436,272],{"className":13437},[86],[35,13439],{"className":13440,"style":2223},[366],[35,13442,1316],{"className":13443},[2227],[35,13445],{"className":13446,"style":2223},[366],[35,13448,13450,13453],{"className":13449},[77],[35,13451],{"className":13452,"style":381},[81],[35,13454,927],{"className":13455},[86],". Siempre la misma regla.",[582,13458,13460],{"id":13459},"los-coeficientes-binomiales","Los coeficientes binomiales",[17,13462,13463,13464,13466],{},"Cada número del triángulo tiene un nombre formal: es un ",[239,13465,4729],{},". Se escribe así:",[17,13468,13470],{"className":13469},[592],[35,13471,13473,13496],{"className":13472},[38],[35,13474,13476],{"className":13475},[42],[44,13477,13478],{"xmlns":46},[48,13479,13480,13493],{},[51,13481,13482,13484,13491],{},[276,13483,1028],{"fence":73},[54,13485,13486,13488],{"linethickness":4321},[57,13487,2197],{},[57,13489,13490],{},"k",[276,13492,1037],{"fence":73},[65,13494,13495],{"encoding":67},"\\binom{n}{k}",[35,13497,13499],{"className":13498,"ariaHidden":73},[72],[35,13500,13502,13506],{"className":13501},[77],[35,13503],{"className":13504,"style":13505},[81],"height:1.2em;vertical-align:-0.35em;",[35,13507,13509,13515,13568],{"className":13508},[86],[35,13510,13512],{"className":13511,"style":1073},[90,1072],[35,13513,1028],{"className":13514},[1077,1078],[35,13516,13518],{"className":13517},[54],[35,13519,13521,13560],{"className":13520},[98,99],[35,13522,13524,13557],{"className":13523},[103],[35,13525,13528,13543],{"className":13526,"style":13527},[107],"height:0.7454em;",[35,13529,13530,13533],{"style":4503},[35,13531],{"className":13532,"style":1173},[115],[35,13534,13536],{"className":13535},[120,121,122,123],[35,13537,13539],{"className":13538},[86,123],[35,13540,13490],{"className":13541,"style":13542},[86,156,123],"margin-right:0.0315em;",[35,13544,13545,13548],{"style":4518},[35,13546],{"className":13547,"style":1173},[115],[35,13549,13551],{"className":13550},[120,121,122,123],[35,13552,13554],{"className":13553},[86,123],[35,13555,2197],{"className":13556},[86,156,123],[35,13558,162],{"className":13559},[161],[35,13561,13563],{"className":13562},[103],[35,13564,13566],{"className":13565,"style":169},[107],[35,13567],{},[35,13569,13571],{"className":13570,"style":1073},[175,1072],[35,13572,1037],{"className":13573},[1077,1078],[17,13575,13576,13577,13635,13636,4759,13664,13692],{},"y se lee \"",[239,13578,13579,4697,13607],{},[35,13580,13582,13595],{"className":13581},[38],[35,13583,13585],{"className":13584},[42],[44,13586,13587],{"xmlns":46},[48,13588,13589,13593],{},[51,13590,13591],{},[57,13592,2197],{},[65,13594,2197],{"encoding":67},[35,13596,13598],{"className":13597,"ariaHidden":73},[72],[35,13599,13601,13604],{"className":13600},[77],[35,13602],{"className":13603,"style":206},[81],[35,13605,2197],{"className":13606},[86,156],[35,13608,13610,13623],{"className":13609},[38],[35,13611,13613],{"className":13612},[42],[44,13614,13615],{"xmlns":46},[48,13616,13617,13621],{},[51,13618,13619],{},[57,13620,13490],{},[65,13622,13490],{"encoding":67},[35,13624,13626],{"className":13625,"ariaHidden":73},[72],[35,13627,13629,13632],{"className":13628},[77],[35,13630],{"className":13631,"style":7026},[81],[35,13633,13490],{"className":13634,"style":13542},[86,156],"\". Su significado es directo: representa el número de formas de elegir ",[35,13637,13639,13652],{"className":13638},[38],[35,13640,13642],{"className":13641},[42],[44,13643,13644],{"xmlns":46},[48,13645,13646,13650],{},[51,13647,13648],{},[57,13649,13490],{},[65,13651,13490],{"encoding":67},[35,13653,13655],{"className":13654,"ariaHidden":73},[72],[35,13656,13658,13661],{"className":13657},[77],[35,13659],{"className":13660,"style":7026},[81],[35,13662,13490],{"className":13663,"style":13542},[86,156],[35,13665,13667,13680],{"className":13666},[38],[35,13668,13670],{"className":13669},[42],[44,13671,13672],{"xmlns":46},[48,13673,13674,13678],{},[51,13675,13676],{},[57,13677,2197],{},[65,13679,2197],{"encoding":67},[35,13681,13683],{"className":13682,"ariaHidden":73},[72],[35,13684,13686,13689],{"className":13685},[77],[35,13687],{"className":13688,"style":206},[81],[35,13690,2197],{"className":13691},[86,156],", sin importar el orden.",[17,13694,13695,13696,13820,13821,176],{},"Por ejemplo, ",[35,13697,13699,13727],{"className":13698},[38],[35,13700,13702],{"className":13701},[42],[44,13703,13704],{"xmlns":46},[48,13705,13706,13724],{},[51,13707,13708,13720,13722],{},[51,13709,13710,13712,13718],{},[276,13711,1028],{"fence":73},[54,13713,13714,13716],{"linethickness":4321},[61,13715,63],{},[61,13717,404],{},[276,13719,1037],{"fence":73},[276,13721,278],{},[61,13723,5442],{},[65,13725,13726],{"encoding":67},"\\binom{4}{2} = 6",[35,13728,13730,13811],{"className":13729,"ariaHidden":73},[72],[35,13731,13733,13736,13802,13805,13808],{"className":13732},[77],[35,13734],{"className":13735,"style":4592},[81],[35,13737,13739,13745,13796],{"className":13738},[86],[35,13740,13742],{"className":13741,"style":1073},[90,1072],[35,13743,1028],{"className":13744},[1077,1078],[35,13746,13748],{"className":13747},[54],[35,13749,13751,13788],{"className":13750},[98,99],[35,13752,13754,13785],{"className":13753},[103],[35,13755,13757,13771],{"className":13756,"style":4500},[107],[35,13758,13759,13762],{"style":4503},[35,13760],{"className":13761,"style":1173},[115],[35,13763,13765],{"className":13764},[120,121,122,123],[35,13766,13768],{"className":13767},[86,123],[35,13769,404],{"className":13770},[86,123],[35,13772,13773,13776],{"style":4518},[35,13774],{"className":13775,"style":1173},[115],[35,13777,13779],{"className":13778},[120,121,122,123],[35,13780,13782],{"className":13781},[86,123],[35,13783,63],{"className":13784},[86,123],[35,13786,162],{"className":13787},[161],[35,13789,13791],{"className":13790},[103],[35,13792,13794],{"className":13793,"style":169},[107],[35,13795],{},[35,13797,13799],{"className":13798,"style":1073},[175,1072],[35,13800,1037],{"className":13801},[1077,1078],[35,13803],{"className":13804,"style":367},[366],[35,13806,278],{"className":13807},[371],[35,13809],{"className":13810,"style":367},[366],[35,13812,13814,13817],{"className":13813},[77],[35,13815],{"className":13816,"style":381},[81],[35,13818,5442],{"className":13819},[86]," significa que hay 6 formas de elegir 2 objetos de entre 4. Y efectivamente, si miras la fila 4 del triángulo, la posición 2 (contando desde 0) es precisamente ",[35,13822,13824,13837],{"className":13823},[38],[35,13825,13827],{"className":13826},[42],[44,13828,13829],{"xmlns":46},[48,13830,13831,13835],{},[51,13832,13833],{},[61,13834,5442],{},[65,13836,5442],{"encoding":67},[35,13838,13840],{"className":13839,"ariaHidden":73},[72],[35,13841,13843,13846],{"className":13842},[77],[35,13844],{"className":13845,"style":381},[81],[35,13847,5442],{"className":13848},[86],[17,13850,13851],{},"La fórmula para calcularlo es:",[17,13853,13855],{"className":13854},[592],[35,13856,13858,13913],{"className":13857},[38],[35,13859,13861],{"className":13860},[42],[44,13862,13863],{"xmlns":46},[48,13864,13865,13910],{},[51,13866,13867,13879,13881],{},[51,13868,13869,13871,13877],{},[276,13870,1028],{"fence":73},[54,13872,13873,13875],{"linethickness":4321},[57,13874,2197],{},[57,13876,13490],{},[276,13878,1037],{"fence":73},[276,13880,278],{},[54,13882,13883,13890],{},[51,13884,13885,13887],{},[57,13886,2197],{},[276,13888,13889],{"stretchy":9816},"!",[51,13891,13892,13894,13896,13898,13900,13902,13904,13906,13908],{},[57,13893,13490],{},[276,13895,13889],{"stretchy":9816},[276,13897,7344],{},[276,13899,1028],{"stretchy":9816},[57,13901,2197],{},[276,13903,1353],{},[57,13905,13490],{},[276,13907,1037],{"stretchy":9816},[276,13909,13889],{"stretchy":9816},[65,13911,13912],{"encoding":67},"\\binom{n}{k} = \\frac{n!}{k! \\cdot (n-k)!}",[35,13914,13916,13997],{"className":13915,"ariaHidden":73},[72],[35,13917,13919,13922,13988,13991,13994],{"className":13918},[77],[35,13920],{"className":13921,"style":13505},[81],[35,13923,13925,13931,13982],{"className":13924},[86],[35,13926,13928],{"className":13927,"style":1073},[90,1072],[35,13929,1028],{"className":13930},[1077,1078],[35,13932,13934],{"className":13933},[54],[35,13935,13937,13974],{"className":13936},[98,99],[35,13938,13940,13971],{"className":13939},[103],[35,13941,13943,13957],{"className":13942,"style":13527},[107],[35,13944,13945,13948],{"style":4503},[35,13946],{"className":13947,"style":1173},[115],[35,13949,13951],{"className":13950},[120,121,122,123],[35,13952,13954],{"className":13953},[86,123],[35,13955,13490],{"className":13956,"style":13542},[86,156,123],[35,13958,13959,13962],{"style":4518},[35,13960],{"className":13961,"style":1173},[115],[35,13963,13965],{"className":13964},[120,121,122,123],[35,13966,13968],{"className":13967},[86,123],[35,13969,2197],{"className":13970},[86,156,123],[35,13972,162],{"className":13973},[161],[35,13975,13977],{"className":13976},[103],[35,13978,13980],{"className":13979,"style":169},[107],[35,13981],{},[35,13983,13985],{"className":13984,"style":1073},[175,1072],[35,13986,1037],{"className":13987},[1077,1078],[35,13989],{"className":13990,"style":367},[366],[35,13992,278],{"className":13993},[371],[35,13995],{"className":13996,"style":367},[366],[35,13998,14000,14004],{"className":13999},[77],[35,14001],{"className":14002,"style":14003},[81],"height:1.4001em;vertical-align:-0.52em;",[35,14005,14007,14010,14095],{"className":14006},[86],[35,14008],{"className":14009},[90,91],[35,14011,14013],{"className":14012},[54],[35,14014,14016,14087],{"className":14015},[98,99],[35,14017,14019,14084],{"className":14018},[103],[35,14020,14023,14059,14067],{"className":14021,"style":14022},[107],"height:0.8801em;",[35,14024,14025,14028],{"style":111},[35,14026],{"className":14027,"style":116},[115],[35,14029,14031],{"className":14030},[120,121,122,123],[35,14032,14034,14037,14040,14043,14046,14049,14052,14055],{"className":14033},[86,123],[35,14035,13490],{"className":14036,"style":13542},[86,156,123],[35,14038,13889],{"className":14039},[175,123],[35,14041,7344],{"className":14042},[2227,123],[35,14044,1028],{"className":14045},[90,123],[35,14047,2197],{"className":14048},[86,156,123],[35,14050,1353],{"className":14051},[2227,123],[35,14053,13490],{"className":14054,"style":13542},[86,156,123],[35,14056,14058],{"className":14057},[175,123],")!",[35,14060,14061,14064],{"style":132},[35,14062],{"className":14063,"style":116},[115],[35,14065],{"className":14066,"style":140},[139],[35,14068,14069,14072],{"style":143},[35,14070],{"className":14071,"style":116},[115],[35,14073,14075],{"className":14074},[120,121,122,123],[35,14076,14078,14081],{"className":14077},[86,123],[35,14079,2197],{"className":14080},[86,156,123],[35,14082,13889],{"className":14083},[175,123],[35,14085,162],{"className":14086},[161],[35,14088,14090],{"className":14089},[103],[35,14091,14093],{"className":14092,"style":10820},[107],[35,14094],{},[35,14096],{"className":14097},[175,91],[17,14099,14100,14101,14135,14136,14139,14140,14168,14169,14197,14198,14343,14344,176],{},"Aquí, el símbolo ",[35,14102,14104,14120],{"className":14103},[38],[35,14105,14107],{"className":14106},[42],[44,14108,14109],{"xmlns":46},[48,14110,14111,14117],{},[51,14112,14113,14115],{},[57,14114,2197],{},[276,14116,13889],{"stretchy":9816},[65,14118,14119],{"encoding":67},"n!",[35,14121,14123],{"className":14122,"ariaHidden":73},[72],[35,14124,14126,14129,14132],{"className":14125},[77],[35,14127],{"className":14128,"style":7026},[81],[35,14130,2197],{"className":14131},[86,156],[35,14133,13889],{"className":14134},[175]," se llama ",[239,14137,14138],{},"factorial"," y significa multiplicar todos los números enteros desde ",[35,14141,14143,14156],{"className":14142},[38],[35,14144,14146],{"className":14145},[42],[44,14147,14148],{"xmlns":46},[48,14149,14150,14154],{},[51,14151,14152],{},[61,14153,272],{},[65,14155,272],{"encoding":67},[35,14157,14159],{"className":14158,"ariaHidden":73},[72],[35,14160,14162,14165],{"className":14161},[77],[35,14163],{"className":14164,"style":381},[81],[35,14166,272],{"className":14167},[86]," hasta ",[35,14170,14172,14185],{"className":14171},[38],[35,14173,14175],{"className":14174},[42],[44,14176,14177],{"xmlns":46},[48,14178,14179,14183],{},[51,14180,14181],{},[57,14182,2197],{},[65,14184,2197],{"encoding":67},[35,14186,14188],{"className":14187,"ariaHidden":73},[72],[35,14189,14191,14194],{"className":14190},[77],[35,14192],{"className":14193,"style":206},[81],[35,14195,2197],{"className":14196},[86,156],". Por ejemplo, ",[35,14199,14201,14238],{"className":14200},[38],[35,14202,14204],{"className":14203},[42],[44,14205,14206],{"xmlns":46},[48,14207,14208,14235],{},[51,14209,14210,14212,14214,14216,14218,14220,14222,14224,14226,14228,14230,14232],{},[61,14211,63],{},[276,14213,13889],{"stretchy":9816},[276,14215,278],{},[61,14217,63],{},[276,14219,6472],{},[61,14221,927],{},[276,14223,6472],{},[61,14225,404],{},[276,14227,6472],{},[61,14229,272],{},[276,14231,278],{},[61,14233,14234],{},"24",[65,14236,14237],{"encoding":67},"4! = 4 \\times 3 \\times 2 \\times 1 = 24",[35,14239,14241,14262,14280,14298,14316,14334],{"className":14240,"ariaHidden":73},[72],[35,14242,14244,14247,14250,14253,14256,14259],{"className":14243},[77],[35,14245],{"className":14246,"style":7026},[81],[35,14248,63],{"className":14249},[86],[35,14251,13889],{"className":14252},[175],[35,14254],{"className":14255,"style":367},[366],[35,14257,278],{"className":14258},[371],[35,14260],{"className":14261,"style":367},[366],[35,14263,14265,14268,14271,14274,14277],{"className":14264},[77],[35,14266],{"className":14267,"style":1331},[81],[35,14269,63],{"className":14270},[86],[35,14272],{"className":14273,"style":2223},[366],[35,14275,6472],{"className":14276},[2227],[35,14278],{"className":14279,"style":2223},[366],[35,14281,14283,14286,14289,14292,14295],{"className":14282},[77],[35,14284],{"className":14285,"style":1331},[81],[35,14287,927],{"className":14288},[86],[35,14290],{"className":14291,"style":2223},[366],[35,14293,6472],{"className":14294},[2227],[35,14296],{"className":14297,"style":2223},[366],[35,14299,14301,14304,14307,14310,14313],{"className":14300},[77],[35,14302],{"className":14303,"style":1331},[81],[35,14305,404],{"className":14306},[86],[35,14308],{"className":14309,"style":2223},[366],[35,14311,6472],{"className":14312},[2227],[35,14314],{"className":14315,"style":2223},[366],[35,14317,14319,14322,14325,14328,14331],{"className":14318},[77],[35,14320],{"className":14321,"style":381},[81],[35,14323,272],{"className":14324},[86],[35,14326],{"className":14327,"style":367},[366],[35,14329,278],{"className":14330},[371],[35,14332],{"className":14333,"style":367},[366],[35,14335,14337,14340],{"className":14336},[77],[35,14338],{"className":14339,"style":381},[81],[35,14341,14234],{"className":14342},[86],". Por convención, ",[35,14345,14347,14367],{"className":14346},[38],[35,14348,14350],{"className":14349},[42],[44,14351,14352],{"xmlns":46},[48,14353,14354,14364],{},[51,14355,14356,14358,14360,14362],{},[61,14357,517],{},[276,14359,13889],{"stretchy":9816},[276,14361,278],{},[61,14363,272],{},[65,14365,14366],{"encoding":67},"0! = 1",[35,14368,14370,14391],{"className":14369,"ariaHidden":73},[72],[35,14371,14373,14376,14379,14382,14385,14388],{"className":14372},[77],[35,14374],{"className":14375,"style":7026},[81],[35,14377,517],{"className":14378},[86],[35,14380,13889],{"className":14381},[175],[35,14383],{"className":14384,"style":367},[366],[35,14386,278],{"className":14387},[371],[35,14389],{"className":14390,"style":367},[366],[35,14392,14394,14397],{"className":14393},[77],[35,14395],{"className":14396,"style":381},[81],[35,14398,272],{"className":14399},[86],[17,14401,14402,14403,14842],{},"Comprobemos: ",[35,14404,14406,14478],{"className":14405},[38],[35,14407,14409],{"className":14408},[42],[44,14410,14411],{"xmlns":46},[48,14412,14413,14475],{},[51,14414,14415,14427,14429,14449,14451,14463,14465,14471,14473],{},[51,14416,14417,14419,14425],{},[276,14418,1028],{"fence":73},[54,14420,14421,14423],{"linethickness":4321},[61,14422,63],{},[61,14424,404],{},[276,14426,1037],{"fence":73},[276,14428,278],{},[54,14430,14431,14437],{},[51,14432,14433,14435],{},[61,14434,63],{},[276,14436,13889],{"stretchy":9816},[51,14438,14439,14441,14443,14445,14447],{},[61,14440,404],{},[276,14442,13889],{"stretchy":9816},[276,14444,7344],{},[61,14446,404],{},[276,14448,13889],{"stretchy":9816},[276,14450,278],{},[54,14452,14453,14455],{},[61,14454,14234],{},[51,14456,14457,14459,14461],{},[61,14458,404],{},[276,14460,6472],{},[61,14462,404],{},[276,14464,278],{},[54,14466,14467,14469],{},[61,14468,14234],{},[61,14470,63],{},[276,14472,278],{},[61,14474,5442],{},[65,14476,14477],{"encoding":67},"\\binom{4}{2} = \\frac{4!}{2! \\cdot 2!} = \\frac{24}{2 \\times 2} = \\frac{24}{4} = 6",[35,14479,14481,14562,14661,14750,14833],{"className":14480,"ariaHidden":73},[72],[35,14482,14484,14487,14553,14556,14559],{"className":14483},[77],[35,14485],{"className":14486,"style":4592},[81],[35,14488,14490,14496,14547],{"className":14489},[86],[35,14491,14493],{"className":14492,"style":1073},[90,1072],[35,14494,1028],{"className":14495},[1077,1078],[35,14497,14499],{"className":14498},[54],[35,14500,14502,14539],{"className":14501},[98,99],[35,14503,14505,14536],{"className":14504},[103],[35,14506,14508,14522],{"className":14507,"style":4500},[107],[35,14509,14510,14513],{"style":4503},[35,14511],{"className":14512,"style":1173},[115],[35,14514,14516],{"className":14515},[120,121,122,123],[35,14517,14519],{"className":14518},[86,123],[35,14520,404],{"className":14521},[86,123],[35,14523,14524,14527],{"style":4518},[35,14525],{"className":14526,"style":1173},[115],[35,14528,14530],{"className":14529},[120,121,122,123],[35,14531,14533],{"className":14532},[86,123],[35,14534,63],{"className":14535},[86,123],[35,14537,162],{"className":14538},[161],[35,14540,14542],{"className":14541},[103],[35,14543,14545],{"className":14544,"style":169},[107],[35,14546],{},[35,14548,14550],{"className":14549,"style":1073},[175,1072],[35,14551,1037],{"className":14552},[1077,1078],[35,14554],{"className":14555,"style":367},[366],[35,14557,278],{"className":14558},[371],[35,14560],{"className":14561,"style":367},[366],[35,14563,14565,14569,14652,14655,14658],{"className":14564},[77],[35,14566],{"className":14567,"style":14568},[81],"height:1.2251em;vertical-align:-0.345em;",[35,14570,14572,14575,14649],{"className":14571},[86],[35,14573],{"className":14574},[90,91],[35,14576,14578],{"className":14577},[54],[35,14579,14581,14641],{"className":14580},[98,99],[35,14582,14584,14638],{"className":14583},[103],[35,14585,14587,14613,14621],{"className":14586,"style":14022},[107],[35,14588,14589,14592],{"style":111},[35,14590],{"className":14591,"style":116},[115],[35,14593,14595],{"className":14594},[120,121,122,123],[35,14596,14598,14601,14604,14607,14610],{"className":14597},[86,123],[35,14599,404],{"className":14600},[86,123],[35,14602,13889],{"className":14603},[175,123],[35,14605,7344],{"className":14606},[2227,123],[35,14608,404],{"className":14609},[86,123],[35,14611,13889],{"className":14612},[175,123],[35,14614,14615,14618],{"style":132},[35,14616],{"className":14617,"style":116},[115],[35,14619],{"className":14620,"style":140},[139],[35,14622,14623,14626],{"style":143},[35,14624],{"className":14625,"style":116},[115],[35,14627,14629],{"className":14628},[120,121,122,123],[35,14630,14632,14635],{"className":14631},[86,123],[35,14633,63],{"className":14634},[86,123],[35,14636,13889],{"className":14637},[175,123],[35,14639,162],{"className":14640},[161],[35,14642,14644],{"className":14643},[103],[35,14645,14647],{"className":14646,"style":169},[107],[35,14648],{},[35,14650],{"className":14651},[175,91],[35,14653],{"className":14654,"style":367},[366],[35,14656,278],{"className":14657},[371],[35,14659],{"className":14660,"style":367},[366],[35,14662,14664,14667,14741,14744,14747],{"className":14663},[77],[35,14665],{"className":14666,"style":3657},[81],[35,14668,14670,14673,14738],{"className":14669},[86],[35,14671],{"className":14672},[90,91],[35,14674,14676],{"className":14675},[54],[35,14677,14679,14730],{"className":14678},[98,99],[35,14680,14682,14727],{"className":14681},[103],[35,14683,14685,14705,14713],{"className":14684,"style":312},[107],[35,14686,14687,14690],{"style":111},[35,14688],{"className":14689,"style":116},[115],[35,14691,14693],{"className":14692},[120,121,122,123],[35,14694,14696,14699,14702],{"className":14695},[86,123],[35,14697,404],{"className":14698},[86,123],[35,14700,6472],{"className":14701},[2227,123],[35,14703,404],{"className":14704},[86,123],[35,14706,14707,14710],{"style":132},[35,14708],{"className":14709,"style":116},[115],[35,14711],{"className":14712,"style":140},[139],[35,14714,14715,14718],{"style":143},[35,14716],{"className":14717,"style":116},[115],[35,14719,14721],{"className":14720},[120,121,122,123],[35,14722,14724],{"className":14723},[86,123],[35,14725,14234],{"className":14726},[86,123],[35,14728,162],{"className":14729},[161],[35,14731,14733],{"className":14732},[103],[35,14734,14736],{"className":14735,"style":3736},[107],[35,14737],{},[35,14739],{"className":14740},[175,91],[35,14742],{"className":14743,"style":367},[366],[35,14745,278],{"className":14746},[371],[35,14748],{"className":14749,"style":367},[366],[35,14751,14753,14756,14824,14827,14830],{"className":14752},[77],[35,14754],{"className":14755,"style":293},[81],[35,14757,14759,14762,14821],{"className":14758},[86],[35,14760],{"className":14761},[90,91],[35,14763,14765],{"className":14764},[54],[35,14766,14768,14813],{"className":14767},[98,99],[35,14769,14771,14810],{"className":14770},[103],[35,14772,14774,14788,14796],{"className":14773,"style":312},[107],[35,14775,14776,14779],{"style":111},[35,14777],{"className":14778,"style":116},[115],[35,14780,14782],{"className":14781},[120,121,122,123],[35,14783,14785],{"className":14784},[86,123],[35,14786,63],{"className":14787},[86,123],[35,14789,14790,14793],{"style":132},[35,14791],{"className":14792,"style":116},[115],[35,14794],{"className":14795,"style":140},[139],[35,14797,14798,14801],{"style":143},[35,14799],{"className":14800,"style":116},[115],[35,14802,14804],{"className":14803},[120,121,122,123],[35,14805,14807],{"className":14806},[86,123],[35,14808,14234],{"className":14809},[86,123],[35,14811,162],{"className":14812},[161],[35,14814,14816],{"className":14815},[103],[35,14817,14819],{"className":14818,"style":169},[107],[35,14820],{},[35,14822],{"className":14823},[175,91],[35,14825],{"className":14826,"style":367},[366],[35,14828,278],{"className":14829},[371],[35,14831],{"className":14832,"style":367},[366],[35,14834,14836,14839],{"className":14835},[77],[35,14837],{"className":14838,"style":381},[81],[35,14840,5442],{"className":14841},[86],". Correcto.",[17,14844,14845,14846,14874],{},"Así, la fila ",[35,14847,14849,14862],{"className":14848},[38],[35,14850,14852],{"className":14851},[42],[44,14853,14854],{"xmlns":46},[48,14855,14856,14860],{},[51,14857,14858],{},[57,14859,2197],{},[65,14861,2197],{"encoding":67},[35,14863,14865],{"className":14864,"ariaHidden":73},[72],[35,14866,14868,14871],{"className":14867},[77],[35,14869],{"className":14870,"style":206},[81],[35,14872,2197],{"className":14873},[86,156]," del triángulo contiene los valores:",[17,14876,14878],{"className":14877},[592],[35,14879,14881,14959],{"className":14880},[38],[35,14882,14884],{"className":14883},[42],[44,14885,14886],{"xmlns":46},[48,14887,14888,14956],{},[51,14889,14890,14902,14904,14906,14918,14920,14922,14934,14936,14938,14940,14942,14944],{},[51,14891,14892,14894,14900],{},[276,14893,1028],{"fence":73},[54,14895,14896,14898],{"linethickness":4321},[57,14897,2197],{},[61,14899,517],{},[276,14901,1037],{"fence":73},[276,14903,525],{"separator":73},[606,14905,1398],{},[51,14907,14908,14910,14916],{},[276,14909,1028],{"fence":73},[54,14911,14912,14914],{"linethickness":4321},[57,14913,2197],{},[61,14915,272],{},[276,14917,1037],{"fence":73},[276,14919,525],{"separator":73},[606,14921,1398],{},[51,14923,14924,14926,14932],{},[276,14925,1028],{"fence":73},[54,14927,14928,14930],{"linethickness":4321},[57,14929,2197],{},[61,14931,404],{},[276,14933,1037],{"fence":73},[276,14935,525],{"separator":73},[606,14937,1398],{},[276,14939,4008],{},[276,14941,525],{"separator":73},[606,14943,1398],{},[51,14945,14946,14948,14954],{},[276,14947,1028],{"fence":73},[54,14949,14950,14952],{"linethickness":4321},[57,14951,2197],{},[57,14953,2197],{},[276,14955,1037],{"fence":73},[65,14957,14958],{"encoding":67},"\\binom{n}{0},\\; \\binom{n}{1},\\; \\binom{n}{2},\\; \\ldots,\\; \\binom{n}{n}",[35,14960,14962],{"className":14961,"ariaHidden":73},[72],[35,14963,14965,14968,15034,15037,15040,15043,15109,15112,15115,15118,15184,15187,15190,15193,15196,15199,15202,15205,15208],{"className":14964},[77],[35,14966],{"className":14967,"style":13505},[81],[35,14969,14971,14977,15028],{"className":14970},[86],[35,14972,14974],{"className":14973,"style":1073},[90,1072],[35,14975,1028],{"className":14976},[1077,1078],[35,14978,14980],{"className":14979},[54],[35,14981,14983,15020],{"className":14982},[98,99],[35,14984,14986,15017],{"className":14985},[103],[35,14987,14989,15003],{"className":14988,"style":13527},[107],[35,14990,14991,14994],{"style":4503},[35,14992],{"className":14993,"style":1173},[115],[35,14995,14997],{"className":14996},[120,121,122,123],[35,14998,15000],{"className":14999},[86,123],[35,15001,517],{"className":15002},[86,123],[35,15004,15005,15008],{"style":4518},[35,15006],{"className":15007,"style":1173},[115],[35,15009,15011],{"className":15010},[120,121,122,123],[35,15012,15014],{"className":15013},[86,123],[35,15015,2197],{"className":15016},[86,156,123],[35,15018,162],{"className":15019},[161],[35,15021,15023],{"className":15022},[103],[35,15024,15026],{"className":15025,"style":169},[107],[35,15027],{},[35,15029,15031],{"className":15030,"style":1073},[175,1072],[35,15032,1037],{"className":15033},[1077,1078],[35,15035,525],{"className":15036},[524],[35,15038],{"className":15039,"style":367},[366],[35,15041],{"className":15042,"style":1452},[366],[35,15044,15046,15052,15103],{"className":15045},[86],[35,15047,15049],{"className":15048,"style":1073},[90,1072],[35,15050,1028],{"className":15051},[1077,1078],[35,15053,15055],{"className":15054},[54],[35,15056,15058,15095],{"className":15057},[98,99],[35,15059,15061,15092],{"className":15060},[103],[35,15062,15064,15078],{"className":15063,"style":13527},[107],[35,15065,15066,15069],{"style":4503},[35,15067],{"className":15068,"style":1173},[115],[35,15070,15072],{"className":15071},[120,121,122,123],[35,15073,15075],{"className":15074},[86,123],[35,15076,272],{"className":15077},[86,123],[35,15079,15080,15083],{"style":4518},[35,15081],{"className":15082,"style":1173},[115],[35,15084,15086],{"className":15085},[120,121,122,123],[35,15087,15089],{"className":15088},[86,123],[35,15090,2197],{"className":15091},[86,156,123],[35,15093,162],{"className":15094},[161],[35,15096,15098],{"className":15097},[103],[35,15099,15101],{"className":15100,"style":169},[107],[35,15102],{},[35,15104,15106],{"className":15105,"style":1073},[175,1072],[35,15107,1037],{"className":15108},[1077,1078],[35,15110,525],{"className":15111},[524],[35,15113],{"className":15114,"style":367},[366],[35,15116],{"className":15117,"style":1452},[366],[35,15119,15121,15127,15178],{"className":15120},[86],[35,15122,15124],{"className":15123,"style":1073},[90,1072],[35,15125,1028],{"className":15126},[1077,1078],[35,15128,15130],{"className":15129},[54],[35,15131,15133,15170],{"className":15132},[98,99],[35,15134,15136,15167],{"className":15135},[103],[35,15137,15139,15153],{"className":15138,"style":13527},[107],[35,15140,15141,15144],{"style":4503},[35,15142],{"className":15143,"style":1173},[115],[35,15145,15147],{"className":15146},[120,121,122,123],[35,15148,15150],{"className":15149},[86,123],[35,15151,404],{"className":15152},[86,123],[35,15154,15155,15158],{"style":4518},[35,15156],{"className":15157,"style":1173},[115],[35,15159,15161],{"className":15160},[120,121,122,123],[35,15162,15164],{"className":15163},[86,123],[35,15165,2197],{"className":15166},[86,156,123],[35,15168,162],{"className":15169},[161],[35,15171,15173],{"className":15172},[103],[35,15174,15176],{"className":15175,"style":169},[107],[35,15177],{},[35,15179,15181],{"className":15180,"style":1073},[175,1072],[35,15182,1037],{"className":15183},[1077,1078],[35,15185,525],{"className":15186},[524],[35,15188],{"className":15189,"style":367},[366],[35,15191],{"className":15192,"style":1452},[366],[35,15194,4008],{"className":15195},[1065],[35,15197],{"className":15198,"style":1452},[366],[35,15200,525],{"className":15201},[524],[35,15203],{"className":15204,"style":367},[366],[35,15206],{"className":15207,"style":1452},[366],[35,15209,15211,15217,15268],{"className":15210},[86],[35,15212,15214],{"className":15213,"style":1073},[90,1072],[35,15215,1028],{"className":15216},[1077,1078],[35,15218,15220],{"className":15219},[54],[35,15221,15223,15260],{"className":15222},[98,99],[35,15224,15226,15257],{"className":15225},[103],[35,15227,15229,15243],{"className":15228,"style":13527},[107],[35,15230,15231,15234],{"style":4503},[35,15232],{"className":15233,"style":1173},[115],[35,15235,15237],{"className":15236},[120,121,122,123],[35,15238,15240],{"className":15239},[86,123],[35,15241,2197],{"className":15242},[86,156,123],[35,15244,15245,15248],{"style":4518},[35,15246],{"className":15247,"style":1173},[115],[35,15249,15251],{"className":15250},[120,121,122,123],[35,15252,15254],{"className":15253},[86,123],[35,15255,2197],{"className":15256},[86,156,123],[35,15258,162],{"className":15259},[161],[35,15261,15263],{"className":15262},[103],[35,15264,15266],{"className":15265,"style":169},[107],[35,15267],{},[35,15269,15271],{"className":15270,"style":1073},[175,1072],[35,15272,1037],{"className":15273},[1077,1078],[561,15275,15277],{"id":15276},"la-combinatoria-el-arte-de-contar-sin-enumerar","La combinatoria: el arte de contar sin enumerar",[17,15279,15280,15281,15284,15285,176],{},"La ",[239,15282,15283],{},"combinatoria"," es la rama de las matemáticas que estudia cuántas maneras distintas existen de organizar, agrupar o elegir elementos. Su poder reside en que nos permite conocer el número de posibilidades ",[212,15286,15287],{},"sin tener que listarlas una por una",[17,15289,15290,15291,15294,15295,15298,15299,15327,15328,15401,15402,176],{},"Para entender por qué esto es útil, pensemos en lanzar una moneda. Cada lanzamiento tiene dos resultados posibles: ",[239,15292,15293],{},"cara"," (O) o ",[239,15296,15297],{},"cruz"," (X). Si lanzamos la moneda una vez, hay ",[35,15300,15302,15315],{"className":15301},[38],[35,15303,15305],{"className":15304},[42],[44,15306,15307],{"xmlns":46},[48,15308,15309,15313],{},[51,15310,15311],{},[61,15312,404],{},[65,15314,404],{"encoding":67},[35,15316,15318],{"className":15317,"ariaHidden":73},[72],[35,15319,15321,15324],{"className":15320},[77],[35,15322],{"className":15323,"style":381},[81],[35,15325,404],{"className":15326},[86]," resultados. Si la lanzamos dos veces, cada uno de los 2 resultados del primer lanzamiento se combina con los 2 del segundo, dando ",[35,15329,15331,15353],{"className":15330},[38],[35,15332,15334],{"className":15333},[42],[44,15335,15336],{"xmlns":46},[48,15337,15338,15350],{},[51,15339,15340,15342,15344,15346,15348],{},[61,15341,404],{},[276,15343,6472],{},[61,15345,404],{},[276,15347,278],{},[61,15349,63],{},[65,15351,15352],{"encoding":67},"2 \\times 2 = 4",[35,15354,15356,15374,15392],{"className":15355,"ariaHidden":73},[72],[35,15357,15359,15362,15365,15368,15371],{"className":15358},[77],[35,15360],{"className":15361,"style":1331},[81],[35,15363,404],{"className":15364},[86],[35,15366],{"className":15367,"style":2223},[366],[35,15369,6472],{"className":15370},[2227],[35,15372],{"className":15373,"style":2223},[366],[35,15375,15377,15380,15383,15386,15389],{"className":15376},[77],[35,15378],{"className":15379,"style":381},[81],[35,15381,404],{"className":15382},[86],[35,15384],{"className":15385,"style":367},[366],[35,15387,278],{"className":15388},[371],[35,15390],{"className":15391,"style":367},[366],[35,15393,15395,15398],{"className":15394},[77],[35,15396],{"className":15397,"style":381},[81],[35,15399,63],{"className":15400},[86]," resultados. Con tres lanzamientos, ",[35,15403,15405,15431],{"className":15404},[38],[35,15406,15408],{"className":15407},[42],[44,15409,15410],{"xmlns":46},[48,15411,15412,15428],{},[51,15413,15414,15416,15418,15420,15422,15424,15426],{},[61,15415,404],{},[276,15417,6472],{},[61,15419,404],{},[276,15421,6472],{},[61,15423,404],{},[276,15425,278],{},[61,15427,1048],{},[65,15429,15430],{"encoding":67},"2 \\times 2 \\times 2 = 8",[35,15432,15434,15452,15470,15488],{"className":15433,"ariaHidden":73},[72],[35,15435,15437,15440,15443,15446,15449],{"className":15436},[77],[35,15438],{"className":15439,"style":1331},[81],[35,15441,404],{"className":15442},[86],[35,15444],{"className":15445,"style":2223},[366],[35,15447,6472],{"className":15448},[2227],[35,15450],{"className":15451,"style":2223},[366],[35,15453,15455,15458,15461,15464,15467],{"className":15454},[77],[35,15456],{"className":15457,"style":1331},[81],[35,15459,404],{"className":15460},[86],[35,15462],{"className":15463,"style":2223},[366],[35,15465,6472],{"className":15466},[2227],[35,15468],{"className":15469,"style":2223},[366],[35,15471,15473,15476,15479,15482,15485],{"className":15472},[77],[35,15474],{"className":15475,"style":381},[81],[35,15477,404],{"className":15478},[86],[35,15480],{"className":15481,"style":367},[366],[35,15483,278],{"className":15484},[371],[35,15486],{"className":15487,"style":367},[366],[35,15489,15491,15494],{"className":15490},[77],[35,15492],{"className":15493,"style":381},[81],[35,15495,1048],{"className":15496},[86],[17,15498,15499,15500,11815],{},"Ese patrón se expresa con un ",[239,15501,15502],{},"exponente",[17,15504,15506],{"className":15505},[592],[35,15507,15509,15527],{"className":15508},[38],[35,15510,15512],{"className":15511},[42],[44,15513,15514],{"xmlns":46},[48,15515,15516,15524],{},[51,15517,15518],{},[1022,15519,15520,15522],{},[61,15521,404],{},[57,15523,2197],{},[65,15525,15526],{"encoding":67},"2^n",[35,15528,15530],{"className":15529,"ariaHidden":73},[72],[35,15531,15533,15537],{"className":15532},[77],[35,15534],{"className":15535,"style":15536},[81],"height:0.6644em;",[35,15538,15540,15543],{"className":15539},[86],[35,15541,404],{"className":15542},[86],[35,15544,15546],{"className":15545},[1156],[35,15547,15549],{"className":15548},[98],[35,15550,15552],{"className":15551},[103],[35,15553,15555],{"className":15554,"style":15536},[107],[35,15556,15558,15561],{"style":15557},"top:-3.063em;margin-right:0.05em;",[35,15559],{"className":15560,"style":1173},[115],[35,15562,15564],{"className":15563},[120,121,122,123],[35,15565,2197],{"className":15566},[86,156,123],[17,15568,4553,15569,15597,15598,15626],{},[35,15570,15572,15585],{"className":15571},[38],[35,15573,15575],{"className":15574},[42],[44,15576,15577],{"xmlns":46},[48,15578,15579,15583],{},[51,15580,15581],{},[57,15582,2197],{},[65,15584,2197],{"encoding":67},[35,15586,15588],{"className":15587,"ariaHidden":73},[72],[35,15589,15591,15594],{"className":15590},[77],[35,15592],{"className":15593,"style":206},[81],[35,15595,2197],{"className":15596},[86,156]," es el número de lanzamientos. El exponente indica cuántas veces multiplicamos la base (",[35,15599,15601,15614],{"className":15600},[38],[35,15602,15604],{"className":15603},[42],[44,15605,15606],{"xmlns":46},[48,15607,15608,15612],{},[51,15609,15610],{},[61,15611,404],{},[65,15613,404],{"encoding":67},[35,15615,15617],{"className":15616,"ariaHidden":73},[72],[35,15618,15620,15623],{"className":15619},[77],[35,15621],{"className":15622,"style":381},[81],[35,15624,404],{"className":15625},[86],") por sí misma. Así:",[2168,15628,15629,15725],{},[2171,15630,15631],{},[2174,15632,15633,15664],{},[2177,15634,15635,15636,1037],{"align":2179},"Lanzamientos (",[35,15637,15639,15652],{"className":15638},[38],[35,15640,15642],{"className":15641},[42],[44,15643,15644],{"xmlns":46},[48,15645,15646,15650],{},[51,15647,15648],{},[57,15649,2197],{},[65,15651,2197],{"encoding":67},[35,15653,15655],{"className":15654,"ariaHidden":73},[72],[35,15656,15658,15661],{"className":15657},[77],[35,15659],{"className":15660,"style":206},[81],[35,15662,2197],{"className":15663},[86,156],[2177,15665,15666,15667,1037],{"align":2179},"Resultados posibles (",[35,15668,15670,15687],{"className":15669},[38],[35,15671,15673],{"className":15672},[42],[44,15674,15675],{"xmlns":46},[48,15676,15677,15685],{},[51,15678,15679],{},[1022,15680,15681,15683],{},[61,15682,404],{},[57,15684,2197],{},[65,15686,15526],{"encoding":67},[35,15688,15690],{"className":15689,"ariaHidden":73},[72],[35,15691,15693,15696],{"className":15692},[77],[35,15694],{"className":15695,"style":15536},[81],[35,15697,15699,15702],{"className":15698},[86],[35,15700,404],{"className":15701},[86],[35,15703,15705],{"className":15704},[1156],[35,15706,15708],{"className":15707},[98],[35,15709,15711],{"className":15710},[103],[35,15712,15714],{"className":15713,"style":15536},[107],[35,15715,15716,15719],{"style":15557},[35,15717],{"className":15718,"style":1173},[115],[35,15720,15722],{"className":15721},[120,121,122,123],[35,15723,2197],{"className":15724},[86,156,123],[2250,15726,15727,15733,15739,15745,15751],{},[2174,15728,15729,15731],{},[2255,15730,272],{"align":2179},[2255,15732,404],{"align":2179},[2174,15734,15735,15737],{},[2255,15736,404],{"align":2179},[2255,15738,63],{"align":2179},[2174,15740,15741,15743],{},[2255,15742,927],{"align":2179},[2255,15744,1048],{"align":2179},[2174,15746,15747,15749],{},[2255,15748,529],{"align":2179},[2255,15750,1847],{"align":2179},[2174,15752,15753,15756],{},[2255,15754,15755],{"align":2179},"10",[2255,15757,15758],{"align":2179},"1.024",[17,15760,15761],{},"Con solo 10 lanzamientos ya hay más de mil combinaciones posibles. La combinatoria nos ayuda a manejar esas cantidades sin perder la cabeza.",[17,15763,15764,15765,176],{},"Si necesitas repasar qué es la probabilidad y cómo se mide, echa un vistazo a ",[21,15766,15769],{"href":15767,"rel":15768},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ffundamentos-de-probabilidad\u002F",[25],"Fundamentos de probabilidad",[561,15771,15773],{"id":15772},"caso-práctico-3-lanzamientos","Caso práctico: 3 lanzamientos",[17,15775,15776,15777,15859],{},"Vamos a conectar todo con un ejemplo concreto. Lanzamos una moneda 3 veces. Sabemos que hay ",[35,15778,15780,15802],{"className":15779},[38],[35,15781,15783],{"className":15782},[42],[44,15784,15785],{"xmlns":46},[48,15786,15787,15799],{},[51,15788,15789,15795,15797],{},[1022,15790,15791,15793],{},[61,15792,404],{},[61,15794,927],{},[276,15796,278],{},[61,15798,1048],{},[65,15800,15801],{"encoding":67},"2^3 = 8",[35,15803,15805,15850],{"className":15804,"ariaHidden":73},[72],[35,15806,15808,15812,15841,15844,15847],{"className":15807},[77],[35,15809],{"className":15810,"style":15811},[81],"height:0.8141em;",[35,15813,15815,15818],{"className":15814},[86],[35,15816,404],{"className":15817},[86],[35,15819,15821],{"className":15820},[1156],[35,15822,15824],{"className":15823},[98],[35,15825,15827],{"className":15826},[103],[35,15828,15830],{"className":15829,"style":15811},[107],[35,15831,15832,15835],{"style":15557},[35,15833],{"className":15834,"style":1173},[115],[35,15836,15838],{"className":15837},[120,121,122,123],[35,15839,927],{"className":15840},[86,123],[35,15842],{"className":15843,"style":367},[366],[35,15845,278],{"className":15846},[371],[35,15848],{"className":15849,"style":367},[366],[35,15851,15853,15856],{"className":15852},[77],[35,15854],{"className":15855,"style":381},[81],[35,15857,1048],{"className":15858},[86]," resultados posibles. Aquí están todos:",[1286,15861,15864],{"className":15862,"code":15863,"language":625,"meta":1291},[1289],"OOO    OOX    OXO    XOO    OXX    XOX    XXO    XXX\n",[1293,15865,15863],{"__ignoreMap":1291},[17,15867,15868,15869,15872],{},"Ahora agrupemos estos resultados según el ",[239,15870,15871],{},"número de caras"," obtenidas:",[2168,15874,15875,15888],{},[2171,15876,15877],{},[2174,15878,15879,15882,15885],{},[2177,15880,15881],{"align":2179},"Caras",[2177,15883,15884],{"align":7764},"Combinaciones",[2177,15886,15887],{"align":2179},"Cantidad",[2250,15889,15890,15899,15908,15917],{},[2174,15891,15892,15894,15897],{},[2255,15893,517],{"align":2179},[2255,15895,15896],{"align":7764},"XXX",[2255,15898,272],{"align":2179},[2174,15900,15901,15903,15906],{},[2255,15902,272],{"align":2179},[2255,15904,15905],{"align":7764},"OXX, XOX, XXO",[2255,15907,927],{"align":2179},[2174,15909,15910,15912,15915],{},[2255,15911,404],{"align":2179},[2255,15913,15914],{"align":7764},"OOX, OXO, XOO",[2255,15916,927],{"align":2179},[2174,15918,15919,15921,15924],{},[2255,15920,927],{"align":2179},[2255,15922,15923],{"align":7764},"OOO",[2255,15925,272],{"align":2179},[17,15927,15928,15929,15997,15998,16001],{},"Mira la columna de cantidades: ",[35,15930,15932,15958],{"className":15931},[38],[35,15933,15935],{"className":15934},[42],[44,15936,15937],{"xmlns":46},[48,15938,15939,15955],{},[51,15940,15941,15943,15945,15947,15949,15951,15953],{},[61,15942,272],{},[276,15944,525],{"separator":73},[61,15946,927],{},[276,15948,525],{"separator":73},[61,15950,927],{},[276,15952,525],{"separator":73},[61,15954,272],{},[65,15956,15957],{"encoding":67},"1, 3, 3, 1",[35,15959,15961],{"className":15960,"ariaHidden":73},[72],[35,15962,15964,15967,15970,15973,15976,15979,15982,15985,15988,15991,15994],{"className":15963},[77],[35,15965],{"className":15966,"style":513},[81],[35,15968,272],{"className":15969},[86],[35,15971,525],{"className":15972},[524],[35,15974],{"className":15975,"style":1452},[366],[35,15977,927],{"className":15978},[86],[35,15980,525],{"className":15981},[524],[35,15983],{"className":15984,"style":1452},[366],[35,15986,927],{"className":15987},[86],[35,15989,525],{"className":15990},[524],[35,15992],{"className":15993,"style":1452},[366],[35,15995,272],{"className":15996},[86],". ¿Te suena? Es exactamente la ",[239,15999,16000],{},"fila 3"," del triángulo de Tartaglia.",[17,16003,16004,16005,4822,16033,16061,16062,16161,16162,16190,16191,16219],{},"Y no es casualidad. El número de formas de obtener exactamente ",[35,16006,16008,16021],{"className":16007},[38],[35,16009,16011],{"className":16010},[42],[44,16012,16013],{"xmlns":46},[48,16014,16015,16019],{},[51,16016,16017],{},[57,16018,13490],{},[65,16020,13490],{"encoding":67},[35,16022,16024],{"className":16023,"ariaHidden":73},[72],[35,16025,16027,16030],{"className":16026},[77],[35,16028],{"className":16029,"style":7026},[81],[35,16031,13490],{"className":16032,"style":13542},[86,156],[35,16034,16036,16049],{"className":16035},[38],[35,16037,16039],{"className":16038},[42],[44,16040,16041],{"xmlns":46},[48,16042,16043,16047],{},[51,16044,16045],{},[57,16046,2197],{},[65,16048,2197],{"encoding":67},[35,16050,16052],{"className":16051,"ariaHidden":73},[72],[35,16053,16055,16058],{"className":16054},[77],[35,16056],{"className":16057,"style":206},[81],[35,16059,2197],{"className":16060},[86,156]," lanzamientos es precisamente ",[35,16063,16065,16086],{"className":16064},[38],[35,16066,16068],{"className":16067},[42],[44,16069,16070],{"xmlns":46},[48,16071,16072,16084],{},[51,16073,16074,16076,16082],{},[276,16075,1028],{"fence":73},[54,16077,16078,16080],{"linethickness":4321},[57,16079,2197],{},[57,16081,13490],{},[276,16083,1037],{"fence":73},[65,16085,13495],{"encoding":67},[35,16087,16089],{"className":16088,"ariaHidden":73},[72],[35,16090,16092,16095],{"className":16091},[77],[35,16093],{"className":16094,"style":13505},[81],[35,16096,16098,16104,16155],{"className":16097},[86],[35,16099,16101],{"className":16100,"style":1073},[90,1072],[35,16102,1028],{"className":16103},[1077,1078],[35,16105,16107],{"className":16106},[54],[35,16108,16110,16147],{"className":16109},[98,99],[35,16111,16113,16144],{"className":16112},[103],[35,16114,16116,16130],{"className":16115,"style":13527},[107],[35,16117,16118,16121],{"style":4503},[35,16119],{"className":16120,"style":1173},[115],[35,16122,16124],{"className":16123},[120,121,122,123],[35,16125,16127],{"className":16126},[86,123],[35,16128,13490],{"className":16129,"style":13542},[86,156,123],[35,16131,16132,16135],{"style":4518},[35,16133],{"className":16134,"style":1173},[115],[35,16136,16138],{"className":16137},[120,121,122,123],[35,16139,16141],{"className":16140},[86,123],[35,16142,2197],{"className":16143},[86,156,123],[35,16145,162],{"className":16146},[161],[35,16148,16150],{"className":16149},[103],[35,16151,16153],{"className":16152,"style":169},[107],[35,16154],{},[35,16156,16158],{"className":16157,"style":1073},[175,1072],[35,16159,1037],{"className":16160},[1077,1078]," — el coeficiente binomial que ocupa la posición ",[35,16163,16165,16178],{"className":16164},[38],[35,16166,16168],{"className":16167},[42],[44,16169,16170],{"xmlns":46},[48,16171,16172,16176],{},[51,16173,16174],{},[57,16175,13490],{},[65,16177,13490],{"encoding":67},[35,16179,16181],{"className":16180,"ariaHidden":73},[72],[35,16182,16184,16187],{"className":16183},[77],[35,16185],{"className":16186,"style":7026},[81],[35,16188,13490],{"className":16189,"style":13542},[86,156]," en la fila ",[35,16192,16194,16207],{"className":16193},[38],[35,16195,16197],{"className":16196},[42],[44,16198,16199],{"xmlns":46},[48,16200,16201,16205],{},[51,16202,16203],{},[57,16204,2197],{},[65,16206,2197],{"encoding":67},[35,16208,16210],{"className":16209,"ariaHidden":73},[72],[35,16211,16213,16216],{"className":16212},[77],[35,16214],{"className":16215,"style":206},[81],[35,16217,2197],{"className":16218},[86,156]," del triángulo.",[582,16221,16223],{"id":16222},"la-probabilidad-entra-en-escena","La probabilidad entra en escena",[17,16225,15280,16226,16228],{},[239,16227,6469],{}," de un suceso mide qué tan posible es que ocurra. Se calcula como:",[17,16230,16232],{"className":16231},[592],[35,16233,16235,16260],{"className":16234},[38],[35,16236,16238],{"className":16237},[42],[44,16239,16240],{"xmlns":46},[48,16241,16242,16257],{},[51,16243,16244,16247,16249],{},[57,16245,16246],{},"P",[276,16248,278],{},[54,16250,16251,16254],{},[606,16252,16253],{},"casos favorables",[606,16255,16256],{},"casos posibles",[65,16258,16259],{"encoding":67},"P = \\frac{\\text{casos favorables}}{\\text{casos posibles}}",[35,16261,16263,16282],{"className":16262,"ariaHidden":73},[72],[35,16264,16266,16269,16273,16276,16279],{"className":16265},[77],[35,16267],{"className":16268,"style":6826},[81],[35,16270,16246],{"className":16271,"style":16272},[86,156],"margin-right:0.1389em;",[35,16274],{"className":16275,"style":367},[366],[35,16277,278],{"className":16278},[371],[35,16280],{"className":16281,"style":367},[366],[35,16283,16285,16289],{"className":16284},[77],[35,16286],{"className":16287,"style":16288},[81],"height:1.3612em;vertical-align:-0.4811em;",[35,16290,16292,16295,16361],{"className":16291},[86],[35,16293],{"className":16294},[90,91],[35,16296,16298],{"className":16297},[54],[35,16299,16301,16352],{"className":16300},[98,99],[35,16302,16304,16349],{"className":16303},[103],[35,16305,16307,16324,16332],{"className":16306,"style":14022},[107],[35,16308,16309,16312],{"style":111},[35,16310],{"className":16311,"style":116},[115],[35,16313,16315],{"className":16314},[120,121,122,123],[35,16316,16318],{"className":16317},[86,123],[35,16319,16321],{"className":16320},[86,625,123],[35,16322,16256],{"className":16323},[86,123],[35,16325,16326,16329],{"style":132},[35,16327],{"className":16328,"style":116},[115],[35,16330],{"className":16331,"style":140},[139],[35,16333,16334,16337],{"style":143},[35,16335],{"className":16336,"style":116},[115],[35,16338,16340],{"className":16339},[120,121,122,123],[35,16341,16343],{"className":16342},[86,123],[35,16344,16346],{"className":16345},[86,625,123],[35,16347,16253],{"className":16348},[86,123],[35,16350,162],{"className":16351},[161],[35,16353,16355],{"className":16354},[103],[35,16356,16359],{"className":16357,"style":16358},[107],"height:0.4811em;",[35,16360],{},[35,16362],{"className":16363},[175,91],[17,16365,16366,16367,16395],{},"Si la moneda es justa, cada uno de los 8 resultados es igual de probable. Así que la probabilidad de obtener exactamente ",[35,16368,16370,16383],{"className":16369},[38],[35,16371,16373],{"className":16372},[42],[44,16374,16375],{"xmlns":46},[48,16376,16377,16381],{},[51,16378,16379],{},[57,16380,13490],{},[65,16382,13490],{"encoding":67},[35,16384,16386],{"className":16385,"ariaHidden":73},[72],[35,16387,16389,16392],{"className":16388},[77],[35,16390],{"className":16391,"style":7026},[81],[35,16393,13490],{"className":16394,"style":13542},[86,156]," caras en 3 lanzamientos es:",[17,16397,16399],{"className":16398},[592],[35,16400,16402,16465],{"className":16401},[38],[35,16403,16405],{"className":16404},[42],[44,16406,16407],{"xmlns":46},[48,16408,16409,16462],{},[51,16410,16411,16413,16415,16417,16420,16422,16424,16444,16446],{},[57,16412,16246],{},[276,16414,1028],{"stretchy":9816},[57,16416,13490],{},[606,16418,16419],{}," caras",[276,16421,1037],{"stretchy":9816},[276,16423,278],{},[54,16425,16426,16438],{},[51,16427,16428,16430,16436],{},[276,16429,1028],{"fence":73},[54,16431,16432,16434],{"linethickness":4321},[61,16433,927],{},[57,16435,13490],{},[276,16437,1037],{"fence":73},[1022,16439,16440,16442],{},[61,16441,404],{},[61,16443,927],{},[276,16445,278],{},[54,16447,16448,16460],{},[51,16449,16450,16452,16458],{},[276,16451,1028],{"fence":73},[54,16453,16454,16456],{"linethickness":4321},[61,16455,927],{},[57,16457,13490],{},[276,16459,1037],{"fence":73},[61,16461,1048],{},[65,16463,16464],{"encoding":67},"P(k \\text{ caras}) = \\frac{\\binom{3}{k}}{2^3} = \\frac{\\binom{3}{k}}{8}",[35,16466,16468,16501,16679],{"className":16467,"ariaHidden":73},[72],[35,16469,16471,16474,16477,16480,16483,16489,16492,16495,16498],{"className":16470},[77],[35,16472],{"className":16473,"style":9837},[81],[35,16475,16246],{"className":16476,"style":16272},[86,156],[35,16478,1028],{"className":16479},[90],[35,16481,13490],{"className":16482,"style":13542},[86,156],[35,16484,16486],{"className":16485},[86,625],[35,16487,16419],{"className":16488},[86],[35,16490,1037],{"className":16491},[175],[35,16493],{"className":16494,"style":367},[366],[35,16496,278],{"className":16497},[371],[35,16499],{"className":16500,"style":367},[366],[35,16502,16504,16507,16670,16673,16676],{"className":16503},[77],[35,16505],{"className":16506,"style":11305},[81],[35,16508,16510,16513,16667],{"className":16509},[86],[35,16511],{"className":16512},[90,91],[35,16514,16516],{"className":16515},[54],[35,16517,16519,16659],{"className":16518},[98,99],[35,16520,16522,16656],{"className":16521},[103],[35,16523,16525,16565,16573],{"className":16524,"style":10248},[107],[35,16526,16527,16530],{"style":111},[35,16528],{"className":16529,"style":116},[115],[35,16531,16533],{"className":16532},[120,121,122,123],[35,16534,16536],{"className":16535},[86,123],[35,16537,16539,16542],{"className":16538},[86,123],[35,16540,404],{"className":16541},[86,123],[35,16543,16545],{"className":16544},[1156],[35,16546,16548],{"className":16547},[98],[35,16549,16551],{"className":16550},[103],[35,16552,16554],{"className":16553,"style":7521},[107],[35,16555,16556,16559],{"style":7524},[35,16557],{"className":16558,"style":7528},[115],[35,16560,16562],{"className":16561},[120,7532,1078,123],[35,16563,927],{"className":16564},[86,123],[35,16566,16567,16570],{"style":132},[35,16568],{"className":16569,"style":116},[115],[35,16571],{"className":16572,"style":140},[139],[35,16574,16575,16578],{"style":10282},[35,16576],{"className":16577,"style":116},[115],[35,16579,16581],{"className":16580},[120,121,122,123],[35,16582,16584],{"className":16583},[86,123],[35,16585,16587,16596,16647],{"className":16586},[86,123],[35,16588,16590],{"className":16589,"style":10299},[90,120,7532,10298,123,1072],[35,16591,16593],{"className":16592},[1077,1078,123],[35,16594,1028],{"className":16595},[123],[35,16597,16599],{"className":16598},[54],[35,16600,16602,16639],{"className":16601},[98,99],[35,16603,16605,16636],{"className":16604},[103],[35,16606,16608,16622],{"className":16607,"style":10318},[107],[35,16609,16610,16613],{"style":10321},[35,16611],{"className":16612,"style":7528},[115],[35,16614,16616],{"className":16615},[120,7532,1078,123],[35,16617,16619],{"className":16618},[86,123],[35,16620,13490],{"className":16621,"style":13542},[86,156,123],[35,16623,16624,16627],{"style":10336},[35,16625],{"className":16626,"style":7528},[115],[35,16628,16630],{"className":16629},[120,7532,1078,123],[35,16631,16633],{"className":16632},[86,123],[35,16634,927],{"className":16635},[86,123],[35,16637,162],{"className":16638},[161],[35,16640,16642],{"className":16641},[103],[35,16643,16645],{"className":16644,"style":10361},[107],[35,16646],{},[35,16648,16650],{"className":16649,"style":10299},[175,120,7532,10298,123,1072],[35,16651,16653],{"className":16652},[1077,1078,123],[35,16654,1037],{"className":16655},[123],[35,16657,162],{"className":16658},[161],[35,16660,16662],{"className":16661},[103],[35,16663,16665],{"className":16664,"style":169},[107],[35,16666],{},[35,16668],{"className":16669},[175,91],[35,16671],{"className":16672,"style":367},[366],[35,16674,278],{"className":16675},[371],[35,16677],{"className":16678,"style":367},[366],[35,16680,16682,16685],{"className":16681},[77],[35,16683],{"className":16684,"style":11305},[81],[35,16686,16688,16691,16819],{"className":16687},[86],[35,16689],{"className":16690},[90,91],[35,16692,16694],{"className":16693},[54],[35,16695,16697,16811],{"className":16696},[98,99],[35,16698,16700,16808],{"className":16699},[103],[35,16701,16703,16717,16725],{"className":16702,"style":10248},[107],[35,16704,16705,16708],{"style":111},[35,16706],{"className":16707,"style":116},[115],[35,16709,16711],{"className":16710},[120,121,122,123],[35,16712,16714],{"className":16713},[86,123],[35,16715,1048],{"className":16716},[86,123],[35,16718,16719,16722],{"style":132},[35,16720],{"className":16721,"style":116},[115],[35,16723],{"className":16724,"style":140},[139],[35,16726,16727,16730],{"style":10282},[35,16728],{"className":16729,"style":116},[115],[35,16731,16733],{"className":16732},[120,121,122,123],[35,16734,16736],{"className":16735},[86,123],[35,16737,16739,16748,16799],{"className":16738},[86,123],[35,16740,16742],{"className":16741,"style":10299},[90,120,7532,10298,123,1072],[35,16743,16745],{"className":16744},[1077,1078,123],[35,16746,1028],{"className":16747},[123],[35,16749,16751],{"className":16750},[54],[35,16752,16754,16791],{"className":16753},[98,99],[35,16755,16757,16788],{"className":16756},[103],[35,16758,16760,16774],{"className":16759,"style":10318},[107],[35,16761,16762,16765],{"style":10321},[35,16763],{"className":16764,"style":7528},[115],[35,16766,16768],{"className":16767},[120,7532,1078,123],[35,16769,16771],{"className":16770},[86,123],[35,16772,13490],{"className":16773,"style":13542},[86,156,123],[35,16775,16776,16779],{"style":10336},[35,16777],{"className":16778,"style":7528},[115],[35,16780,16782],{"className":16781},[120,7532,1078,123],[35,16783,16785],{"className":16784},[86,123],[35,16786,927],{"className":16787},[86,123],[35,16789,162],{"className":16790},[161],[35,16792,16794],{"className":16793},[103],[35,16795,16797],{"className":16796,"style":10361},[107],[35,16798],{},[35,16800,16802],{"className":16801,"style":10299},[175,120,7532,10298,123,1072],[35,16803,16805],{"className":16804},[1077,1078,123],[35,16806,1037],{"className":16807},[123],[35,16809,162],{"className":16810},[161],[35,16812,16814],{"className":16813},[103],[35,16815,16817],{"className":16816,"style":169},[107],[35,16818],{},[35,16820],{"className":16821},[175,91],[17,16823,16824],{},"Calculemos cada caso:",[2168,16826,16827,16970],{},[2171,16828,16829],{},[2174,16830,16831,16862,16964,16967],{},[2177,16832,16833,16861],{"align":2179},[35,16834,16836,16849],{"className":16835},[38],[35,16837,16839],{"className":16838},[42],[44,16840,16841],{"xmlns":46},[48,16842,16843,16847],{},[51,16844,16845],{},[57,16846,13490],{},[65,16848,13490],{"encoding":67},[35,16850,16852],{"className":16851,"ariaHidden":73},[72],[35,16853,16855,16858],{"className":16854},[77],[35,16856],{"className":16857,"style":7026},[81],[35,16859,13490],{"className":16860,"style":13542},[86,156]," caras",[2177,16863,16864],{"align":2179},[35,16865,16867,16889],{"className":16866},[38],[35,16868,16870],{"className":16869},[42],[44,16871,16872],{"xmlns":46},[48,16873,16874,16886],{},[51,16875,16876,16878,16884],{},[276,16877,1028],{"fence":73},[54,16879,16880,16882],{"linethickness":4321},[61,16881,927],{},[57,16883,13490],{},[276,16885,1037],{"fence":73},[65,16887,16888],{"encoding":67},"\\binom{3}{k}",[35,16890,16892],{"className":16891,"ariaHidden":73},[72],[35,16893,16895,16898],{"className":16894},[77],[35,16896],{"className":16897,"style":4592},[81],[35,16899,16901,16907,16958],{"className":16900},[86],[35,16902,16904],{"className":16903,"style":1073},[90,1072],[35,16905,1028],{"className":16906},[1077,1078],[35,16908,16910],{"className":16909},[54],[35,16911,16913,16950],{"className":16912},[98,99],[35,16914,16916,16947],{"className":16915},[103],[35,16917,16919,16933],{"className":16918,"style":4500},[107],[35,16920,16921,16924],{"style":4503},[35,16922],{"className":16923,"style":1173},[115],[35,16925,16927],{"className":16926},[120,121,122,123],[35,16928,16930],{"className":16929},[86,123],[35,16931,13490],{"className":16932,"style":13542},[86,156,123],[35,16934,16935,16938],{"style":4518},[35,16936],{"className":16937,"style":1173},[115],[35,16939,16941],{"className":16940},[120,121,122,123],[35,16942,16944],{"className":16943},[86,123],[35,16945,927],{"className":16946},[86,123],[35,16948,162],{"className":16949},[161],[35,16951,16953],{"className":16952},[103],[35,16954,16956],{"className":16955,"style":169},[107],[35,16957],{},[35,16959,16961],{"className":16960,"style":1073},[175,1072],[35,16962,1037],{"className":16963},[1077,1078],[2177,16965,16966],{"align":2179},"Probabilidad",[2177,16968,16969],{"align":2179},"Porcentaje",[2250,16971,16972,17114,17256,17394],{},[2174,16973,16974,16976,16978,17111],{},[2255,16975,517],{"align":2179},[2255,16977,272],{"align":2179},[2255,16979,16980],{"align":2179},[35,16981,16983,17006],{"className":16982},[38],[35,16984,16986],{"className":16985},[42],[44,16987,16988],{"xmlns":46},[48,16989,16990,17003],{},[51,16991,16992,16998,17000],{},[54,16993,16994,16996],{},[61,16995,272],{},[61,16997,1048],{},[276,16999,278],{},[61,17001,17002],{},"0,125",[65,17004,17005],{"encoding":67},"\\frac{1}{8} = 0{,}125",[35,17007,17009,17092],{"className":17008,"ariaHidden":73},[72],[35,17010,17012,17015,17083,17086,17089],{"className":17011},[77],[35,17013],{"className":17014,"style":293},[81],[35,17016,17018,17021,17080],{"className":17017},[86],[35,17019],{"className":17020},[90,91],[35,17022,17024],{"className":17023},[54],[35,17025,17027,17072],{"className":17026},[98,99],[35,17028,17030,17069],{"className":17029},[103],[35,17031,17033,17047,17055],{"className":17032,"style":312},[107],[35,17034,17035,17038],{"style":111},[35,17036],{"className":17037,"style":116},[115],[35,17039,17041],{"className":17040},[120,121,122,123],[35,17042,17044],{"className":17043},[86,123],[35,17045,1048],{"className":17046},[86,123],[35,17048,17049,17052],{"style":132},[35,17050],{"className":17051,"style":116},[115],[35,17053],{"className":17054,"style":140},[139],[35,17056,17057,17060],{"style":143},[35,17058],{"className":17059,"style":116},[115],[35,17061,17063],{"className":17062},[120,121,122,123],[35,17064,17066],{"className":17065},[86,123],[35,17067,272],{"className":17068},[86,123],[35,17070,162],{"className":17071},[161],[35,17073,17075],{"className":17074},[103],[35,17076,17078],{"className":17077,"style":169},[107],[35,17079],{},[35,17081],{"className":17082},[175,91],[35,17084],{"className":17085,"style":367},[366],[35,17087,278],{"className":17088},[371],[35,17090],{"className":17091,"style":367},[366],[35,17093,17095,17098,17101,17107],{"className":17094},[77],[35,17096],{"className":17097,"style":513},[81],[35,17099,517],{"className":17100},[86],[35,17102,17104],{"className":17103},[86],[35,17105,525],{"className":17106},[524],[35,17108,17110],{"className":17109},[86],"125",[2255,17112,17113],{"align":2179},"12,5 %",[2174,17115,17116,17118,17120,17253],{},[2255,17117,272],{"align":2179},[2255,17119,927],{"align":2179},[2255,17121,17122],{"align":2179},[35,17123,17125,17148],{"className":17124},[38],[35,17126,17128],{"className":17127},[42],[44,17129,17130],{"xmlns":46},[48,17131,17132,17145],{},[51,17133,17134,17140,17142],{},[54,17135,17136,17138],{},[61,17137,927],{},[61,17139,1048],{},[276,17141,278],{},[61,17143,17144],{},"0,375",[65,17146,17147],{"encoding":67},"\\frac{3}{8} = 0{,}375",[35,17149,17151,17234],{"className":17150,"ariaHidden":73},[72],[35,17152,17154,17157,17225,17228,17231],{"className":17153},[77],[35,17155],{"className":17156,"style":293},[81],[35,17158,17160,17163,17222],{"className":17159},[86],[35,17161],{"className":17162},[90,91],[35,17164,17166],{"className":17165},[54],[35,17167,17169,17214],{"className":17168},[98,99],[35,17170,17172,17211],{"className":17171},[103],[35,17173,17175,17189,17197],{"className":17174,"style":312},[107],[35,17176,17177,17180],{"style":111},[35,17178],{"className":17179,"style":116},[115],[35,17181,17183],{"className":17182},[120,121,122,123],[35,17184,17186],{"className":17185},[86,123],[35,17187,1048],{"className":17188},[86,123],[35,17190,17191,17194],{"style":132},[35,17192],{"className":17193,"style":116},[115],[35,17195],{"className":17196,"style":140},[139],[35,17198,17199,17202],{"style":143},[35,17200],{"className":17201,"style":116},[115],[35,17203,17205],{"className":17204},[120,121,122,123],[35,17206,17208],{"className":17207},[86,123],[35,17209,927],{"className":17210},[86,123],[35,17212,162],{"className":17213},[161],[35,17215,17217],{"className":17216},[103],[35,17218,17220],{"className":17219,"style":169},[107],[35,17221],{},[35,17223],{"className":17224},[175,91],[35,17226],{"className":17227,"style":367},[366],[35,17229,278],{"className":17230},[371],[35,17232],{"className":17233,"style":367},[366],[35,17235,17237,17240,17243,17249],{"className":17236},[77],[35,17238],{"className":17239,"style":513},[81],[35,17241,517],{"className":17242},[86],[35,17244,17246],{"className":17245},[86],[35,17247,525],{"className":17248},[524],[35,17250,17252],{"className":17251},[86],"375",[2255,17254,17255],{"align":2179},"37,5 %",[2174,17257,17258,17260,17262,17392],{},[2255,17259,404],{"align":2179},[2255,17261,927],{"align":2179},[2255,17263,17264],{"align":2179},[35,17265,17267,17288],{"className":17266},[38],[35,17268,17270],{"className":17269},[42],[44,17271,17272],{"xmlns":46},[48,17273,17274,17286],{},[51,17275,17276,17282,17284],{},[54,17277,17278,17280],{},[61,17279,927],{},[61,17281,1048],{},[276,17283,278],{},[61,17285,17144],{},[65,17287,17147],{"encoding":67},[35,17289,17291,17374],{"className":17290,"ariaHidden":73},[72],[35,17292,17294,17297,17365,17368,17371],{"className":17293},[77],[35,17295],{"className":17296,"style":293},[81],[35,17298,17300,17303,17362],{"className":17299},[86],[35,17301],{"className":17302},[90,91],[35,17304,17306],{"className":17305},[54],[35,17307,17309,17354],{"className":17308},[98,99],[35,17310,17312,17351],{"className":17311},[103],[35,17313,17315,17329,17337],{"className":17314,"style":312},[107],[35,17316,17317,17320],{"style":111},[35,17318],{"className":17319,"style":116},[115],[35,17321,17323],{"className":17322},[120,121,122,123],[35,17324,17326],{"className":17325},[86,123],[35,17327,1048],{"className":17328},[86,123],[35,17330,17331,17334],{"style":132},[35,17332],{"className":17333,"style":116},[115],[35,17335],{"className":17336,"style":140},[139],[35,17338,17339,17342],{"style":143},[35,17340],{"className":17341,"style":116},[115],[35,17343,17345],{"className":17344},[120,121,122,123],[35,17346,17348],{"className":17347},[86,123],[35,17349,927],{"className":17350},[86,123],[35,17352,162],{"className":17353},[161],[35,17355,17357],{"className":17356},[103],[35,17358,17360],{"className":17359,"style":169},[107],[35,17361],{},[35,17363],{"className":17364},[175,91],[35,17366],{"className":17367,"style":367},[366],[35,17369,278],{"className":17370},[371],[35,17372],{"className":17373,"style":367},[366],[35,17375,17377,17380,17383,17389],{"className":17376},[77],[35,17378],{"className":17379,"style":513},[81],[35,17381,517],{"className":17382},[86],[35,17384,17386],{"className":17385},[86],[35,17387,525],{"className":17388},[524],[35,17390,17252],{"className":17391},[86],[2255,17393,17255],{"align":2179},[2174,17395,17396,17398,17400,17530],{},[2255,17397,927],{"align":2179},[2255,17399,272],{"align":2179},[2255,17401,17402],{"align":2179},[35,17403,17405,17426],{"className":17404},[38],[35,17406,17408],{"className":17407},[42],[44,17409,17410],{"xmlns":46},[48,17411,17412,17424],{},[51,17413,17414,17420,17422],{},[54,17415,17416,17418],{},[61,17417,272],{},[61,17419,1048],{},[276,17421,278],{},[61,17423,17002],{},[65,17425,17005],{"encoding":67},[35,17427,17429,17512],{"className":17428,"ariaHidden":73},[72],[35,17430,17432,17435,17503,17506,17509],{"className":17431},[77],[35,17433],{"className":17434,"style":293},[81],[35,17436,17438,17441,17500],{"className":17437},[86],[35,17439],{"className":17440},[90,91],[35,17442,17444],{"className":17443},[54],[35,17445,17447,17492],{"className":17446},[98,99],[35,17448,17450,17489],{"className":17449},[103],[35,17451,17453,17467,17475],{"className":17452,"style":312},[107],[35,17454,17455,17458],{"style":111},[35,17456],{"className":17457,"style":116},[115],[35,17459,17461],{"className":17460},[120,121,122,123],[35,17462,17464],{"className":17463},[86,123],[35,17465,1048],{"className":17466},[86,123],[35,17468,17469,17472],{"style":132},[35,17470],{"className":17471,"style":116},[115],[35,17473],{"className":17474,"style":140},[139],[35,17476,17477,17480],{"style":143},[35,17478],{"className":17479,"style":116},[115],[35,17481,17483],{"className":17482},[120,121,122,123],[35,17484,17486],{"className":17485},[86,123],[35,17487,272],{"className":17488},[86,123],[35,17490,162],{"className":17491},[161],[35,17493,17495],{"className":17494},[103],[35,17496,17498],{"className":17497,"style":169},[107],[35,17499],{},[35,17501],{"className":17502},[175,91],[35,17504],{"className":17505,"style":367},[366],[35,17507,278],{"className":17508},[371],[35,17510],{"className":17511,"style":367},[366],[35,17513,17515,17518,17521,17527],{"className":17514},[77],[35,17516],{"className":17517,"style":513},[81],[35,17519,517],{"className":17520},[86],[35,17522,17524],{"className":17523},[86],[35,17525,525],{"className":17526},[524],[35,17528,17110],{"className":17529},[86],[2255,17531,17113],{"align":2179},[17,17533,17534,17535,17688,17689,17717],{},"Si sumamos todas las probabilidades: ",[35,17536,17538,17568],{"className":17537},[38],[35,17539,17541],{"className":17540},[42],[44,17542,17543],{"xmlns":46},[48,17544,17545,17565],{},[51,17546,17547,17549,17551,17553,17555,17557,17559,17561,17563],{},[61,17548,17002],{},[276,17550,1316],{},[61,17552,17144],{},[276,17554,1316],{},[61,17556,17144],{},[276,17558,1316],{},[61,17560,17002],{},[276,17562,278],{},[61,17564,272],{},[65,17566,17567],{"encoding":67},"0{,}125 + 0{,}375 + 0{,}375 + 0{,}125 = 1",[35,17569,17571,17598,17625,17652,17679],{"className":17570,"ariaHidden":73},[72],[35,17572,17574,17577,17580,17586,17589,17592,17595],{"className":17573},[77],[35,17575],{"className":17576,"style":513},[81],[35,17578,517],{"className":17579},[86],[35,17581,17583],{"className":17582},[86],[35,17584,525],{"className":17585},[524],[35,17587,17110],{"className":17588},[86],[35,17590],{"className":17591,"style":2223},[366],[35,17593,1316],{"className":17594},[2227],[35,17596],{"className":17597,"style":2223},[366],[35,17599,17601,17604,17607,17613,17616,17619,17622],{"className":17600},[77],[35,17602],{"className":17603,"style":513},[81],[35,17605,517],{"className":17606},[86],[35,17608,17610],{"className":17609},[86],[35,17611,525],{"className":17612},[524],[35,17614,17252],{"className":17615},[86],[35,17617],{"className":17618,"style":2223},[366],[35,17620,1316],{"className":17621},[2227],[35,17623],{"className":17624,"style":2223},[366],[35,17626,17628,17631,17634,17640,17643,17646,17649],{"className":17627},[77],[35,17629],{"className":17630,"style":513},[81],[35,17632,517],{"className":17633},[86],[35,17635,17637],{"className":17636},[86],[35,17638,525],{"className":17639},[524],[35,17641,17252],{"className":17642},[86],[35,17644],{"className":17645,"style":2223},[366],[35,17647,1316],{"className":17648},[2227],[35,17650],{"className":17651,"style":2223},[366],[35,17653,17655,17658,17661,17667,17670,17673,17676],{"className":17654},[77],[35,17656],{"className":17657,"style":513},[81],[35,17659,517],{"className":17660},[86],[35,17662,17664],{"className":17663},[86],[35,17665,525],{"className":17666},[524],[35,17668,17110],{"className":17669},[86],[35,17671],{"className":17672,"style":367},[366],[35,17674,278],{"className":17675},[371],[35,17677],{"className":17678,"style":367},[366],[35,17680,17682,17685],{"className":17681},[77],[35,17683],{"className":17684,"style":381},[81],[35,17686,272],{"className":17687},[86],". La suma siempre da ",[35,17690,17692,17705],{"className":17691},[38],[35,17693,17695],{"className":17694},[42],[44,17696,17697],{"xmlns":46},[48,17698,17699,17703],{},[51,17700,17701],{},[61,17702,272],{},[65,17704,272],{"encoding":67},[35,17706,17708],{"className":17707,"ariaHidden":73},[72],[35,17709,17711,17714],{"className":17710},[77],[35,17712],{"className":17713,"style":381},[81],[35,17715,272],{"className":17716},[86],", porque entre todos los casos se cubre el 100 % de las posibilidades.",[17,17719,17720],{},"La fórmula general que acabamos de descubrir funciona para cualquier número de lanzamientos:",[17,17722,17724],{"className":17723},[592],[35,17725,17727,17777],{"className":17726},[38],[35,17728,17730],{"className":17729},[42],[44,17731,17732],{"xmlns":46},[48,17733,17734,17774],{},[51,17735,17736,17738,17740,17742,17745,17747,17750,17752,17754],{},[57,17737,16246],{},[276,17739,1028],{"stretchy":9816},[57,17741,13490],{},[606,17743,17744],{}," caras en ",[57,17746,2197],{},[606,17748,17749],{}," lanzamientos",[276,17751,1037],{"stretchy":9816},[276,17753,278],{},[54,17755,17756,17768],{},[51,17757,17758,17760,17766],{},[276,17759,1028],{"fence":73},[54,17761,17762,17764],{"linethickness":4321},[57,17763,2197],{},[57,17765,13490],{},[276,17767,1037],{"fence":73},[1022,17769,17770,17772],{},[61,17771,404],{},[57,17773,2197],{},[65,17775,17776],{"encoding":67},"P(k \\text{ caras en } n \\text{ lanzamientos}) = \\frac{\\binom{n}{k}}{2^n}",[35,17778,17780,17822],{"className":17779,"ariaHidden":73},[72],[35,17781,17783,17786,17789,17792,17795,17801,17804,17810,17813,17816,17819],{"className":17782},[77],[35,17784],{"className":17785,"style":9837},[81],[35,17787,16246],{"className":17788,"style":16272},[86,156],[35,17790,1028],{"className":17791},[90],[35,17793,13490],{"className":17794,"style":13542},[86,156],[35,17796,17798],{"className":17797},[86,625],[35,17799,17744],{"className":17800},[86],[35,17802,2197],{"className":17803},[86,156],[35,17805,17807],{"className":17806},[86,625],[35,17808,17749],{"className":17809},[86],[35,17811,1037],{"className":17812},[175],[35,17814],{"className":17815,"style":367},[366],[35,17817,278],{"className":17818},[371],[35,17820],{"className":17821,"style":367},[366],[35,17823,17825,17828],{"className":17824},[77],[35,17826],{"className":17827,"style":11305},[81],[35,17829,17831,17834,17989],{"className":17830},[86],[35,17832],{"className":17833},[90,91],[35,17835,17837],{"className":17836},[54],[35,17838,17840,17981],{"className":17839},[98,99],[35,17841,17843,17978],{"className":17842},[103],[35,17844,17846,17886,17894],{"className":17845,"style":10248},[107],[35,17847,17848,17851],{"style":111},[35,17849],{"className":17850,"style":116},[115],[35,17852,17854],{"className":17853},[120,121,122,123],[35,17855,17857],{"className":17856},[86,123],[35,17858,17860,17863],{"className":17859},[86,123],[35,17861,404],{"className":17862},[86,123],[35,17864,17866],{"className":17865},[1156],[35,17867,17869],{"className":17868},[98],[35,17870,17872],{"className":17871},[103],[35,17873,17875],{"className":17874,"style":11025},[107],[35,17876,17877,17880],{"style":7524},[35,17878],{"className":17879,"style":7528},[115],[35,17881,17883],{"className":17882},[120,7532,1078,123],[35,17884,2197],{"className":17885},[86,156,123],[35,17887,17888,17891],{"style":132},[35,17889],{"className":17890,"style":116},[115],[35,17892],{"className":17893,"style":140},[139],[35,17895,17896,17899],{"style":10282},[35,17897],{"className":17898,"style":116},[115],[35,17900,17902],{"className":17901},[120,121,122,123],[35,17903,17905],{"className":17904},[86,123],[35,17906,17908,17917,17969],{"className":17907},[86,123],[35,17909,17911],{"className":17910,"style":10299},[90,120,7532,10298,123,1072],[35,17912,17914],{"className":17913},[1077,1078,123],[35,17915,1028],{"className":17916},[123],[35,17918,17920],{"className":17919},[54],[35,17921,17923,17961],{"className":17922},[98,99],[35,17924,17926,17958],{"className":17925},[103],[35,17927,17930,17944],{"className":17928,"style":17929},[107],"height:0.7785em;",[35,17931,17932,17935],{"style":10321},[35,17933],{"className":17934,"style":7528},[115],[35,17936,17938],{"className":17937},[120,7532,1078,123],[35,17939,17941],{"className":17940},[86,123],[35,17942,13490],{"className":17943,"style":13542},[86,156,123],[35,17945,17946,17949],{"style":10336},[35,17947],{"className":17948,"style":7528},[115],[35,17950,17952],{"className":17951},[120,7532,1078,123],[35,17953,17955],{"className":17954},[86,123],[35,17956,2197],{"className":17957},[86,156,123],[35,17959,162],{"className":17960},[161],[35,17962,17964],{"className":17963},[103],[35,17965,17967],{"className":17966,"style":10361},[107],[35,17968],{},[35,17970,17972],{"className":17971,"style":10299},[175,120,7532,10298,123,1072],[35,17973,17975],{"className":17974},[1077,1078,123],[35,17976,1037],{"className":17977},[123],[35,17979,162],{"className":17980},[161],[35,17982,17984],{"className":17983},[103],[35,17985,17987],{"className":17986,"style":169},[107],[35,17988],{},[35,17990],{"className":17991},[175,91],[561,17993,17995],{"id":17994},"caso-práctico-5-lanzamientos","Caso práctico: 5 lanzamientos",[17,17997,17998],{},"Subamos la apuesta. Con 5 lanzamientos, la fila correspondiente del triángulo es:",[17,18000,18002],{"className":18001},[592],[35,18003,18005,18040],{"className":18004},[38],[35,18006,18008],{"className":18007},[42],[44,18009,18010],{"xmlns":46},[48,18011,18012,18037],{},[51,18013,18014,18016,18019,18021,18023,18025,18027,18029,18031,18033,18035],{},[61,18015,272],{},[366,18017],{"width":18018},"1em",[61,18020,529],{},[366,18022],{"width":18018},[61,18024,15755],{},[366,18026],{"width":18018},[61,18028,15755],{},[366,18030],{"width":18018},[61,18032,529],{},[366,18034],{"width":18018},[61,18036,272],{},[65,18038,18039],{"encoding":67},"1 \\quad 5 \\quad 10 \\quad 10 \\quad 5 \\quad 1",[35,18041,18043],{"className":18042,"ariaHidden":73},[72],[35,18044,18046,18049,18052,18056,18059,18062,18065,18068,18071,18074,18077,18080],{"className":18045},[77],[35,18047],{"className":18048,"style":381},[81],[35,18050,272],{"className":18051},[86],[35,18053],{"className":18054,"style":18055},[366],"margin-right:1em;",[35,18057,529],{"className":18058},[86],[35,18060],{"className":18061,"style":18055},[366],[35,18063,15755],{"className":18064},[86],[35,18066],{"className":18067,"style":18055},[366],[35,18069,15755],{"className":18070},[86],[35,18072],{"className":18073,"style":18055},[366],[35,18075,529],{"className":18076},[86],[35,18078],{"className":18079,"style":18055},[366],[35,18081,272],{"className":18082},[86],[17,18084,18085,18086,176],{},"y el total de resultados posibles es ",[35,18087,18089,18111],{"className":18088},[38],[35,18090,18092],{"className":18091},[42],[44,18093,18094],{"xmlns":46},[48,18095,18096,18108],{},[51,18097,18098,18104,18106],{},[1022,18099,18100,18102],{},[61,18101,404],{},[61,18103,529],{},[276,18105,278],{},[61,18107,1847],{},[65,18109,18110],{"encoding":67},"2^5 = 32",[35,18112,18114,18158],{"className":18113,"ariaHidden":73},[72],[35,18115,18117,18120,18149,18152,18155],{"className":18116},[77],[35,18118],{"className":18119,"style":15811},[81],[35,18121,18123,18126],{"className":18122},[86],[35,18124,404],{"className":18125},[86],[35,18127,18129],{"className":18128},[1156],[35,18130,18132],{"className":18131},[98],[35,18133,18135],{"className":18134},[103],[35,18136,18138],{"className":18137,"style":15811},[107],[35,18139,18140,18143],{"style":15557},[35,18141],{"className":18142,"style":1173},[115],[35,18144,18146],{"className":18145},[120,121,122,123],[35,18147,529],{"className":18148},[86,123],[35,18150],{"className":18151,"style":367},[366],[35,18153,278],{"className":18154},[371],[35,18156],{"className":18157,"style":367},[366],[35,18159,18161,18164],{"className":18160},[77],[35,18162],{"className":18163,"style":381},[81],[35,18165,1847],{"className":18166},[86],[17,18168,18169,18170,18173,18174,18298],{},"Veamos algunos ejemplos concretos. ¿De cuántas formas podemos obtener exactamente ",[239,18171,18172],{},"2 caras","? El triángulo nos dice ",[35,18175,18177,18205],{"className":18176},[38],[35,18178,18180],{"className":18179},[42],[44,18181,18182],{"xmlns":46},[48,18183,18184,18202],{},[51,18185,18186,18198,18200],{},[51,18187,18188,18190,18196],{},[276,18189,1028],{"fence":73},[54,18191,18192,18194],{"linethickness":4321},[61,18193,529],{},[61,18195,404],{},[276,18197,1037],{"fence":73},[276,18199,278],{},[61,18201,15755],{},[65,18203,18204],{"encoding":67},"\\binom{5}{2} = 10",[35,18206,18208,18289],{"className":18207,"ariaHidden":73},[72],[35,18209,18211,18214,18280,18283,18286],{"className":18210},[77],[35,18212],{"className":18213,"style":4592},[81],[35,18215,18217,18223,18274],{"className":18216},[86],[35,18218,18220],{"className":18219,"style":1073},[90,1072],[35,18221,1028],{"className":18222},[1077,1078],[35,18224,18226],{"className":18225},[54],[35,18227,18229,18266],{"className":18228},[98,99],[35,18230,18232,18263],{"className":18231},[103],[35,18233,18235,18249],{"className":18234,"style":4500},[107],[35,18236,18237,18240],{"style":4503},[35,18238],{"className":18239,"style":1173},[115],[35,18241,18243],{"className":18242},[120,121,122,123],[35,18244,18246],{"className":18245},[86,123],[35,18247,404],{"className":18248},[86,123],[35,18250,18251,18254],{"style":4518},[35,18252],{"className":18253,"style":1173},[115],[35,18255,18257],{"className":18256},[120,121,122,123],[35,18258,18260],{"className":18259},[86,123],[35,18261,529],{"className":18262},[86,123],[35,18264,162],{"className":18265},[161],[35,18267,18269],{"className":18268},[103],[35,18270,18272],{"className":18271,"style":169},[107],[35,18273],{},[35,18275,18277],{"className":18276,"style":1073},[175,1072],[35,18278,1037],{"className":18279},[1077,1078],[35,18281],{"className":18282,"style":367},[366],[35,18284,278],{"className":18285},[371],[35,18287],{"className":18288,"style":367},[366],[35,18290,18292,18295],{"className":18291},[77],[35,18293],{"className":18294,"style":381},[81],[35,18296,15755],{"className":18297},[86],". Estas son las 10 combinaciones:",[1286,18300,18303],{"className":18301,"code":18302,"language":625,"meta":1291},[1289],"OOXXX   OXOXX   OXXOX   OXXXO   XOOXX\nXOXOX   XOXXO   XXOOX   XXOXO   XXXOO\n",[1293,18304,18302],{"__ignoreMap":1291},[17,18306,18307,18308,18311,18312,18436],{},"¿Y exactamente ",[239,18309,18310],{},"3 caras","? También ",[35,18313,18315,18343],{"className":18314},[38],[35,18316,18318],{"className":18317},[42],[44,18319,18320],{"xmlns":46},[48,18321,18322,18340],{},[51,18323,18324,18336,18338],{},[51,18325,18326,18328,18334],{},[276,18327,1028],{"fence":73},[54,18329,18330,18332],{"linethickness":4321},[61,18331,529],{},[61,18333,927],{},[276,18335,1037],{"fence":73},[276,18337,278],{},[61,18339,15755],{},[65,18341,18342],{"encoding":67},"\\binom{5}{3} = 10",[35,18344,18346,18427],{"className":18345,"ariaHidden":73},[72],[35,18347,18349,18352,18418,18421,18424],{"className":18348},[77],[35,18350],{"className":18351,"style":4592},[81],[35,18353,18355,18361,18412],{"className":18354},[86],[35,18356,18358],{"className":18357,"style":1073},[90,1072],[35,18359,1028],{"className":18360},[1077,1078],[35,18362,18364],{"className":18363},[54],[35,18365,18367,18404],{"className":18366},[98,99],[35,18368,18370,18401],{"className":18369},[103],[35,18371,18373,18387],{"className":18372,"style":4500},[107],[35,18374,18375,18378],{"style":4503},[35,18376],{"className":18377,"style":1173},[115],[35,18379,18381],{"className":18380},[120,121,122,123],[35,18382,18384],{"className":18383},[86,123],[35,18385,927],{"className":18386},[86,123],[35,18388,18389,18392],{"style":4518},[35,18390],{"className":18391,"style":1173},[115],[35,18393,18395],{"className":18394},[120,121,122,123],[35,18396,18398],{"className":18397},[86,123],[35,18399,529],{"className":18400},[86,123],[35,18402,162],{"className":18403},[161],[35,18405,18407],{"className":18406},[103],[35,18408,18410],{"className":18409,"style":169},[107],[35,18411],{},[35,18413,18415],{"className":18414,"style":1073},[175,1072],[35,18416,1037],{"className":18417},[1077,1078],[35,18419],{"className":18420,"style":367},[366],[35,18422,278],{"className":18423},[371],[35,18425],{"className":18426,"style":367},[366],[35,18428,18430,18433],{"className":18429},[77],[35,18431],{"className":18432,"style":381},[81],[35,18434,15755],{"className":18435},[86],". El triángulo es simétrico: hay tantas formas de obtener 2 caras (y 3 cruces) como 3 caras (y 2 cruces).",[17,18438,18439],{},"La tabla completa de probabilidades:",[2168,18441,18442,18582],{},[2171,18443,18444],{},[2174,18445,18446,18476,18578,18580],{},[2177,18447,18448,16861],{"align":2179},[35,18449,18451,18464],{"className":18450},[38],[35,18452,18454],{"className":18453},[42],[44,18455,18456],{"xmlns":46},[48,18457,18458,18462],{},[51,18459,18460],{},[57,18461,13490],{},[65,18463,13490],{"encoding":67},[35,18465,18467],{"className":18466,"ariaHidden":73},[72],[35,18468,18470,18473],{"className":18469},[77],[35,18471],{"className":18472,"style":7026},[81],[35,18474,13490],{"className":18475,"style":13542},[86,156],[2177,18477,18478],{"align":2179},[35,18479,18481,18503],{"className":18480},[38],[35,18482,18484],{"className":18483},[42],[44,18485,18486],{"xmlns":46},[48,18487,18488,18500],{},[51,18489,18490,18492,18498],{},[276,18491,1028],{"fence":73},[54,18493,18494,18496],{"linethickness":4321},[61,18495,529],{},[57,18497,13490],{},[276,18499,1037],{"fence":73},[65,18501,18502],{"encoding":67},"\\binom{5}{k}",[35,18504,18506],{"className":18505,"ariaHidden":73},[72],[35,18507,18509,18512],{"className":18508},[77],[35,18510],{"className":18511,"style":4592},[81],[35,18513,18515,18521,18572],{"className":18514},[86],[35,18516,18518],{"className":18517,"style":1073},[90,1072],[35,18519,1028],{"className":18520},[1077,1078],[35,18522,18524],{"className":18523},[54],[35,18525,18527,18564],{"className":18526},[98,99],[35,18528,18530,18561],{"className":18529},[103],[35,18531,18533,18547],{"className":18532,"style":4500},[107],[35,18534,18535,18538],{"style":4503},[35,18536],{"className":18537,"style":1173},[115],[35,18539,18541],{"className":18540},[120,121,122,123],[35,18542,18544],{"className":18543},[86,123],[35,18545,13490],{"className":18546,"style":13542},[86,156,123],[35,18548,18549,18552],{"style":4518},[35,18550],{"className":18551,"style":1173},[115],[35,18553,18555],{"className":18554},[120,121,122,123],[35,18556,18558],{"className":18557},[86,123],[35,18559,529],{"className":18560},[86,123],[35,18562,162],{"className":18563},[161],[35,18565,18567],{"className":18566},[103],[35,18568,18570],{"className":18569,"style":169},[107],[35,18571],{},[35,18573,18575],{"className":18574,"style":1073},[175,1072],[35,18576,1037],{"className":18577},[1077,1078],[2177,18579,16966],{"align":2179},[2177,18581,16969],{"align":2179},[2250,18583,18584,18692,18801,18910,19017,19124],{},[2174,18585,18586,18588,18590,18689],{},[2255,18587,517],{"align":2179},[2255,18589,272],{"align":2179},[2255,18591,18592],{"align":2179},[35,18593,18595,18612],{"className":18594},[38],[35,18596,18598],{"className":18597},[42],[44,18599,18600],{"xmlns":46},[48,18601,18602,18610],{},[51,18603,18604],{},[54,18605,18606,18608],{},[61,18607,272],{},[61,18609,1847],{},[65,18611,2946],{"encoding":67},[35,18613,18615],{"className":18614,"ariaHidden":73},[72],[35,18616,18618,18621],{"className":18617},[77],[35,18619],{"className":18620,"style":293},[81],[35,18622,18624,18627,18686],{"className":18623},[86],[35,18625],{"className":18626},[90,91],[35,18628,18630],{"className":18629},[54],[35,18631,18633,18678],{"className":18632},[98,99],[35,18634,18636,18675],{"className":18635},[103],[35,18637,18639,18653,18661],{"className":18638,"style":312},[107],[35,18640,18641,18644],{"style":111},[35,18642],{"className":18643,"style":116},[115],[35,18645,18647],{"className":18646},[120,121,122,123],[35,18648,18650],{"className":18649},[86,123],[35,18651,1847],{"className":18652},[86,123],[35,18654,18655,18658],{"style":132},[35,18656],{"className":18657,"style":116},[115],[35,18659],{"className":18660,"style":140},[139],[35,18662,18663,18666],{"style":143},[35,18664],{"className":18665,"style":116},[115],[35,18667,18669],{"className":18668},[120,121,122,123],[35,18670,18672],{"className":18671},[86,123],[35,18673,272],{"className":18674},[86,123],[35,18676,162],{"className":18677},[161],[35,18679,18681],{"className":18680},[103],[35,18682,18684],{"className":18683,"style":169},[107],[35,18685],{},[35,18687],{"className":18688},[175,91],[2255,18690,18691],{"align":2179},"3,1 %",[2174,18693,18694,18696,18698,18798],{},[2255,18695,272],{"align":2179},[2255,18697,529],{"align":2179},[2255,18699,18700],{"align":2179},[35,18701,18703,18721],{"className":18702},[38],[35,18704,18706],{"className":18705},[42],[44,18707,18708],{"xmlns":46},[48,18709,18710,18718],{},[51,18711,18712],{},[54,18713,18714,18716],{},[61,18715,529],{},[61,18717,1847],{},[65,18719,18720],{"encoding":67},"\\frac{5}{32}",[35,18722,18724],{"className":18723,"ariaHidden":73},[72],[35,18725,18727,18730],{"className":18726},[77],[35,18728],{"className":18729,"style":293},[81],[35,18731,18733,18736,18795],{"className":18732},[86],[35,18734],{"className":18735},[90,91],[35,18737,18739],{"className":18738},[54],[35,18740,18742,18787],{"className":18741},[98,99],[35,18743,18745,18784],{"className":18744},[103],[35,18746,18748,18762,18770],{"className":18747,"style":312},[107],[35,18749,18750,18753],{"style":111},[35,18751],{"className":18752,"style":116},[115],[35,18754,18756],{"className":18755},[120,121,122,123],[35,18757,18759],{"className":18758},[86,123],[35,18760,1847],{"className":18761},[86,123],[35,18763,18764,18767],{"style":132},[35,18765],{"className":18766,"style":116},[115],[35,18768],{"className":18769,"style":140},[139],[35,18771,18772,18775],{"style":143},[35,18773],{"className":18774,"style":116},[115],[35,18776,18778],{"className":18777},[120,121,122,123],[35,18779,18781],{"className":18780},[86,123],[35,18782,529],{"className":18783},[86,123],[35,18785,162],{"className":18786},[161],[35,18788,18790],{"className":18789},[103],[35,18791,18793],{"className":18792,"style":169},[107],[35,18794],{},[35,18796],{"className":18797},[175,91],[2255,18799,18800],{"align":2179},"15,6 %",[2174,18802,18803,18805,18807,18907],{},[2255,18804,404],{"align":2179},[2255,18806,15755],{"align":2179},[2255,18808,18809],{"align":2179},[35,18810,18812,18830],{"className":18811},[38],[35,18813,18815],{"className":18814},[42],[44,18816,18817],{"xmlns":46},[48,18818,18819,18827],{},[51,18820,18821],{},[54,18822,18823,18825],{},[61,18824,15755],{},[61,18826,1847],{},[65,18828,18829],{"encoding":67},"\\frac{10}{32}",[35,18831,18833],{"className":18832,"ariaHidden":73},[72],[35,18834,18836,18839],{"className":18835},[77],[35,18837],{"className":18838,"style":293},[81],[35,18840,18842,18845,18904],{"className":18841},[86],[35,18843],{"className":18844},[90,91],[35,18846,18848],{"className":18847},[54],[35,18849,18851,18896],{"className":18850},[98,99],[35,18852,18854,18893],{"className":18853},[103],[35,18855,18857,18871,18879],{"className":18856,"style":312},[107],[35,18858,18859,18862],{"style":111},[35,18860],{"className":18861,"style":116},[115],[35,18863,18865],{"className":18864},[120,121,122,123],[35,18866,18868],{"className":18867},[86,123],[35,18869,1847],{"className":18870},[86,123],[35,18872,18873,18876],{"style":132},[35,18874],{"className":18875,"style":116},[115],[35,18877],{"className":18878,"style":140},[139],[35,18880,18881,18884],{"style":143},[35,18882],{"className":18883,"style":116},[115],[35,18885,18887],{"className":18886},[120,121,122,123],[35,18888,18890],{"className":18889},[86,123],[35,18891,15755],{"className":18892},[86,123],[35,18894,162],{"className":18895},[161],[35,18897,18899],{"className":18898},[103],[35,18900,18902],{"className":18901,"style":169},[107],[35,18903],{},[35,18905],{"className":18906},[175,91],[2255,18908,18909],{"align":2179},"31,3 %",[2174,18911,18912,18914,18916,19015],{},[2255,18913,927],{"align":2179},[2255,18915,15755],{"align":2179},[2255,18917,18918],{"align":2179},[35,18919,18921,18938],{"className":18920},[38],[35,18922,18924],{"className":18923},[42],[44,18925,18926],{"xmlns":46},[48,18927,18928,18936],{},[51,18929,18930],{},[54,18931,18932,18934],{},[61,18933,15755],{},[61,18935,1847],{},[65,18937,18829],{"encoding":67},[35,18939,18941],{"className":18940,"ariaHidden":73},[72],[35,18942,18944,18947],{"className":18943},[77],[35,18945],{"className":18946,"style":293},[81],[35,18948,18950,18953,19012],{"className":18949},[86],[35,18951],{"className":18952},[90,91],[35,18954,18956],{"className":18955},[54],[35,18957,18959,19004],{"className":18958},[98,99],[35,18960,18962,19001],{"className":18961},[103],[35,18963,18965,18979,18987],{"className":18964,"style":312},[107],[35,18966,18967,18970],{"style":111},[35,18968],{"className":18969,"style":116},[115],[35,18971,18973],{"className":18972},[120,121,122,123],[35,18974,18976],{"className":18975},[86,123],[35,18977,1847],{"className":18978},[86,123],[35,18980,18981,18984],{"style":132},[35,18982],{"className":18983,"style":116},[115],[35,18985],{"className":18986,"style":140},[139],[35,18988,18989,18992],{"style":143},[35,18990],{"className":18991,"style":116},[115],[35,18993,18995],{"className":18994},[120,121,122,123],[35,18996,18998],{"className":18997},[86,123],[35,18999,15755],{"className":19000},[86,123],[35,19002,162],{"className":19003},[161],[35,19005,19007],{"className":19006},[103],[35,19008,19010],{"className":19009,"style":169},[107],[35,19011],{},[35,19013],{"className":19014},[175,91],[2255,19016,18909],{"align":2179},[2174,19018,19019,19021,19023,19122],{},[2255,19020,63],{"align":2179},[2255,19022,529],{"align":2179},[2255,19024,19025],{"align":2179},[35,19026,19028,19045],{"className":19027},[38],[35,19029,19031],{"className":19030},[42],[44,19032,19033],{"xmlns":46},[48,19034,19035,19043],{},[51,19036,19037],{},[54,19038,19039,19041],{},[61,19040,529],{},[61,19042,1847],{},[65,19044,18720],{"encoding":67},[35,19046,19048],{"className":19047,"ariaHidden":73},[72],[35,19049,19051,19054],{"className":19050},[77],[35,19052],{"className":19053,"style":293},[81],[35,19055,19057,19060,19119],{"className":19056},[86],[35,19058],{"className":19059},[90,91],[35,19061,19063],{"className":19062},[54],[35,19064,19066,19111],{"className":19065},[98,99],[35,19067,19069,19108],{"className":19068},[103],[35,19070,19072,19086,19094],{"className":19071,"style":312},[107],[35,19073,19074,19077],{"style":111},[35,19075],{"className":19076,"style":116},[115],[35,19078,19080],{"className":19079},[120,121,122,123],[35,19081,19083],{"className":19082},[86,123],[35,19084,1847],{"className":19085},[86,123],[35,19087,19088,19091],{"style":132},[35,19089],{"className":19090,"style":116},[115],[35,19092],{"className":19093,"style":140},[139],[35,19095,19096,19099],{"style":143},[35,19097],{"className":19098,"style":116},[115],[35,19100,19102],{"className":19101},[120,121,122,123],[35,19103,19105],{"className":19104},[86,123],[35,19106,529],{"className":19107},[86,123],[35,19109,162],{"className":19110},[161],[35,19112,19114],{"className":19113},[103],[35,19115,19117],{"className":19116,"style":169},[107],[35,19118],{},[35,19120],{"className":19121},[175,91],[2255,19123,18800],{"align":2179},[2174,19125,19126,19128,19130,19229],{},[2255,19127,529],{"align":2179},[2255,19129,272],{"align":2179},[2255,19131,19132],{"align":2179},[35,19133,19135,19152],{"className":19134},[38],[35,19136,19138],{"className":19137},[42],[44,19139,19140],{"xmlns":46},[48,19141,19142,19150],{},[51,19143,19144],{},[54,19145,19146,19148],{},[61,19147,272],{},[61,19149,1847],{},[65,19151,2946],{"encoding":67},[35,19153,19155],{"className":19154,"ariaHidden":73},[72],[35,19156,19158,19161],{"className":19157},[77],[35,19159],{"className":19160,"style":293},[81],[35,19162,19164,19167,19226],{"className":19163},[86],[35,19165],{"className":19166},[90,91],[35,19168,19170],{"className":19169},[54],[35,19171,19173,19218],{"className":19172},[98,99],[35,19174,19176,19215],{"className":19175},[103],[35,19177,19179,19193,19201],{"className":19178,"style":312},[107],[35,19180,19181,19184],{"style":111},[35,19182],{"className":19183,"style":116},[115],[35,19185,19187],{"className":19186},[120,121,122,123],[35,19188,19190],{"className":19189},[86,123],[35,19191,1847],{"className":19192},[86,123],[35,19194,19195,19198],{"style":132},[35,19196],{"className":19197,"style":116},[115],[35,19199],{"className":19200,"style":140},[139],[35,19202,19203,19206],{"style":143},[35,19204],{"className":19205,"style":116},[115],[35,19207,19209],{"className":19208},[120,121,122,123],[35,19210,19212],{"className":19211},[86,123],[35,19213,272],{"className":19214},[86,123],[35,19216,162],{"className":19217},[161],[35,19219,19221],{"className":19220},[103],[35,19222,19224],{"className":19223,"style":169},[107],[35,19225],{},[35,19227],{"className":19228},[175,91],[2255,19230,18691],{"align":2179},[17,19232,19233],{},"Si lo dibujamos como un histograma, la distribución tiene una forma familiar:",[1286,19235,19238],{"className":19236,"code":19237,"language":625,"meta":1291},[1289],"0 caras  ▌              3,1 %\n1 cara   ▌▌▌▌▌         15,6 %\n2 caras  ▌▌▌▌▌▌▌▌▌▌    31,3 %\n3 caras  ▌▌▌▌▌▌▌▌▌▌    31,3 %\n4 caras  ▌▌▌▌▌         15,6 %\n5 caras  ▌              3,1 %\n",[1293,19239,19237],{"__ignoreMap":1291},[17,19241,19242],{},"Observa tres cosas:",[229,19244,19245,19251,19257],{},[232,19246,19247,19250],{},[239,19248,19249],{},"Simetría."," La distribución es un espejo perfecto alrededor del centro. Obtener 0 caras es tan improbable como obtener 5; obtener 1 es tan probable como obtener 4.",[232,19252,19253,19256],{},[239,19254,19255],{},"Concentración central."," Los resultados más probables están en el centro (2 o 3 caras), mientras que los extremos (todo caras o todo cruces) son muy raros.",[232,19258,19259,19262],{},[239,19260,19261],{},"Forma de campana."," La silueta recuerda a la famosa curva de Gauss. Esto no es accidental: a medida que aumentamos el número de lanzamientos, la distribución se aproxima cada vez más a esa curva.",[17,19264,19265,19266,176],{},"Si te interesa explorar más la estadística detrás de los lanzamientos, lee ",[21,19267,19270],{"href":19268,"rel":19269},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fla-estadistica-inesperada-de-los-lanzamientos-de-moneda\u002F",[25],"La estadística inesperada de los lanzamientos de moneda",[561,19272,19274],{"id":19273},"la-conexión-con-el-álgebra-el-binomio-de-newton","La conexión con el álgebra: el binomio de Newton",[17,19276,19277,19278,19281],{},"Hasta ahora hemos hablado de triángulos y monedas. Pero hay un tercer protagonista que conecta todo: el ",[239,19279,19280],{},"álgebra",". Concretamente, una expresión que quizá recuerdes del colegio:",[17,19283,19285],{"className":19284},[592],[35,19286,19288,19315],{"className":19287},[38],[35,19289,19291],{"className":19290},[42],[44,19292,19293],{"xmlns":46},[48,19294,19295,19312],{},[51,19296,19297,19299,19301,19303,19306],{},[276,19298,1028],{"stretchy":9816},[57,19300,21],{},[276,19302,1316],{},[57,19304,19305],{},"b",[1022,19307,19308,19310],{},[276,19309,1037],{"stretchy":9816},[57,19311,2197],{},[65,19313,19314],{"encoding":67},"(a + b)^n",[35,19316,19318,19339],{"className":19317,"ariaHidden":73},[72],[35,19319,19321,19324,19327,19330,19333,19336],{"className":19320},[77],[35,19322],{"className":19323,"style":9837},[81],[35,19325,1028],{"className":19326},[90],[35,19328,21],{"className":19329},[86,156],[35,19331],{"className":19332,"style":2223},[366],[35,19334,1316],{"className":19335},[2227],[35,19337],{"className":19338,"style":2223},[366],[35,19340,19342,19345,19348],{"className":19341},[77],[35,19343],{"className":19344,"style":9837},[81],[35,19346,19305],{"className":19347},[86,156],[35,19349,19351,19354],{"className":19350},[175],[35,19352,1037],{"className":19353},[175],[35,19355,19357],{"className":19356},[1156],[35,19358,19360],{"className":19359},[98],[35,19361,19363],{"className":19362},[103],[35,19364,19366],{"className":19365,"style":15536},[107],[35,19367,19368,19371],{"style":15557},[35,19369],{"className":19370,"style":1173},[115],[35,19372,19374],{"className":19373},[120,121,122,123],[35,19375,2197],{"className":19376},[86,156,123],[17,19378,19379],{},"Antes de desarrollarla, aclaremos un par de conceptos:",[3549,19381,19382,19503],{},[232,19383,19384,19385,19388,19389,176],{},"Un ",[239,19386,19387],{},"polinomio"," es una expresión matemática formada por sumas de términos con variables elevadas a distintas potencias. Por ejemplo: ",[35,19390,19392,19422],{"className":19391},[38],[35,19393,19395],{"className":19394},[42],[44,19396,19397],{"xmlns":46},[48,19398,19399,19419],{},[51,19400,19401,19403,19409,19411,19413,19415,19417],{},[61,19402,927],{},[1022,19404,19405,19407],{},[57,19406,11195],{},[61,19408,404],{},[276,19410,1316],{},[61,19412,404],{},[57,19414,11195],{},[276,19416,1316],{},[61,19418,272],{},[65,19420,19421],{"encoding":67},"3x^2 + 2x + 1",[35,19423,19425,19473,19494],{"className":19424,"ariaHidden":73},[72],[35,19426,19428,19432,19435,19464,19467,19470],{"className":19427},[77],[35,19429],{"className":19430,"style":19431},[81],"height:0.8974em;vertical-align:-0.0833em;",[35,19433,927],{"className":19434},[86],[35,19436,19438,19441],{"className":19437},[86],[35,19439,11195],{"className":19440},[86,156],[35,19442,19444],{"className":19443},[1156],[35,19445,19447],{"className":19446},[98],[35,19448,19450],{"className":19449},[103],[35,19451,19453],{"className":19452,"style":15811},[107],[35,19454,19455,19458],{"style":15557},[35,19456],{"className":19457,"style":1173},[115],[35,19459,19461],{"className":19460},[120,121,122,123],[35,19462,404],{"className":19463},[86,123],[35,19465],{"className":19466,"style":2223},[366],[35,19468,1316],{"className":19469},[2227],[35,19471],{"className":19472,"style":2223},[366],[35,19474,19476,19479,19482,19485,19488,19491],{"className":19475},[77],[35,19477],{"className":19478,"style":1331},[81],[35,19480,404],{"className":19481},[86],[35,19483,11195],{"className":19484},[86,156],[35,19486],{"className":19487,"style":2223},[366],[35,19489,1316],{"className":19490},[2227],[35,19492],{"className":19493,"style":2223},[366],[35,19495,19497,19500],{"className":19496},[77],[35,19498],{"className":19499,"style":381},[81],[35,19501,272],{"className":19502},[86],[232,19504,19384,19505,19508,19509,19512,19513,176],{},[239,19506,19507],{},"binomio"," es un caso particular de polinomio: tiene exactamente ",[239,19510,19511],{},"dos"," términos. Por ejemplo: ",[35,19514,19516,19538],{"className":19515},[38],[35,19517,19519],{"className":19518},[42],[44,19520,19521],{"xmlns":46},[48,19522,19523,19535],{},[51,19524,19525,19527,19529,19531,19533],{},[276,19526,1028],{"stretchy":9816},[57,19528,21],{},[276,19530,1316],{},[57,19532,19305],{},[276,19534,1037],{"stretchy":9816},[65,19536,19537],{"encoding":67},"(a + b)",[35,19539,19541,19562],{"className":19540,"ariaHidden":73},[72],[35,19542,19544,19547,19550,19553,19556,19559],{"className":19543},[77],[35,19545],{"className":19546,"style":9837},[81],[35,19548,1028],{"className":19549},[90],[35,19551,21],{"className":19552},[86,156],[35,19554],{"className":19555,"style":2223},[366],[35,19557,1316],{"className":19558},[2227],[35,19560],{"className":19561,"style":2223},[366],[35,19563,19565,19568,19571],{"className":19564},[77],[35,19566],{"className":19567,"style":9837},[81],[35,19569,19305],{"className":19570},[86,156],[35,19572,1037],{"className":19573},[175],[17,19575,19576,19577,19605],{},"Elevar un binomio a una potencia significa multiplicarlo por sí mismo ",[35,19578,19580,19593],{"className":19579},[38],[35,19581,19583],{"className":19582},[42],[44,19584,19585],{"xmlns":46},[48,19586,19587,19591],{},[51,19588,19589],{},[57,19590,2197],{},[65,19592,2197],{"encoding":67},[35,19594,19596],{"className":19595,"ariaHidden":73},[72],[35,19597,19599,19602],{"className":19598},[77],[35,19600],{"className":19601,"style":206},[81],[35,19603,2197],{"className":19604},[86,156]," veces. Empecemos con los casos más sencillos:",[17,19607,19608],{},[239,19609,19610],{},"Potencia 2:",[17,19612,19614],{"className":19613},[592],[35,19615,19617,19667],{"className":19616},[38],[35,19618,19620],{"className":19619},[42],[44,19621,19622],{"xmlns":46},[48,19623,19624,19664],{},[51,19625,19626,19628,19630,19632,19634,19640,19642,19648,19650,19652,19654,19656,19658],{},[276,19627,1028],{"stretchy":9816},[57,19629,21],{},[276,19631,1316],{},[57,19633,19305],{},[1022,19635,19636,19638],{},[276,19637,1037],{"stretchy":9816},[61,19639,404],{},[276,19641,278],{},[1022,19643,19644,19646],{},[57,19645,21],{},[61,19647,404],{},[276,19649,1316],{},[61,19651,404],{},[57,19653,21],{},[57,19655,19305],{},[276,19657,1316],{},[1022,19659,19660,19662],{},[57,19661,19305],{},[61,19663,404],{},[65,19665,19666],{"encoding":67},"(a + b)^2 = a^2 + 2ab + b^2",[35,19668,19670,19691,19739,19783,19806],{"className":19669,"ariaHidden":73},[72],[35,19671,19673,19676,19679,19682,19685,19688],{"className":19672},[77],[35,19674],{"className":19675,"style":9837},[81],[35,19677,1028],{"className":19678},[90],[35,19680,21],{"className":19681},[86,156],[35,19683],{"className":19684,"style":2223},[366],[35,19686,1316],{"className":19687},[2227],[35,19689],{"className":19690,"style":2223},[366],[35,19692,19694,19698,19701,19730,19733,19736],{"className":19693},[77],[35,19695],{"className":19696,"style":19697},[81],"height:1.0641em;vertical-align:-0.25em;",[35,19699,19305],{"className":19700},[86,156],[35,19702,19704,19707],{"className":19703},[175],[35,19705,1037],{"className":19706},[175],[35,19708,19710],{"className":19709},[1156],[35,19711,19713],{"className":19712},[98],[35,19714,19716],{"className":19715},[103],[35,19717,19719],{"className":19718,"style":15811},[107],[35,19720,19721,19724],{"style":15557},[35,19722],{"className":19723,"style":1173},[115],[35,19725,19727],{"className":19726},[120,121,122,123],[35,19728,404],{"className":19729},[86,123],[35,19731],{"className":19732,"style":367},[366],[35,19734,278],{"className":19735},[371],[35,19737],{"className":19738,"style":367},[366],[35,19740,19742,19745,19774,19777,19780],{"className":19741},[77],[35,19743],{"className":19744,"style":19431},[81],[35,19746,19748,19751],{"className":19747},[86],[35,19749,21],{"className":19750},[86,156],[35,19752,19754],{"className":19753},[1156],[35,19755,19757],{"className":19756},[98],[35,19758,19760],{"className":19759},[103],[35,19761,19763],{"className":19762,"style":15811},[107],[35,19764,19765,19768],{"style":15557},[35,19766],{"className":19767,"style":1173},[115],[35,19769,19771],{"className":19770},[120,121,122,123],[35,19772,404],{"className":19773},[86,123],[35,19775],{"className":19776,"style":2223},[366],[35,19778,1316],{"className":19779},[2227],[35,19781],{"className":19782,"style":2223},[366],[35,19784,19786,19790,19793,19797,19800,19803],{"className":19785},[77],[35,19787],{"className":19788,"style":19789},[81],"height:0.7778em;vertical-align:-0.0833em;",[35,19791,404],{"className":19792},[86],[35,19794,19796],{"className":19795},[86,156],"ab",[35,19798],{"className":19799,"style":2223},[366],[35,19801,1316],{"className":19802},[2227],[35,19804],{"className":19805,"style":2223},[366],[35,19807,19809,19812],{"className":19808},[77],[35,19810],{"className":19811,"style":15811},[81],[35,19813,19815,19818],{"className":19814},[86],[35,19816,19305],{"className":19817},[86,156],[35,19819,19821],{"className":19820},[1156],[35,19822,19824],{"className":19823},[98],[35,19825,19827],{"className":19826},[103],[35,19828,19830],{"className":19829,"style":15811},[107],[35,19831,19832,19835],{"style":15557},[35,19833],{"className":19834,"style":1173},[115],[35,19836,19838],{"className":19837},[120,121,122,123],[35,19839,404],{"className":19840},[86,123],[17,19842,19843,19844,19899],{},"Los coeficientes son ",[35,19845,19847,19869],{"className":19846},[38],[35,19848,19850],{"className":19849},[42],[44,19851,19852],{"xmlns":46},[48,19853,19854,19866],{},[51,19855,19856,19858,19860,19862,19864],{},[61,19857,272],{},[276,19859,525],{"separator":73},[61,19861,404],{},[276,19863,525],{"separator":73},[61,19865,272],{},[65,19867,19868],{"encoding":67},"1, 2, 1",[35,19870,19872],{"className":19871,"ariaHidden":73},[72],[35,19873,19875,19878,19881,19884,19887,19890,19893,19896],{"className":19874},[77],[35,19876],{"className":19877,"style":513},[81],[35,19879,272],{"className":19880},[86],[35,19882,525],{"className":19883},[524],[35,19885],{"className":19886,"style":1452},[366],[35,19888,404],{"className":19889},[86],[35,19891,525],{"className":19892},[524],[35,19894],{"className":19895,"style":1452},[366],[35,19897,272],{"className":19898},[86]," — la fila 2 del triángulo.",[17,19901,19902],{},[239,19903,19904],{},"Potencia 3:",[17,19906,19908],{"className":19907},[592],[35,19909,19911,19977],{"className":19910},[38],[35,19912,19914],{"className":19913},[42],[44,19915,19916],{"xmlns":46},[48,19917,19918,19974],{},[51,19919,19920,19922,19924,19926,19928,19934,19936,19942,19944,19946,19952,19954,19956,19958,19960,19966,19968],{},[276,19921,1028],{"stretchy":9816},[57,19923,21],{},[276,19925,1316],{},[57,19927,19305],{},[1022,19929,19930,19932],{},[276,19931,1037],{"stretchy":9816},[61,19933,927],{},[276,19935,278],{},[1022,19937,19938,19940],{},[57,19939,21],{},[61,19941,927],{},[276,19943,1316],{},[61,19945,927],{},[1022,19947,19948,19950],{},[57,19949,21],{},[61,19951,404],{},[57,19953,19305],{},[276,19955,1316],{},[61,19957,927],{},[57,19959,21],{},[1022,19961,19962,19964],{},[57,19963,19305],{},[61,19965,404],{},[276,19967,1316],{},[1022,19969,19970,19972],{},[57,19971,19305],{},[61,19973,927],{},[65,19975,19976],{"encoding":67},"(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3",[35,19978,19980,20001,20048,20092,20142,20192],{"className":19979,"ariaHidden":73},[72],[35,19981,19983,19986,19989,19992,19995,19998],{"className":19982},[77],[35,19984],{"className":19985,"style":9837},[81],[35,19987,1028],{"className":19988},[90],[35,19990,21],{"className":19991},[86,156],[35,19993],{"className":19994,"style":2223},[366],[35,19996,1316],{"className":19997},[2227],[35,19999],{"className":20000,"style":2223},[366],[35,20002,20004,20007,20010,20039,20042,20045],{"className":20003},[77],[35,20005],{"className":20006,"style":19697},[81],[35,20008,19305],{"className":20009},[86,156],[35,20011,20013,20016],{"className":20012},[175],[35,20014,1037],{"className":20015},[175],[35,20017,20019],{"className":20018},[1156],[35,20020,20022],{"className":20021},[98],[35,20023,20025],{"className":20024},[103],[35,20026,20028],{"className":20027,"style":15811},[107],[35,20029,20030,20033],{"style":15557},[35,20031],{"className":20032,"style":1173},[115],[35,20034,20036],{"className":20035},[120,121,122,123],[35,20037,927],{"className":20038},[86,123],[35,20040],{"className":20041,"style":367},[366],[35,20043,278],{"className":20044},[371],[35,20046],{"className":20047,"style":367},[366],[35,20049,20051,20054,20083,20086,20089],{"className":20050},[77],[35,20052],{"className":20053,"style":19431},[81],[35,20055,20057,20060],{"className":20056},[86],[35,20058,21],{"className":20059},[86,156],[35,20061,20063],{"className":20062},[1156],[35,20064,20066],{"className":20065},[98],[35,20067,20069],{"className":20068},[103],[35,20070,20072],{"className":20071,"style":15811},[107],[35,20073,20074,20077],{"style":15557},[35,20075],{"className":20076,"style":1173},[115],[35,20078,20080],{"className":20079},[120,121,122,123],[35,20081,927],{"className":20082},[86,123],[35,20084],{"className":20085,"style":2223},[366],[35,20087,1316],{"className":20088},[2227],[35,20090],{"className":20091,"style":2223},[366],[35,20093,20095,20098,20101,20130,20133,20136,20139],{"className":20094},[77],[35,20096],{"className":20097,"style":19431},[81],[35,20099,927],{"className":20100},[86],[35,20102,20104,20107],{"className":20103},[86],[35,20105,21],{"className":20106},[86,156],[35,20108,20110],{"className":20109},[1156],[35,20111,20113],{"className":20112},[98],[35,20114,20116],{"className":20115},[103],[35,20117,20119],{"className":20118,"style":15811},[107],[35,20120,20121,20124],{"style":15557},[35,20122],{"className":20123,"style":1173},[115],[35,20125,20127],{"className":20126},[120,121,122,123],[35,20128,404],{"className":20129},[86,123],[35,20131,19305],{"className":20132},[86,156],[35,20134],{"className":20135,"style":2223},[366],[35,20137,1316],{"className":20138},[2227],[35,20140],{"className":20141,"style":2223},[366],[35,20143,20145,20148,20151,20154,20183,20186,20189],{"className":20144},[77],[35,20146],{"className":20147,"style":19431},[81],[35,20149,927],{"className":20150},[86],[35,20152,21],{"className":20153},[86,156],[35,20155,20157,20160],{"className":20156},[86],[35,20158,19305],{"className":20159},[86,156],[35,20161,20163],{"className":20162},[1156],[35,20164,20166],{"className":20165},[98],[35,20167,20169],{"className":20168},[103],[35,20170,20172],{"className":20171,"style":15811},[107],[35,20173,20174,20177],{"style":15557},[35,20175],{"className":20176,"style":1173},[115],[35,20178,20180],{"className":20179},[120,121,122,123],[35,20181,404],{"className":20182},[86,123],[35,20184],{"className":20185,"style":2223},[366],[35,20187,1316],{"className":20188},[2227],[35,20190],{"className":20191,"style":2223},[366],[35,20193,20195,20198],{"className":20194},[77],[35,20196],{"className":20197,"style":15811},[81],[35,20199,20201,20204],{"className":20200},[86],[35,20202,19305],{"className":20203},[86,156],[35,20205,20207],{"className":20206},[1156],[35,20208,20210],{"className":20209},[98],[35,20211,20213],{"className":20212},[103],[35,20214,20216],{"className":20215,"style":15811},[107],[35,20217,20218,20221],{"style":15557},[35,20219],{"className":20220,"style":1173},[115],[35,20222,20224],{"className":20223},[120,121,122,123],[35,20225,927],{"className":20226},[86,123],[17,20228,19843,20229,20296],{},[35,20230,20232,20257],{"className":20231},[38],[35,20233,20235],{"className":20234},[42],[44,20236,20237],{"xmlns":46},[48,20238,20239,20255],{},[51,20240,20241,20243,20245,20247,20249,20251,20253],{},[61,20242,272],{},[276,20244,525],{"separator":73},[61,20246,927],{},[276,20248,525],{"separator":73},[61,20250,927],{},[276,20252,525],{"separator":73},[61,20254,272],{},[65,20256,15957],{"encoding":67},[35,20258,20260],{"className":20259,"ariaHidden":73},[72],[35,20261,20263,20266,20269,20272,20275,20278,20281,20284,20287,20290,20293],{"className":20262},[77],[35,20264],{"className":20265,"style":513},[81],[35,20267,272],{"className":20268},[86],[35,20270,525],{"className":20271},[524],[35,20273],{"className":20274,"style":1452},[366],[35,20276,927],{"className":20277},[86],[35,20279,525],{"className":20280},[524],[35,20282],{"className":20283,"style":1452},[366],[35,20285,927],{"className":20286},[86],[35,20288,525],{"className":20289},[524],[35,20291],{"className":20292,"style":1452},[366],[35,20294,272],{"className":20295},[86]," — la fila 3.",[17,20298,20299],{},[239,20300,20301],{},"Potencia 5:",[17,20303,20305],{"className":20304},[592],[35,20306,20308,20406],{"className":20307},[38],[35,20309,20311],{"className":20310},[42],[44,20312,20313],{"xmlns":46},[48,20314,20315,20403],{},[51,20316,20317,20319,20321,20323,20325,20331,20333,20339,20341,20343,20349,20351,20353,20355,20361,20367,20369,20371,20377,20383,20385,20387,20389,20395,20397],{},[276,20318,1028],{"stretchy":9816},[57,20320,21],{},[276,20322,1316],{},[57,20324,19305],{},[1022,20326,20327,20329],{},[276,20328,1037],{"stretchy":9816},[61,20330,529],{},[276,20332,278],{},[1022,20334,20335,20337],{},[57,20336,21],{},[61,20338,529],{},[276,20340,1316],{},[61,20342,529],{},[1022,20344,20345,20347],{},[57,20346,21],{},[61,20348,63],{},[57,20350,19305],{},[276,20352,1316],{},[61,20354,15755],{},[1022,20356,20357,20359],{},[57,20358,21],{},[61,20360,927],{},[1022,20362,20363,20365],{},[57,20364,19305],{},[61,20366,404],{},[276,20368,1316],{},[61,20370,15755],{},[1022,20372,20373,20375],{},[57,20374,21],{},[61,20376,404],{},[1022,20378,20379,20381],{},[57,20380,19305],{},[61,20382,927],{},[276,20384,1316],{},[61,20386,529],{},[57,20388,21],{},[1022,20390,20391,20393],{},[57,20392,19305],{},[61,20394,63],{},[276,20396,1316],{},[1022,20398,20399,20401],{},[57,20400,19305],{},[61,20402,529],{},[65,20404,20405],{"encoding":67},"(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5",[35,20407,20409,20430,20477,20521,20571,20647,20723,20773],{"className":20408,"ariaHidden":73},[72],[35,20410,20412,20415,20418,20421,20424,20427],{"className":20411},[77],[35,20413],{"className":20414,"style":9837},[81],[35,20416,1028],{"className":20417},[90],[35,20419,21],{"className":20420},[86,156],[35,20422],{"className":20423,"style":2223},[366],[35,20425,1316],{"className":20426},[2227],[35,20428],{"className":20429,"style":2223},[366],[35,20431,20433,20436,20439,20468,20471,20474],{"className":20432},[77],[35,20434],{"className":20435,"style":19697},[81],[35,20437,19305],{"className":20438},[86,156],[35,20440,20442,20445],{"className":20441},[175],[35,20443,1037],{"className":20444},[175],[35,20446,20448],{"className":20447},[1156],[35,20449,20451],{"className":20450},[98],[35,20452,20454],{"className":20453},[103],[35,20455,20457],{"className":20456,"style":15811},[107],[35,20458,20459,20462],{"style":15557},[35,20460],{"className":20461,"style":1173},[115],[35,20463,20465],{"className":20464},[120,121,122,123],[35,20466,529],{"className":20467},[86,123],[35,20469],{"className":20470,"style":367},[366],[35,20472,278],{"className":20473},[371],[35,20475],{"className":20476,"style":367},[366],[35,20478,20480,20483,20512,20515,20518],{"className":20479},[77],[35,20481],{"className":20482,"style":19431},[81],[35,20484,20486,20489],{"className":20485},[86],[35,20487,21],{"className":20488},[86,156],[35,20490,20492],{"className":20491},[1156],[35,20493,20495],{"className":20494},[98],[35,20496,20498],{"className":20497},[103],[35,20499,20501],{"className":20500,"style":15811},[107],[35,20502,20503,20506],{"style":15557},[35,20504],{"className":20505,"style":1173},[115],[35,20507,20509],{"className":20508},[120,121,122,123],[35,20510,529],{"className":20511},[86,123],[35,20513],{"className":20514,"style":2223},[366],[35,20516,1316],{"className":20517},[2227],[35,20519],{"className":20520,"style":2223},[366],[35,20522,20524,20527,20530,20559,20562,20565,20568],{"className":20523},[77],[35,20525],{"className":20526,"style":19431},[81],[35,20528,529],{"className":20529},[86],[35,20531,20533,20536],{"className":20532},[86],[35,20534,21],{"className":20535},[86,156],[35,20537,20539],{"className":20538},[1156],[35,20540,20542],{"className":20541},[98],[35,20543,20545],{"className":20544},[103],[35,20546,20548],{"className":20547,"style":15811},[107],[35,20549,20550,20553],{"style":15557},[35,20551],{"className":20552,"style":1173},[115],[35,20554,20556],{"className":20555},[120,121,122,123],[35,20557,63],{"className":20558},[86,123],[35,20560,19305],{"className":20561},[86,156],[35,20563],{"className":20564,"style":2223},[366],[35,20566,1316],{"className":20567},[2227],[35,20569],{"className":20570,"style":2223},[366],[35,20572,20574,20577,20580,20609,20638,20641,20644],{"className":20573},[77],[35,20575],{"className":20576,"style":19431},[81],[35,20578,15755],{"className":20579},[86],[35,20581,20583,20586],{"className":20582},[86],[35,20584,21],{"className":20585},[86,156],[35,20587,20589],{"className":20588},[1156],[35,20590,20592],{"className":20591},[98],[35,20593,20595],{"className":20594},[103],[35,20596,20598],{"className":20597,"style":15811},[107],[35,20599,20600,20603],{"style":15557},[35,20601],{"className":20602,"style":1173},[115],[35,20604,20606],{"className":20605},[120,121,122,123],[35,20607,927],{"className":20608},[86,123],[35,20610,20612,20615],{"className":20611},[86],[35,20613,19305],{"className":20614},[86,156],[35,20616,20618],{"className":20617},[1156],[35,20619,20621],{"className":20620},[98],[35,20622,20624],{"className":20623},[103],[35,20625,20627],{"className":20626,"style":15811},[107],[35,20628,20629,20632],{"style":15557},[35,20630],{"className":20631,"style":1173},[115],[35,20633,20635],{"className":20634},[120,121,122,123],[35,20636,404],{"className":20637},[86,123],[35,20639],{"className":20640,"style":2223},[366],[35,20642,1316],{"className":20643},[2227],[35,20645],{"className":20646,"style":2223},[366],[35,20648,20650,20653,20656,20685,20714,20717,20720],{"className":20649},[77],[35,20651],{"className":20652,"style":19431},[81],[35,20654,15755],{"className":20655},[86],[35,20657,20659,20662],{"className":20658},[86],[35,20660,21],{"className":20661},[86,156],[35,20663,20665],{"className":20664},[1156],[35,20666,20668],{"className":20667},[98],[35,20669,20671],{"className":20670},[103],[35,20672,20674],{"className":20673,"style":15811},[107],[35,20675,20676,20679],{"style":15557},[35,20677],{"className":20678,"style":1173},[115],[35,20680,20682],{"className":20681},[120,121,122,123],[35,20683,404],{"className":20684},[86,123],[35,20686,20688,20691],{"className":20687},[86],[35,20689,19305],{"className":20690},[86,156],[35,20692,20694],{"className":20693},[1156],[35,20695,20697],{"className":20696},[98],[35,20698,20700],{"className":20699},[103],[35,20701,20703],{"className":20702,"style":15811},[107],[35,20704,20705,20708],{"style":15557},[35,20706],{"className":20707,"style":1173},[115],[35,20709,20711],{"className":20710},[120,121,122,123],[35,20712,927],{"className":20713},[86,123],[35,20715],{"className":20716,"style":2223},[366],[35,20718,1316],{"className":20719},[2227],[35,20721],{"className":20722,"style":2223},[366],[35,20724,20726,20729,20732,20735,20764,20767,20770],{"className":20725},[77],[35,20727],{"className":20728,"style":19431},[81],[35,20730,529],{"className":20731},[86],[35,20733,21],{"className":20734},[86,156],[35,20736,20738,20741],{"className":20737},[86],[35,20739,19305],{"className":20740},[86,156],[35,20742,20744],{"className":20743},[1156],[35,20745,20747],{"className":20746},[98],[35,20748,20750],{"className":20749},[103],[35,20751,20753],{"className":20752,"style":15811},[107],[35,20754,20755,20758],{"style":15557},[35,20756],{"className":20757,"style":1173},[115],[35,20759,20761],{"className":20760},[120,121,122,123],[35,20762,63],{"className":20763},[86,123],[35,20765],{"className":20766,"style":2223},[366],[35,20768,1316],{"className":20769},[2227],[35,20771],{"className":20772,"style":2223},[366],[35,20774,20776,20779],{"className":20775},[77],[35,20777],{"className":20778,"style":15811},[81],[35,20780,20782,20785],{"className":20781},[86],[35,20783,19305],{"className":20784},[86,156],[35,20786,20788],{"className":20787},[1156],[35,20789,20791],{"className":20790},[98],[35,20792,20794],{"className":20793},[103],[35,20795,20797],{"className":20796,"style":15811},[107],[35,20798,20799,20802],{"style":15557},[35,20800],{"className":20801,"style":1173},[115],[35,20803,20805],{"className":20804},[120,121,122,123],[35,20806,529],{"className":20807},[86,123],[17,20809,19843,20810,20904],{},[35,20811,20813,20847],{"className":20812},[38],[35,20814,20816],{"className":20815},[42],[44,20817,20818],{"xmlns":46},[48,20819,20820,20844],{},[51,20821,20822,20824,20826,20828,20830,20832,20834,20836,20838,20840,20842],{},[61,20823,272],{},[276,20825,525],{"separator":73},[61,20827,529],{},[276,20829,525],{"separator":73},[61,20831,15755],{},[276,20833,525],{"separator":73},[61,20835,15755],{},[276,20837,525],{"separator":73},[61,20839,529],{},[276,20841,525],{"separator":73},[61,20843,272],{},[65,20845,20846],{"encoding":67},"1, 5, 10, 10, 5, 1",[35,20848,20850],{"className":20849,"ariaHidden":73},[72],[35,20851,20853,20856,20859,20862,20865,20868,20871,20874,20877,20880,20883,20886,20889,20892,20895,20898,20901],{"className":20852},[77],[35,20854],{"className":20855,"style":513},[81],[35,20857,272],{"className":20858},[86],[35,20860,525],{"className":20861},[524],[35,20863],{"className":20864,"style":1452},[366],[35,20866,529],{"className":20867},[86],[35,20869,525],{"className":20870},[524],[35,20872],{"className":20873,"style":1452},[366],[35,20875,15755],{"className":20876},[86],[35,20878,525],{"className":20879},[524],[35,20881],{"className":20882,"style":1452},[366],[35,20884,15755],{"className":20885},[86],[35,20887,525],{"className":20888},[524],[35,20890],{"className":20891,"style":1452},[366],[35,20893,529],{"className":20894},[86],[35,20896,525],{"className":20897},[524],[35,20899],{"className":20900,"style":1452},[366],[35,20902,272],{"className":20903},[86]," — la fila 5.",[17,20906,20907,20908,20911,20912,11815],{},"¿Ves el patrón? ",[239,20909,20910],{},"Las filas del triángulo de Tartaglia son exactamente los coeficientes del desarrollo del binomio."," Esta relación se formaliza con el ",[239,20913,20914],{},"teorema del binomio",[17,20916,20918],{"className":20917},[592],[35,20919,20921,20998],{"className":20920},[38],[35,20922,20924],{"className":20923},[42],[44,20925,20926],{"xmlns":46},[48,20927,20928,20995],{},[51,20929,20930,20932,20934,20936,20938,20944,20946,20960,20972,20975,20987,20989],{},[276,20931,1028],{"stretchy":9816},[57,20933,21],{},[276,20935,1316],{},[57,20937,19305],{},[1022,20939,20940,20942],{},[276,20941,1037],{"stretchy":9816},[57,20943,2197],{},[276,20945,278],{},[7303,20947,20948,20950,20958],{},[276,20949,7307],{},[51,20951,20952,20954,20956],{},[57,20953,13490],{},[276,20955,278],{},[61,20957,517],{},[57,20959,2197],{},[51,20961,20962,20964,20970],{},[276,20963,1028],{"fence":73},[54,20965,20966,20968],{"linethickness":4321},[57,20967,2197],{},[57,20969,13490],{},[276,20971,1037],{"fence":73},[606,20973,20974],{}," ",[1022,20976,20977,20979],{},[57,20978,21],{},[51,20980,20981,20983,20985],{},[57,20982,2197],{},[276,20984,1353],{},[57,20986,13490],{},[606,20988,20974],{},[1022,20990,20991,20993],{},[57,20992,19305],{},[57,20994,13490],{},[65,20996,20997],{"encoding":67},"(a+b)^n = \\sum_{k=0}^{n} \\binom{n}{k} \\, a^{n-k} \\, b^k",[35,20999,21001,21022,21069],{"className":21000,"ariaHidden":73},[72],[35,21002,21004,21007,21010,21013,21016,21019],{"className":21003},[77],[35,21005],{"className":21006,"style":9837},[81],[35,21008,1028],{"className":21009},[90],[35,21011,21],{"className":21012},[86,156],[35,21014],{"className":21015,"style":2223},[366],[35,21017,1316],{"className":21018},[2227],[35,21020],{"className":21021,"style":2223},[366],[35,21023,21025,21028,21031,21060,21063,21066],{"className":21024},[77],[35,21026],{"className":21027,"style":9837},[81],[35,21029,19305],{"className":21030},[86,156],[35,21032,21034,21037],{"className":21033},[175],[35,21035,1037],{"className":21036},[175],[35,21038,21040],{"className":21039},[1156],[35,21041,21043],{"className":21042},[98],[35,21044,21046],{"className":21045},[103],[35,21047,21049],{"className":21048,"style":15536},[107],[35,21050,21051,21054],{"style":15557},[35,21052],{"className":21053,"style":1173},[115],[35,21055,21057],{"className":21056},[120,121,122,123],[35,21058,2197],{"className":21059},[86,156,123],[35,21061],{"className":21062,"style":367},[366],[35,21064,278],{"className":21065},[371],[35,21067],{"className":21068,"style":367},[366],[35,21070,21072,21075,21138,21141,21207,21210,21249,21252],{"className":21071},[77],[35,21073],{"className":21074,"style":13505},[81],[35,21076,21078,21081],{"className":21077},[7402],[35,21079,7307],{"className":21080,"style":7408},[7402,7406,7407],[35,21082,21084],{"className":21083},[1156],[35,21085,21087,21130],{"className":21086},[98,99],[35,21088,21090,21127],{"className":21089},[103],[35,21091,21093,21113],{"className":21092,"style":7421},[107],[35,21094,21095,21098],{"style":7424},[35,21096],{"className":21097,"style":1173},[115],[35,21099,21101],{"className":21100},[120,121,122,123],[35,21102,21104,21107,21110],{"className":21103},[86,123],[35,21105,13490],{"className":21106,"style":13542},[86,156,123],[35,21108,278],{"className":21109},[371,123],[35,21111,517],{"className":21112},[86,123],[35,21114,21115,21118],{"style":7445},[35,21116],{"className":21117,"style":1173},[115],[35,21119,21121],{"className":21120},[120,121,122,123],[35,21122,21124],{"className":21123},[86,123],[35,21125,2197],{"className":21126},[86,156,123],[35,21128,162],{"className":21129},[161],[35,21131,21133],{"className":21132},[103],[35,21134,21136],{"className":21135,"style":7467},[107],[35,21137],{},[35,21139],{"className":21140,"style":1452},[366],[35,21142,21144,21150,21201],{"className":21143},[86],[35,21145,21147],{"className":21146,"style":1073},[90,1072],[35,21148,1028],{"className":21149},[1077,1078],[35,21151,21153],{"className":21152},[54],[35,21154,21156,21193],{"className":21155},[98,99],[35,21157,21159,21190],{"className":21158},[103],[35,21160,21162,21176],{"className":21161,"style":13527},[107],[35,21163,21164,21167],{"style":4503},[35,21165],{"className":21166,"style":1173},[115],[35,21168,21170],{"className":21169},[120,121,122,123],[35,21171,21173],{"className":21172},[86,123],[35,21174,13490],{"className":21175,"style":13542},[86,156,123],[35,21177,21178,21181],{"style":4518},[35,21179],{"className":21180,"style":1173},[115],[35,21182,21184],{"className":21183},[120,121,122,123],[35,21185,21187],{"className":21186},[86,123],[35,21188,2197],{"className":21189},[86,156,123],[35,21191,162],{"className":21192},[161],[35,21194,21196],{"className":21195},[103],[35,21197,21199],{"className":21198,"style":169},[107],[35,21200],{},[35,21202,21204],{"className":21203,"style":1073},[175,1072],[35,21205,1037],{"className":21206},[1077,1078],[35,21208],{"className":21209,"style":1452},[366],[35,21211,21213,21216],{"className":21212},[86],[35,21214,21],{"className":21215},[86,156],[35,21217,21219],{"className":21218},[1156],[35,21220,21222],{"className":21221},[98],[35,21223,21225],{"className":21224},[103],[35,21226,21229],{"className":21227,"style":21228},[107],"height:0.8491em;",[35,21230,21231,21234],{"style":15557},[35,21232],{"className":21233,"style":1173},[115],[35,21235,21237],{"className":21236},[120,121,122,123],[35,21238,21240,21243,21246],{"className":21239},[86,123],[35,21241,2197],{"className":21242},[86,156,123],[35,21244,1353],{"className":21245},[2227,123],[35,21247,13490],{"className":21248,"style":13542},[86,156,123],[35,21250],{"className":21251,"style":1452},[366],[35,21253,21255,21258],{"className":21254},[86],[35,21256,19305],{"className":21257},[86,156],[35,21259,21261],{"className":21260},[1156],[35,21262,21264],{"className":21263},[98],[35,21265,21267],{"className":21266},[103],[35,21268,21270],{"className":21269,"style":21228},[107],[35,21271,21272,21275],{"style":15557},[35,21273],{"className":21274,"style":1173},[115],[35,21276,21278],{"className":21277},[120,121,122,123],[35,21279,13490],{"className":21280,"style":13542},[86,156,123],[17,21282,21283,21284,21313,21314,14168,21365,21416,21417,21516],{},"El símbolo ",[35,21285,21287,21301],{"className":21286},[38],[35,21288,21290],{"className":21289},[42],[44,21291,21292],{"xmlns":46},[48,21293,21294,21298],{},[51,21295,21296],{},[276,21297,7307],{},[65,21299,21300],{"encoding":67},"\\sum",[35,21302,21304],{"className":21303,"ariaHidden":73},[72],[35,21305,21307,21310],{"className":21306},[77],[35,21308],{"className":21309,"style":9837},[81],[35,21311,7307],{"className":21312,"style":7408},[7402,7406,7407]," (sigma) simplemente significa «suma todos los términos desde ",[35,21315,21317,21335],{"className":21316},[38],[35,21318,21320],{"className":21319},[42],[44,21321,21322],{"xmlns":46},[48,21323,21324,21332],{},[51,21325,21326,21328,21330],{},[57,21327,13490],{},[276,21329,278],{},[61,21331,517],{},[65,21333,21334],{"encoding":67},"k = 0",[35,21336,21338,21356],{"className":21337,"ariaHidden":73},[72],[35,21339,21341,21344,21347,21350,21353],{"className":21340},[77],[35,21342],{"className":21343,"style":7026},[81],[35,21345,13490],{"className":21346,"style":13542},[86,156],[35,21348],{"className":21349,"style":367},[366],[35,21351,278],{"className":21352},[371],[35,21354],{"className":21355,"style":367},[366],[35,21357,21359,21362],{"className":21358},[77],[35,21360],{"className":21361,"style":381},[81],[35,21363,517],{"className":21364},[86],[35,21366,21368,21386],{"className":21367},[38],[35,21369,21371],{"className":21370},[42],[44,21372,21373],{"xmlns":46},[48,21374,21375,21383],{},[51,21376,21377,21379,21381],{},[57,21378,13490],{},[276,21380,278],{},[57,21382,2197],{},[65,21384,21385],{"encoding":67},"k = n",[35,21387,21389,21407],{"className":21388,"ariaHidden":73},[72],[35,21390,21392,21395,21398,21401,21404],{"className":21391},[77],[35,21393],{"className":21394,"style":7026},[81],[35,21396,13490],{"className":21397,"style":13542},[86,156],[35,21399],{"className":21400,"style":367},[366],[35,21402,278],{"className":21403},[371],[35,21405],{"className":21406,"style":367},[366],[35,21408,21410,21413],{"className":21409},[77],[35,21411],{"className":21412,"style":206},[81],[35,21414,2197],{"className":21415},[86,156],"». Cada término tiene un coeficiente binomial ",[35,21418,21420,21441],{"className":21419},[38],[35,21421,21423],{"className":21422},[42],[44,21424,21425],{"xmlns":46},[48,21426,21427,21439],{},[51,21428,21429,21431,21437],{},[276,21430,1028],{"fence":73},[54,21432,21433,21435],{"linethickness":4321},[57,21434,2197],{},[57,21436,13490],{},[276,21438,1037],{"fence":73},[65,21440,13495],{"encoding":67},[35,21442,21444],{"className":21443,"ariaHidden":73},[72],[35,21445,21447,21450],{"className":21446},[77],[35,21448],{"className":21449,"style":13505},[81],[35,21451,21453,21459,21510],{"className":21452},[86],[35,21454,21456],{"className":21455,"style":1073},[90,1072],[35,21457,1028],{"className":21458},[1077,1078],[35,21460,21462],{"className":21461},[54],[35,21463,21465,21502],{"className":21464},[98,99],[35,21466,21468,21499],{"className":21467},[103],[35,21469,21471,21485],{"className":21470,"style":13527},[107],[35,21472,21473,21476],{"style":4503},[35,21474],{"className":21475,"style":1173},[115],[35,21477,21479],{"className":21478},[120,121,122,123],[35,21480,21482],{"className":21481},[86,123],[35,21483,13490],{"className":21484,"style":13542},[86,156,123],[35,21486,21487,21490],{"style":4518},[35,21488],{"className":21489,"style":1173},[115],[35,21491,21493],{"className":21492},[120,121,122,123],[35,21494,21496],{"className":21495},[86,123],[35,21497,2197],{"className":21498},[86,156,123],[35,21500,162],{"className":21501},[161],[35,21503,21505],{"className":21504},[103],[35,21506,21508],{"className":21507,"style":169},[107],[35,21509],{},[35,21511,21513],{"className":21512,"style":1073},[175,1072],[35,21514,1037],{"className":21515},[1077,1078]," que viene directamente del triángulo.",[582,21518,21520],{"id":21519},"el-puente-entre-álgebra-y-probabilidad","El puente entre álgebra y probabilidad",[17,21522,21523,21524,21552,21553,530,21555,21552,21583,11815],{},"Ahora viene lo fascinante. Sustituye ",[35,21525,21527,21540],{"className":21526},[38],[35,21528,21530],{"className":21529},[42],[44,21531,21532],{"xmlns":46},[48,21533,21534,21538],{},[51,21535,21536],{},[57,21537,21],{},[65,21539,21],{"encoding":67},[35,21541,21543],{"className":21542,"ariaHidden":73},[72],[35,21544,21546,21549],{"className":21545},[77],[35,21547],{"className":21548,"style":206},[81],[35,21550,21],{"className":21551},[86,156]," por la probabilidad de obtener ",[239,21554,15293],{},[35,21556,21558,21571],{"className":21557},[38],[35,21559,21561],{"className":21560},[42],[44,21562,21563],{"xmlns":46},[48,21564,21565,21569],{},[51,21566,21567],{},[57,21568,19305],{},[65,21570,19305],{"encoding":67},[35,21572,21574],{"className":21573,"ariaHidden":73},[72],[35,21575,21577,21580],{"className":21576},[77],[35,21578],{"className":21579,"style":7026},[81],[35,21581,19305],{"className":21582},[86,156],[239,21584,15297],{},[17,21586,21588],{"className":21587},[592],[35,21589,21591,21647],{"className":21590},[38],[35,21592,21594],{"className":21593},[42],[44,21595,21596],{"xmlns":46},[48,21597,21598,21644],{},[51,21599,21600,21602,21604,21606,21608,21610,21612,21614,21620,21622,21624,21626,21628,21630,21632,21634,21636,21638],{},[57,21601,21],{},[276,21603,278],{},[57,21605,16246],{},[276,21607,1028],{"stretchy":9816},[606,21609,15293],{},[276,21611,1037],{"stretchy":9816},[276,21613,278],{},[54,21615,21616,21618],{},[61,21617,272],{},[61,21619,404],{},[276,21621,525],{"separator":73},[366,21623],{"width":18018},[57,21625,19305],{},[276,21627,278],{},[57,21629,16246],{},[276,21631,1028],{"stretchy":9816},[606,21633,15297],{},[276,21635,1037],{"stretchy":9816},[276,21637,278],{},[54,21639,21640,21642],{},[61,21641,272],{},[61,21643,404],{},[65,21645,21646],{"encoding":67},"a = P(\\text{cara}) = \\frac{1}{2}, \\quad b = P(\\text{cruz}) = \\frac{1}{2}",[35,21648,21650,21668,21698,21793,21823],{"className":21649,"ariaHidden":73},[72],[35,21651,21653,21656,21659,21662,21665],{"className":21652},[77],[35,21654],{"className":21655,"style":206},[81],[35,21657,21],{"className":21658},[86,156],[35,21660],{"className":21661,"style":367},[366],[35,21663,278],{"className":21664},[371],[35,21666],{"className":21667,"style":367},[366],[35,21669,21671,21674,21677,21680,21686,21689,21692,21695],{"className":21670},[77],[35,21672],{"className":21673,"style":9837},[81],[35,21675,16246],{"className":21676,"style":16272},[86,156],[35,21678,1028],{"className":21679},[90],[35,21681,21683],{"className":21682},[86,625],[35,21684,15293],{"className":21685},[86],[35,21687,1037],{"className":21688},[175],[35,21690],{"className":21691,"style":367},[366],[35,21693,278],{"className":21694},[371],[35,21696],{"className":21697,"style":367},[366],[35,21699,21701,21704,21772,21775,21778,21781,21784,21787,21790],{"className":21700},[77],[35,21702],{"className":21703,"style":293},[81],[35,21705,21707,21710,21769],{"className":21706},[86],[35,21708],{"className":21709},[90,91],[35,21711,21713],{"className":21712},[54],[35,21714,21716,21761],{"className":21715},[98,99],[35,21717,21719,21758],{"className":21718},[103],[35,21720,21722,21736,21744],{"className":21721,"style":312},[107],[35,21723,21724,21727],{"style":111},[35,21725],{"className":21726,"style":116},[115],[35,21728,21730],{"className":21729},[120,121,122,123],[35,21731,21733],{"className":21732},[86,123],[35,21734,404],{"className":21735},[86,123],[35,21737,21738,21741],{"style":132},[35,21739],{"className":21740,"style":116},[115],[35,21742],{"className":21743,"style":140},[139],[35,21745,21746,21749],{"style":143},[35,21747],{"className":21748,"style":116},[115],[35,21750,21752],{"className":21751},[120,121,122,123],[35,21753,21755],{"className":21754},[86,123],[35,21756,272],{"className":21757},[86,123],[35,21759,162],{"className":21760},[161],[35,21762,21764],{"className":21763},[103],[35,21765,21767],{"className":21766,"style":169},[107],[35,21768],{},[35,21770],{"className":21771},[175,91],[35,21773,525],{"className":21774},[524],[35,21776],{"className":21777,"style":18055},[366],[35,21779],{"className":21780,"style":1452},[366],[35,21782,19305],{"className":21783},[86,156],[35,21785],{"className":21786,"style":367},[366],[35,21788,278],{"className":21789},[371],[35,21791],{"className":21792,"style":367},[366],[35,21794,21796,21799,21802,21805,21811,21814,21817,21820],{"className":21795},[77],[35,21797],{"className":21798,"style":9837},[81],[35,21800,16246],{"className":21801,"style":16272},[86,156],[35,21803,1028],{"className":21804},[90],[35,21806,21808],{"className":21807},[86,625],[35,21809,15297],{"className":21810},[86],[35,21812,1037],{"className":21813},[175],[35,21815],{"className":21816,"style":367},[366],[35,21818,278],{"className":21819},[371],[35,21821],{"className":21822,"style":367},[366],[35,21824,21826,21829],{"className":21825},[77],[35,21827],{"className":21828,"style":293},[81],[35,21830,21832,21835,21894],{"className":21831},[86],[35,21833],{"className":21834},[90,91],[35,21836,21838],{"className":21837},[54],[35,21839,21841,21886],{"className":21840},[98,99],[35,21842,21844,21883],{"className":21843},[103],[35,21845,21847,21861,21869],{"className":21846,"style":312},[107],[35,21848,21849,21852],{"style":111},[35,21850],{"className":21851,"style":116},[115],[35,21853,21855],{"className":21854},[120,121,122,123],[35,21856,21858],{"className":21857},[86,123],[35,21859,404],{"className":21860},[86,123],[35,21862,21863,21866],{"style":132},[35,21864],{"className":21865,"style":116},[115],[35,21867],{"className":21868,"style":140},[139],[35,21870,21871,21874],{"style":143},[35,21872],{"className":21873,"style":116},[115],[35,21875,21877],{"className":21876},[120,121,122,123],[35,21878,21880],{"className":21879},[86,123],[35,21881,272],{"className":21882},[86,123],[35,21884,162],{"className":21885},[161],[35,21887,21889],{"className":21888},[103],[35,21890,21892],{"className":21891,"style":169},[107],[35,21893],{},[35,21895],{"className":21896},[175,91],[17,21898,21899],{},"El binomio se convierte en:",[17,21901,21903],{"className":21902},[592],[35,21904,21906,22044],{"className":21905},[38],[35,21907,21909],{"className":21908},[42],[44,21910,21911],{"xmlns":46},[48,21912,21913,22041],{},[51,21914,21915,21939,21941,21955,21967,21989,22005,22007,22021],{},[1022,21916,21917,21937],{},[51,21918,21919,21921,21927,21929,21935],{},[276,21920,1028],{"fence":73},[54,21922,21923,21925],{},[61,21924,272],{},[61,21926,404],{},[276,21928,1316],{},[54,21930,21931,21933],{},[61,21932,272],{},[61,21934,404],{},[276,21936,1037],{"fence":73},[57,21938,2197],{},[276,21940,278],{},[7303,21942,21943,21945,21953],{},[276,21944,7307],{},[51,21946,21947,21949,21951],{},[57,21948,13490],{},[276,21950,278],{},[61,21952,517],{},[57,21954,2197],{},[51,21956,21957,21959,21965],{},[276,21958,1028],{"fence":73},[54,21960,21961,21963],{"linethickness":4321},[57,21962,2197],{},[57,21964,13490],{},[276,21966,1037],{"fence":73},[1022,21968,21969,21981],{},[51,21970,21971,21973,21979],{},[276,21972,1028],{"fence":73},[54,21974,21975,21977],{},[61,21976,272],{},[61,21978,404],{},[276,21980,1037],{"fence":73},[51,21982,21983,21985,21987],{},[57,21984,2197],{},[276,21986,1353],{},[57,21988,13490],{},[1022,21990,21991,22003],{},[51,21992,21993,21995,22001],{},[276,21994,1028],{"fence":73},[54,21996,21997,21999],{},[61,21998,272],{},[61,22000,404],{},[276,22002,1037],{"fence":73},[57,22004,13490],{},[276,22006,278],{},[7303,22008,22009,22011,22019],{},[276,22010,7307],{},[51,22012,22013,22015,22017],{},[57,22014,13490],{},[276,22016,278],{},[61,22018,517],{},[57,22020,2197],{},[54,22022,22023,22035],{},[51,22024,22025,22027,22033],{},[276,22026,1028],{"fence":73},[54,22028,22029,22031],{"linethickness":4321},[57,22030,2197],{},[57,22032,13490],{},[276,22034,1037],{"fence":73},[1022,22036,22037,22039],{},[61,22038,404],{},[57,22040,2197],{},[65,22042,22043],{"encoding":67},"\\left(\\frac{1}{2} + \\frac{1}{2}\\right)^n = \\sum_{k=0}^{n} \\binom{n}{k} \\left(\\frac{1}{2}\\right)^{n-k} \\left(\\frac{1}{2}\\right)^k = \\sum_{k=0}^{n} \\frac{\\binom{n}{k}}{2^n}",[35,22045,22047,22250,22632],{"className":22046,"ariaHidden":73},[72],[35,22048,22050,22054,22241,22244,22247],{"className":22049},[77],[35,22051],{"className":22052,"style":22053},[81],"height:1.2543em;vertical-align:-0.35em;",[35,22055,22057,22217],{"className":22056},[1065],[35,22058,22060,22066,22134,22137,22140,22143,22211],{"className":22059},[1065],[35,22061,22063],{"className":22062,"style":1073},[90,1072],[35,22064,1028],{"className":22065},[1077,1078],[35,22067,22069,22072,22131],{"className":22068},[86],[35,22070],{"className":22071},[90,91],[35,22073,22075],{"className":22074},[54],[35,22076,22078,22123],{"className":22077},[98,99],[35,22079,22081,22120],{"className":22080},[103],[35,22082,22084,22098,22106],{"className":22083,"style":312},[107],[35,22085,22086,22089],{"style":111},[35,22087],{"className":22088,"style":116},[115],[35,22090,22092],{"className":22091},[120,121,122,123],[35,22093,22095],{"className":22094},[86,123],[35,22096,404],{"className":22097},[86,123],[35,22099,22100,22103],{"style":132},[35,22101],{"className":22102,"style":116},[115],[35,22104],{"className":22105,"style":140},[139],[35,22107,22108,22111],{"style":143},[35,22109],{"className":22110,"style":116},[115],[35,22112,22114],{"className":22113},[120,121,122,123],[35,22115,22117],{"className":22116},[86,123],[35,22118,272],{"className":22119},[86,123],[35,22121,162],{"className":22122},[161],[35,22124,22126],{"className":22125},[103],[35,22127,22129],{"className":22128,"style":169},[107],[35,22130],{},[35,22132],{"className":22133},[175,91],[35,22135],{"className":22136,"style":2223},[366],[35,22138,1316],{"className":22139},[2227],[35,22141],{"className":22142,"style":2223},[366],[35,22144,22146,22149,22208],{"className":22145},[86],[35,22147],{"className":22148},[90,91],[35,22150,22152],{"className":22151},[54],[35,22153,22155,22200],{"className":22154},[98,99],[35,22156,22158,22197],{"className":22157},[103],[35,22159,22161,22175,22183],{"className":22160,"style":312},[107],[35,22162,22163,22166],{"style":111},[35,22164],{"className":22165,"style":116},[115],[35,22167,22169],{"className":22168},[120,121,122,123],[35,22170,22172],{"className":22171},[86,123],[35,22173,404],{"className":22174},[86,123],[35,22176,22177,22180],{"style":132},[35,22178],{"className":22179,"style":116},[115],[35,22181],{"className":22182,"style":140},[139],[35,22184,22185,22188],{"style":143},[35,22186],{"className":22187,"style":116},[115],[35,22189,22191],{"className":22190},[120,121,122,123],[35,22192,22194],{"className":22193},[86,123],[35,22195,272],{"className":22196},[86,123],[35,22198,162],{"className":22199},[161],[35,22201,22203],{"className":22202},[103],[35,22204,22206],{"className":22205,"style":169},[107],[35,22207],{},[35,22209],{"className":22210},[175,91],[35,22212,22214],{"className":22213,"style":1073},[175,1072],[35,22215,1037],{"className":22216},[1077,1078],[35,22218,22220],{"className":22219},[1156],[35,22221,22223],{"className":22222},[98],[35,22224,22226],{"className":22225},[103],[35,22227,22230],{"className":22228,"style":22229},[107],"height:0.9043em;",[35,22231,22232,22235],{"style":1169},[35,22233],{"className":22234,"style":1173},[115],[35,22236,22238],{"className":22237},[120,121,122,123],[35,22239,2197],{"className":22240},[86,156,123],[35,22242],{"className":22243,"style":367},[366],[35,22245,278],{"className":22246},[371],[35,22248],{"className":22249,"style":367},[366],[35,22251,22253,22257,22320,22323,22389,22392,22511,22514,22623,22626,22629],{"className":22252},[77],[35,22254],{"className":22255,"style":22256},[81],"height:1.439em;vertical-align:-0.35em;",[35,22258,22260,22263],{"className":22259},[7402],[35,22261,7307],{"className":22262,"style":7408},[7402,7406,7407],[35,22264,22266],{"className":22265},[1156],[35,22267,22269,22312],{"className":22268},[98,99],[35,22270,22272,22309],{"className":22271},[103],[35,22273,22275,22295],{"className":22274,"style":7421},[107],[35,22276,22277,22280],{"style":7424},[35,22278],{"className":22279,"style":1173},[115],[35,22281,22283],{"className":22282},[120,121,122,123],[35,22284,22286,22289,22292],{"className":22285},[86,123],[35,22287,13490],{"className":22288,"style":13542},[86,156,123],[35,22290,278],{"className":22291},[371,123],[35,22293,517],{"className":22294},[86,123],[35,22296,22297,22300],{"style":7445},[35,22298],{"className":22299,"style":1173},[115],[35,22301,22303],{"className":22302},[120,121,122,123],[35,22304,22306],{"className":22305},[86,123],[35,22307,2197],{"className":22308},[86,156,123],[35,22310,162],{"className":22311},[161],[35,22313,22315],{"className":22314},[103],[35,22316,22318],{"className":22317,"style":7467},[107],[35,22319],{},[35,22321],{"className":22322,"style":1452},[366],[35,22324,22326,22332,22383],{"className":22325},[86],[35,22327,22329],{"className":22328,"style":1073},[90,1072],[35,22330,1028],{"className":22331},[1077,1078],[35,22333,22335],{"className":22334},[54],[35,22336,22338,22375],{"className":22337},[98,99],[35,22339,22341,22372],{"className":22340},[103],[35,22342,22344,22358],{"className":22343,"style":13527},[107],[35,22345,22346,22349],{"style":4503},[35,22347],{"className":22348,"style":1173},[115],[35,22350,22352],{"className":22351},[120,121,122,123],[35,22353,22355],{"className":22354},[86,123],[35,22356,13490],{"className":22357,"style":13542},[86,156,123],[35,22359,22360,22363],{"style":4518},[35,22361],{"className":22362,"style":1173},[115],[35,22364,22366],{"className":22365},[120,121,122,123],[35,22367,22369],{"className":22368},[86,123],[35,22370,2197],{"className":22371},[86,156,123],[35,22373,162],{"className":22374},[161],[35,22376,22378],{"className":22377},[103],[35,22379,22381],{"className":22380,"style":169},[107],[35,22382],{},[35,22384,22386],{"className":22385,"style":1073},[175,1072],[35,22387,1037],{"className":22388},[1077,1078],[35,22390],{"className":22391,"style":1452},[366],[35,22393,22395,22478],{"className":22394},[1065],[35,22396,22398,22404,22472],{"className":22397},[1065],[35,22399,22401],{"className":22400,"style":1073},[90,1072],[35,22402,1028],{"className":22403},[1077,1078],[35,22405,22407,22410,22469],{"className":22406},[86],[35,22408],{"className":22409},[90,91],[35,22411,22413],{"className":22412},[54],[35,22414,22416,22461],{"className":22415},[98,99],[35,22417,22419,22458],{"className":22418},[103],[35,22420,22422,22436,22444],{"className":22421,"style":312},[107],[35,22423,22424,22427],{"style":111},[35,22425],{"className":22426,"style":116},[115],[35,22428,22430],{"className":22429},[120,121,122,123],[35,22431,22433],{"className":22432},[86,123],[35,22434,404],{"className":22435},[86,123],[35,22437,22438,22441],{"style":132},[35,22439],{"className":22440,"style":116},[115],[35,22442],{"className":22443,"style":140},[139],[35,22445,22446,22449],{"style":143},[35,22447],{"className":22448,"style":116},[115],[35,22450,22452],{"className":22451},[120,121,122,123],[35,22453,22455],{"className":22454},[86,123],[35,22456,272],{"className":22457},[86,123],[35,22459,162],{"className":22460},[161],[35,22462,22464],{"className":22463},[103],[35,22465,22467],{"className":22466,"style":169},[107],[35,22468],{},[35,22470],{"className":22471},[175,91],[35,22473,22475],{"className":22474,"style":1073},[175,1072],[35,22476,1037],{"className":22477},[1077,1078],[35,22479,22481],{"className":22480},[1156],[35,22482,22484],{"className":22483},[98],[35,22485,22487],{"className":22486},[103],[35,22488,22491],{"className":22489,"style":22490},[107],"height:1.089em;",[35,22492,22493,22496],{"style":1169},[35,22494],{"className":22495,"style":1173},[115],[35,22497,22499],{"className":22498},[120,121,122,123],[35,22500,22502,22505,22508],{"className":22501},[86,123],[35,22503,2197],{"className":22504},[86,156,123],[35,22506,1353],{"className":22507},[2227,123],[35,22509,13490],{"className":22510,"style":13542},[86,156,123],[35,22512],{"className":22513,"style":1452},[366],[35,22515,22517,22600],{"className":22516},[1065],[35,22518,22520,22526,22594],{"className":22519},[1065],[35,22521,22523],{"className":22522,"style":1073},[90,1072],[35,22524,1028],{"className":22525},[1077,1078],[35,22527,22529,22532,22591],{"className":22528},[86],[35,22530],{"className":22531},[90,91],[35,22533,22535],{"className":22534},[54],[35,22536,22538,22583],{"className":22537},[98,99],[35,22539,22541,22580],{"className":22540},[103],[35,22542,22544,22558,22566],{"className":22543,"style":312},[107],[35,22545,22546,22549],{"style":111},[35,22547],{"className":22548,"style":116},[115],[35,22550,22552],{"className":22551},[120,121,122,123],[35,22553,22555],{"className":22554},[86,123],[35,22556,404],{"className":22557},[86,123],[35,22559,22560,22563],{"style":132},[35,22561],{"className":22562,"style":116},[115],[35,22564],{"className":22565,"style":140},[139],[35,22567,22568,22571],{"style":143},[35,22569],{"className":22570,"style":116},[115],[35,22572,22574],{"className":22573},[120,121,122,123],[35,22575,22577],{"className":22576},[86,123],[35,22578,272],{"className":22579},[86,123],[35,22581,162],{"className":22582},[161],[35,22584,22586],{"className":22585},[103],[35,22587,22589],{"className":22588,"style":169},[107],[35,22590],{},[35,22592],{"className":22593},[175,91],[35,22595,22597],{"className":22596,"style":1073},[175,1072],[35,22598,1037],{"className":22599},[1077,1078],[35,22601,22603],{"className":22602},[1156],[35,22604,22606],{"className":22605},[98],[35,22607,22609],{"className":22608},[103],[35,22610,22612],{"className":22611,"style":22490},[107],[35,22613,22614,22617],{"style":1169},[35,22615],{"className":22616,"style":1173},[115],[35,22618,22620],{"className":22619},[120,121,122,123],[35,22621,13490],{"className":22622,"style":13542},[86,156,123],[35,22624],{"className":22625,"style":367},[366],[35,22627,278],{"className":22628},[371],[35,22630],{"className":22631,"style":367},[366],[35,22633,22635,22638,22701,22704],{"className":22634},[77],[35,22636],{"className":22637,"style":11305},[81],[35,22639,22641,22644],{"className":22640},[7402],[35,22642,7307],{"className":22643,"style":7408},[7402,7406,7407],[35,22645,22647],{"className":22646},[1156],[35,22648,22650,22693],{"className":22649},[98,99],[35,22651,22653,22690],{"className":22652},[103],[35,22654,22656,22676],{"className":22655,"style":7421},[107],[35,22657,22658,22661],{"style":7424},[35,22659],{"className":22660,"style":1173},[115],[35,22662,22664],{"className":22663},[120,121,122,123],[35,22665,22667,22670,22673],{"className":22666},[86,123],[35,22668,13490],{"className":22669,"style":13542},[86,156,123],[35,22671,278],{"className":22672},[371,123],[35,22674,517],{"className":22675},[86,123],[35,22677,22678,22681],{"style":7445},[35,22679],{"className":22680,"style":1173},[115],[35,22682,22684],{"className":22683},[120,121,122,123],[35,22685,22687],{"className":22686},[86,123],[35,22688,2197],{"className":22689},[86,156,123],[35,22691,162],{"className":22692},[161],[35,22694,22696],{"className":22695},[103],[35,22697,22699],{"className":22698,"style":7467},[107],[35,22700],{},[35,22702],{"className":22703,"style":1452},[366],[35,22705,22707,22710,22864],{"className":22706},[86],[35,22708],{"className":22709},[90,91],[35,22711,22713],{"className":22712},[54],[35,22714,22716,22856],{"className":22715},[98,99],[35,22717,22719,22853],{"className":22718},[103],[35,22720,22722,22762,22770],{"className":22721,"style":10248},[107],[35,22723,22724,22727],{"style":111},[35,22725],{"className":22726,"style":116},[115],[35,22728,22730],{"className":22729},[120,121,122,123],[35,22731,22733],{"className":22732},[86,123],[35,22734,22736,22739],{"className":22735},[86,123],[35,22737,404],{"className":22738},[86,123],[35,22740,22742],{"className":22741},[1156],[35,22743,22745],{"className":22744},[98],[35,22746,22748],{"className":22747},[103],[35,22749,22751],{"className":22750,"style":11025},[107],[35,22752,22753,22756],{"style":7524},[35,22754],{"className":22755,"style":7528},[115],[35,22757,22759],{"className":22758},[120,7532,1078,123],[35,22760,2197],{"className":22761},[86,156,123],[35,22763,22764,22767],{"style":132},[35,22765],{"className":22766,"style":116},[115],[35,22768],{"className":22769,"style":140},[139],[35,22771,22772,22775],{"style":10282},[35,22773],{"className":22774,"style":116},[115],[35,22776,22778],{"className":22777},[120,121,122,123],[35,22779,22781],{"className":22780},[86,123],[35,22782,22784,22793,22844],{"className":22783},[86,123],[35,22785,22787],{"className":22786,"style":10299},[90,120,7532,10298,123,1072],[35,22788,22790],{"className":22789},[1077,1078,123],[35,22791,1028],{"className":22792},[123],[35,22794,22796],{"className":22795},[54],[35,22797,22799,22836],{"className":22798},[98,99],[35,22800,22802,22833],{"className":22801},[103],[35,22803,22805,22819],{"className":22804,"style":17929},[107],[35,22806,22807,22810],{"style":10321},[35,22808],{"className":22809,"style":7528},[115],[35,22811,22813],{"className":22812},[120,7532,1078,123],[35,22814,22816],{"className":22815},[86,123],[35,22817,13490],{"className":22818,"style":13542},[86,156,123],[35,22820,22821,22824],{"style":10336},[35,22822],{"className":22823,"style":7528},[115],[35,22825,22827],{"className":22826},[120,7532,1078,123],[35,22828,22830],{"className":22829},[86,123],[35,22831,2197],{"className":22832},[86,156,123],[35,22834,162],{"className":22835},[161],[35,22837,22839],{"className":22838},[103],[35,22840,22842],{"className":22841,"style":10361},[107],[35,22843],{},[35,22845,22847],{"className":22846,"style":10299},[175,120,7532,10298,123,1072],[35,22848,22850],{"className":22849},[1077,1078,123],[35,22851,1037],{"className":22852},[123],[35,22854,162],{"className":22855},[161],[35,22857,22859],{"className":22858},[103],[35,22860,22862],{"className":22861,"style":169},[107],[35,22863],{},[35,22865],{"className":22866},[175,91],[17,22868,22869,22870,22951,22952,23159,23160,23188,23189,23241],{},"El lado izquierdo es simplemente ",[35,22871,22873,22895],{"className":22872},[38],[35,22874,22876],{"className":22875},[42],[44,22877,22878],{"xmlns":46},[48,22879,22880,22892],{},[51,22881,22882,22888,22890],{},[1022,22883,22884,22886],{},[61,22885,272],{},[57,22887,2197],{},[276,22889,278],{},[61,22891,272],{},[65,22893,22894],{"encoding":67},"1^n = 1",[35,22896,22898,22942],{"className":22897,"ariaHidden":73},[72],[35,22899,22901,22904,22933,22936,22939],{"className":22900},[77],[35,22902],{"className":22903,"style":15536},[81],[35,22905,22907,22910],{"className":22906},[86],[35,22908,272],{"className":22909},[86],[35,22911,22913],{"className":22912},[1156],[35,22914,22916],{"className":22915},[98],[35,22917,22919],{"className":22918},[103],[35,22920,22922],{"className":22921,"style":15536},[107],[35,22923,22924,22927],{"style":15557},[35,22925],{"className":22926,"style":1173},[115],[35,22928,22930],{"className":22929},[120,121,122,123],[35,22931,2197],{"className":22932},[86,156,123],[35,22934],{"className":22935,"style":367},[366],[35,22937,278],{"className":22938},[371],[35,22940],{"className":22941,"style":367},[366],[35,22943,22945,22948],{"className":22944},[77],[35,22946],{"className":22947,"style":381},[81],[35,22949,272],{"className":22950},[86],": la certeza total. El lado derecho es la suma de todas las probabilidades individuales. Y cada término ",[35,22953,22955,22987],{"className":22954},[38],[35,22956,22958],{"className":22957},[42],[44,22959,22960],{"xmlns":46},[48,22961,22962,22984],{},[51,22963,22964],{},[54,22965,22966,22978],{},[51,22967,22968,22970,22976],{},[276,22969,1028],{"fence":73},[54,22971,22972,22974],{"linethickness":4321},[57,22973,2197],{},[57,22975,13490],{},[276,22977,1037],{"fence":73},[1022,22979,22980,22982],{},[61,22981,404],{},[57,22983,2197],{},[65,22985,22986],{"encoding":67},"\\frac{\\binom{n}{k}}{2^n}",[35,22988,22990],{"className":22989,"ariaHidden":73},[72],[35,22991,22993,22996],{"className":22992},[77],[35,22994],{"className":22995,"style":11305},[81],[35,22997,22999,23002,23156],{"className":22998},[86],[35,23000],{"className":23001},[90,91],[35,23003,23005],{"className":23004},[54],[35,23006,23008,23148],{"className":23007},[98,99],[35,23009,23011,23145],{"className":23010},[103],[35,23012,23014,23054,23062],{"className":23013,"style":10248},[107],[35,23015,23016,23019],{"style":111},[35,23017],{"className":23018,"style":116},[115],[35,23020,23022],{"className":23021},[120,121,122,123],[35,23023,23025],{"className":23024},[86,123],[35,23026,23028,23031],{"className":23027},[86,123],[35,23029,404],{"className":23030},[86,123],[35,23032,23034],{"className":23033},[1156],[35,23035,23037],{"className":23036},[98],[35,23038,23040],{"className":23039},[103],[35,23041,23043],{"className":23042,"style":11025},[107],[35,23044,23045,23048],{"style":7524},[35,23046],{"className":23047,"style":7528},[115],[35,23049,23051],{"className":23050},[120,7532,1078,123],[35,23052,2197],{"className":23053},[86,156,123],[35,23055,23056,23059],{"style":132},[35,23057],{"className":23058,"style":116},[115],[35,23060],{"className":23061,"style":140},[139],[35,23063,23064,23067],{"style":10282},[35,23065],{"className":23066,"style":116},[115],[35,23068,23070],{"className":23069},[120,121,122,123],[35,23071,23073],{"className":23072},[86,123],[35,23074,23076,23085,23136],{"className":23075},[86,123],[35,23077,23079],{"className":23078,"style":10299},[90,120,7532,10298,123,1072],[35,23080,23082],{"className":23081},[1077,1078,123],[35,23083,1028],{"className":23084},[123],[35,23086,23088],{"className":23087},[54],[35,23089,23091,23128],{"className":23090},[98,99],[35,23092,23094,23125],{"className":23093},[103],[35,23095,23097,23111],{"className":23096,"style":17929},[107],[35,23098,23099,23102],{"style":10321},[35,23100],{"className":23101,"style":7528},[115],[35,23103,23105],{"className":23104},[120,7532,1078,123],[35,23106,23108],{"className":23107},[86,123],[35,23109,13490],{"className":23110,"style":13542},[86,156,123],[35,23112,23113,23116],{"style":10336},[35,23114],{"className":23115,"style":7528},[115],[35,23117,23119],{"className":23118},[120,7532,1078,123],[35,23120,23122],{"className":23121},[86,123],[35,23123,2197],{"className":23124},[86,156,123],[35,23126,162],{"className":23127},[161],[35,23129,23131],{"className":23130},[103],[35,23132,23134],{"className":23133,"style":10361},[107],[35,23135],{},[35,23137,23139],{"className":23138,"style":10299},[175,120,7532,10298,123,1072],[35,23140,23142],{"className":23141},[1077,1078,123],[35,23143,1037],{"className":23144},[123],[35,23146,162],{"className":23147},[161],[35,23149,23151],{"className":23150},[103],[35,23152,23154],{"className":23153,"style":169},[107],[35,23155],{},[35,23157],{"className":23158},[175,91]," es exactamente la probabilidad de obtener ",[35,23161,23163,23176],{"className":23162},[38],[35,23164,23166],{"className":23165},[42],[44,23167,23168],{"xmlns":46},[48,23169,23170,23174],{},[51,23171,23172],{},[57,23173,13490],{},[65,23175,13490],{"encoding":67},[35,23177,23179],{"className":23178,"ariaHidden":73},[72],[35,23180,23182,23185],{"className":23181},[77],[35,23183],{"className":23184,"style":7026},[81],[35,23186,13490],{"className":23187,"style":13542},[86,156]," cruces (o equivalentemente, ",[35,23190,23192,23210],{"className":23191},[38],[35,23193,23195],{"className":23194},[42],[44,23196,23197],{"xmlns":46},[48,23198,23199,23207],{},[51,23200,23201,23203,23205],{},[57,23202,2197],{},[276,23204,1353],{},[57,23206,13490],{},[65,23208,23209],{"encoding":67},"n - k",[35,23211,23213,23232],{"className":23212,"ariaHidden":73},[72],[35,23214,23216,23220,23223,23226,23229],{"className":23215},[77],[35,23217],{"className":23218,"style":23219},[81],"height:0.6667em;vertical-align:-0.0833em;",[35,23221,2197],{"className":23222},[86,156],[35,23224],{"className":23225,"style":2223},[366],[35,23227,1353],{"className":23228},[2227],[35,23230],{"className":23231,"style":2223},[366],[35,23233,23235,23238],{"className":23234},[77],[35,23236],{"className":23237,"style":7026},[81],[35,23239,13490],{"className":23240,"style":13542},[86,156]," caras).",[17,23243,23244],{},"Comprobémoslo con 5 lanzamientos:",[17,23246,23248],{"className":23247},[592],[35,23249,23251,23303],{"className":23250},[38],[35,23252,23254],{"className":23253},[42],[44,23255,23256],{"xmlns":46},[48,23257,23258,23300],{},[51,23259,23260,23288,23290,23296,23298],{},[54,23261,23262,23286],{},[51,23263,23264,23266,23268,23270,23272,23274,23276,23278,23280,23282,23284],{},[61,23265,272],{},[276,23267,1316],{},[61,23269,529],{},[276,23271,1316],{},[61,23273,15755],{},[276,23275,1316],{},[61,23277,15755],{},[276,23279,1316],{},[61,23281,529],{},[276,23283,1316],{},[61,23285,272],{},[61,23287,1847],{},[276,23289,278],{},[54,23291,23292,23294],{},[61,23293,1847],{},[61,23295,1847],{},[276,23297,278],{},[61,23299,272],{},[65,23301,23302],{"encoding":67},"\\frac{1 + 5 + 10 + 10 + 5 + 1}{32} = \\frac{32}{32} = 1",[35,23304,23306,23419,23502],{"className":23305,"ariaHidden":73},[72],[35,23307,23309,23312,23410,23413,23416],{"className":23308},[77],[35,23310],{"className":23311,"style":293},[81],[35,23313,23315,23318,23407],{"className":23314},[86],[35,23316],{"className":23317},[90,91],[35,23319,23321],{"className":23320},[54],[35,23322,23324,23399],{"className":23323},[98,99],[35,23325,23327,23396],{"className":23326},[103],[35,23328,23330,23344,23352],{"className":23329,"style":312},[107],[35,23331,23332,23335],{"style":111},[35,23333],{"className":23334,"style":116},[115],[35,23336,23338],{"className":23337},[120,121,122,123],[35,23339,23341],{"className":23340},[86,123],[35,23342,1847],{"className":23343},[86,123],[35,23345,23346,23349],{"style":132},[35,23347],{"className":23348,"style":116},[115],[35,23350],{"className":23351,"style":140},[139],[35,23353,23354,23357],{"style":143},[35,23355],{"className":23356,"style":116},[115],[35,23358,23360],{"className":23359},[120,121,122,123],[35,23361,23363,23366,23369,23372,23375,23378,23381,23384,23387,23390,23393],{"className":23362},[86,123],[35,23364,272],{"className":23365},[86,123],[35,23367,1316],{"className":23368},[2227,123],[35,23370,529],{"className":23371},[86,123],[35,23373,1316],{"className":23374},[2227,123],[35,23376,15755],{"className":23377},[86,123],[35,23379,1316],{"className":23380},[2227,123],[35,23382,15755],{"className":23383},[86,123],[35,23385,1316],{"className":23386},[2227,123],[35,23388,529],{"className":23389},[86,123],[35,23391,1316],{"className":23392},[2227,123],[35,23394,272],{"className":23395},[86,123],[35,23397,162],{"className":23398},[161],[35,23400,23402],{"className":23401},[103],[35,23403,23405],{"className":23404,"style":169},[107],[35,23406],{},[35,23408],{"className":23409},[175,91],[35,23411],{"className":23412,"style":367},[366],[35,23414,278],{"className":23415},[371],[35,23417],{"className":23418,"style":367},[366],[35,23420,23422,23425,23493,23496,23499],{"className":23421},[77],[35,23423],{"className":23424,"style":293},[81],[35,23426,23428,23431,23490],{"className":23427},[86],[35,23429],{"className":23430},[90,91],[35,23432,23434],{"className":23433},[54],[35,23435,23437,23482],{"className":23436},[98,99],[35,23438,23440,23479],{"className":23439},[103],[35,23441,23443,23457,23465],{"className":23442,"style":312},[107],[35,23444,23445,23448],{"style":111},[35,23446],{"className":23447,"style":116},[115],[35,23449,23451],{"className":23450},[120,121,122,123],[35,23452,23454],{"className":23453},[86,123],[35,23455,1847],{"className":23456},[86,123],[35,23458,23459,23462],{"style":132},[35,23460],{"className":23461,"style":116},[115],[35,23463],{"className":23464,"style":140},[139],[35,23466,23467,23470],{"style":143},[35,23468],{"className":23469,"style":116},[115],[35,23471,23473],{"className":23472},[120,121,122,123],[35,23474,23476],{"className":23475},[86,123],[35,23477,1847],{"className":23478},[86,123],[35,23480,162],{"className":23481},[161],[35,23483,23485],{"className":23484},[103],[35,23486,23488],{"className":23487,"style":169},[107],[35,23489],{},[35,23491],{"className":23492},[175,91],[35,23494],{"className":23495,"style":367},[366],[35,23497,278],{"className":23498},[371],[35,23500],{"className":23501,"style":367},[366],[35,23503,23505,23508],{"className":23504},[77],[35,23506],{"className":23507,"style":381},[81],[35,23509,272],{"className":23510},[86],[17,23512,23513,23514,23573,23574,23602,23603,176],{},"Cada número de la fila 5 dividido entre ",[35,23515,23517,23535],{"className":23516},[38],[35,23518,23520],{"className":23519},[42],[44,23521,23522],{"xmlns":46},[48,23523,23524,23532],{},[51,23525,23526],{},[1022,23527,23528,23530],{},[61,23529,404],{},[61,23531,529],{},[65,23533,23534],{"encoding":67},"2^5",[35,23536,23538],{"className":23537,"ariaHidden":73},[72],[35,23539,23541,23544],{"className":23540},[77],[35,23542],{"className":23543,"style":15811},[81],[35,23545,23547,23550],{"className":23546},[86],[35,23548,404],{"className":23549},[86],[35,23551,23553],{"className":23552},[1156],[35,23554,23556],{"className":23555},[98],[35,23557,23559],{"className":23558},[103],[35,23560,23562],{"className":23561,"style":15811},[107],[35,23563,23564,23567],{"style":15557},[35,23565],{"className":23566,"style":1173},[115],[35,23568,23570],{"className":23569},[120,121,122,123],[35,23571,529],{"className":23572},[86,123]," nos da una probabilidad, y juntas suman ",[35,23575,23577,23590],{"className":23576},[38],[35,23578,23580],{"className":23579},[42],[44,23581,23582],{"xmlns":46},[48,23583,23584,23588],{},[51,23585,23586],{},[61,23587,272],{},[65,23589,272],{"encoding":67},[35,23591,23593],{"className":23592,"ariaHidden":73},[72],[35,23594,23596,23599],{"className":23595},[77],[35,23597],{"className":23598,"style":381},[81],[35,23600,272],{"className":23601},[86],". El triángulo, la combinatoria y el álgebra no son tres historias diferentes: ",[239,23604,23605],{},"son la misma historia contada en tres idiomas distintos",[561,23607,12840],{"id":12839},[17,23609,23610],{},"A lo largo de este artículo hemos seguido un hilo que conecta tres ramas de las matemáticas a través de un mismo objeto:",[3549,23612,23613,23618,23623],{},[232,23614,15280,23615,23617],{},[239,23616,15283],{}," nos dice de cuántas maneras podemos organizar un conjunto de resultados.",[232,23619,4248,23620,23622],{},[239,23621,19280],{}," del binomio genera esas mismas cantidades como coeficientes de una fórmula.",[232,23624,15280,23625,23627],{},[239,23626,6469],{}," transforma esos números en predicciones sobre el mundo real.",[17,23629,23630,23631,23633],{},"Y el ",[239,23632,13145],{}," es el objeto que las une: una tabla visual donde las tres disciplinas convergen.",[17,23635,23636],{},"Lo que empezó como un triángulo para contar combinaciones — dibujado en la China del siglo XI — termina describiendo con precisión qué ocurre cuando lanzas una moneda cinco, diez o cien veces. Una estructura abstracta creada para contar objetos resulta describir fenómenos aleatorios.",[17,23638,23639],{},"Y hay un detalle más que merece una mención. A medida que el número de lanzamientos crece, las filas del triángulo se parecen cada vez más a la curva de campana de Gauss: esa curva ubicua que aparece en la estadística, la física y la biología. La razón profunda de esa convergencia es otra historia, pero el hecho de que un triángulo de sumas tan simples contenga la semilla de una de las distribuciones más importantes de la ciencia es, como mínimo, hermoso.",[17,23641,23642],{},"Las matemáticas tienen esa cualidad única: revelan patrones ocultos detrás de lo que parece puro azar. Un lanzamiento de moneda es impredecible. Diez lanzamientos ya insinúan un orden. Cien lanzamientos dibujan una curva casi perfecta. Y todo estaba escrito, desde el principio, en un triángulo.",[17,23644,12947,23645,23648],{},[21,23646,12951],{"href":578,"rel":23647},[25]," y observa la combinatoria en acción.",{"title":1291,"searchDepth":12954,"depth":12954,"links":23650},[23651,23652,23655,23656,23659,23660,23663],{"id":13138,"depth":12954,"text":13139},{"id":13166,"depth":12954,"text":13167,"children":23653},[23654],{"id":13459,"depth":12959,"text":13460},{"id":15276,"depth":12954,"text":15277},{"id":15772,"depth":12954,"text":15773,"children":23657},[23658],{"id":16222,"depth":12959,"text":16223},{"id":17994,"depth":12954,"text":17995},{"id":19273,"depth":12954,"text":19274,"children":23661},[23662],{"id":21519,"depth":12959,"text":21520},{"id":12839,"depth":12954,"text":12840},"2026-05-24T10:00:00Z","Descubre cómo el triángulo de Tartaglia conecta la combinatoria, el álgebra y la probabilidad — explicado paso a paso con lanzamientos de moneda.",{"excerpt":23667},{"type":10,"value":23668},[23669,23671],[13,23670,13127],{"id":13132},[17,23672,13135],{},"\u002Fblog\u002Fpascals-triangle-and-probability.es",{"title":13127,"description":23665},"triangulo-de-tartaglia-combinatoria-y-probabilidad","blog\u002Fpascals-triangle-and-probability.es","qZQyHpms2FZH6Ff6w11iThobO-PzvEaP5Ib4ymGhG00",{"id":23679,"title":23680,"_locale":8,"body":23681,"date":24101,"description":24102,"extension":12971,"meta":24103,"navigation":13119,"path":24110,"published":13119,"seo":24111,"slug":24112,"stem":24113,"__hash__":24114},"blog\u002Fblog\u002Fheads-or-tails.es.md","Cara o Cruz — La Guía Completa sobre el Lanzamiento de Moneda",{"type":10,"value":23682,"toc":24093},[23683,23686,23689,23693,23699,23713,23720,23724,23731,23816,23901,23974,23986,24004,24008,24015,24026,24029,24033,24039,24045,24051,24057,24061,24068,24071,24074,24078,24084],[13,23684,23680],{"id":23685},"cara-o-cruz-la-guía-completa-sobre-el-lanzamiento-de-moneda",[17,23687,23688],{},"\"¿Cara o cruz?\" Tres palabras que han resuelto disputas, iniciado juegos y tomado decisiones durante miles de años. Pero ¿de dónde viene esa frase? ¿Y es la moneda realmente justa?",[561,23690,23692],{"id":23691},"de-dónde-vienen-cara-y-cruz","¿De dónde vienen \"cara\" y \"cruz\"?",[17,23694,23695,23698],{},[239,23696,23697],{},"Cara"," es sencillo: las monedas han tenido retratos desde la antigüedad. Las monedas griegas y romanas mostraban el perfil de un gobernante, deidad o emperador — una cara literal.",[17,23700,23701,23704,23705,23708,23709,23712],{},[239,23702,23703],{},"Cruz"," tiene una explicación más local. En la tradición hispánica, el reverso de muchas monedas mostraba una cruz — el símbolo religioso que marcaba el \"otro lado\" de la moneda. Así quedó establecida la dicotomía cara\u002Fcruz que usamos hoy. En latín, el juego se llamaba ",[212,23706,23707],{},"caput aut navis"," (\"cabeza o barco\"), porque el reverso mostraba la proa de una nave. En inglés moderno se dice ",[212,23710,23711],{},"heads or tails"," (\"cabeza o cola\"), refiriéndose a los animales representados en el reverso de las monedas británicas.",[17,23714,23715,23716,23719],{},"La versión romana evolucionó hasta el juego que Julio César hizo famoso: cuando las monedas empezaron a llevar su retrato, el juego pasó a llamarse ",[212,23717,23718],{},"capita aut navia"," (\"cabeza o barco\") en su honor.",[561,23721,23723],{"id":23722},"las-matemáticas-del-lanzamiento-de-moneda","Las matemáticas del lanzamiento de moneda",[17,23725,23726,23727,23730],{},"En teoría de la probabilidad, el lanzamiento de una moneda justa es el ejemplo por excelencia de un ",[239,23728,23729],{},"ensayo de Bernoulli",": un experimento aleatorio con exactamente dos resultados igualmente probables (1 ó 0). La probabilidad de cara es 0,5, la probabilidad de cruz es 0,5, y cada lanzamiento es independiente de todos los demás. En notación matemática, se representa como:",[17,23732,23734],{"className":23733},[592],[35,23735,23737,23765],{"className":23736},[38],[35,23738,23740],{"className":23739},[42],[44,23741,23742],{"xmlns":46},[48,23743,23744,23762],{},[51,23745,23746,23748,23750,23752,23754,23756,23758,23760],{},[57,23747,16246],{},[276,23749,1028],{"stretchy":9816},[606,23751,15293],{},[276,23753,1037],{"stretchy":9816},[276,23755,278],{},[61,23757,517],{},[276,23759,525],{"separator":73},[61,23761,529],{},[65,23763,23764],{"encoding":67},"P(\\text{cara}) = 0,5",[35,23766,23768,23798],{"className":23767,"ariaHidden":73},[72],[35,23769,23771,23774,23777,23780,23786,23789,23792,23795],{"className":23770},[77],[35,23772],{"className":23773,"style":9837},[81],[35,23775,16246],{"className":23776,"style":16272},[86,156],[35,23778,1028],{"className":23779},[90],[35,23781,23783],{"className":23782},[86,625],[35,23784,15293],{"className":23785},[86],[35,23787,1037],{"className":23788},[175],[35,23790],{"className":23791,"style":367},[366],[35,23793,278],{"className":23794},[371],[35,23796],{"className":23797,"style":367},[366],[35,23799,23801,23804,23807,23810,23813],{"className":23800},[77],[35,23802],{"className":23803,"style":513},[81],[35,23805,517],{"className":23806},[86],[35,23808,525],{"className":23809},[524],[35,23811],{"className":23812,"style":1452},[366],[35,23814,529],{"className":23815},[86],[17,23817,23819],{"className":23818},[592],[35,23820,23822,23850],{"className":23821},[38],[35,23823,23825],{"className":23824},[42],[44,23826,23827],{"xmlns":46},[48,23828,23829,23847],{},[51,23830,23831,23833,23835,23837,23839,23841,23843,23845],{},[57,23832,16246],{},[276,23834,1028],{"stretchy":9816},[606,23836,15297],{},[276,23838,1037],{"stretchy":9816},[276,23840,278],{},[61,23842,517],{},[276,23844,525],{"separator":73},[61,23846,529],{},[65,23848,23849],{"encoding":67},"P(\\text{cruz}) = 0,5",[35,23851,23853,23883],{"className":23852,"ariaHidden":73},[72],[35,23854,23856,23859,23862,23865,23871,23874,23877,23880],{"className":23855},[77],[35,23857],{"className":23858,"style":9837},[81],[35,23860,16246],{"className":23861,"style":16272},[86,156],[35,23863,1028],{"className":23864},[90],[35,23866,23868],{"className":23867},[86,625],[35,23869,15297],{"className":23870},[86],[35,23872,1037],{"className":23873},[175],[35,23875],{"className":23876,"style":367},[366],[35,23878,278],{"className":23879},[371],[35,23881],{"className":23882,"style":367},[366],[35,23884,23886,23889,23892,23895,23898],{"className":23885},[77],[35,23887],{"className":23888,"style":513},[81],[35,23890,517],{"className":23891},[86],[35,23893,525],{"className":23894},[524],[35,23896],{"className":23897,"style":1452},[366],[35,23899,529],{"className":23900},[86],[17,23902,23904],{"className":23903},[592],[35,23905,23907,23932],{"className":23906},[38],[35,23908,23910],{"className":23909},[42],[44,23911,23912],{"xmlns":46},[48,23913,23914,23929],{},[51,23915,23916,23918,23920,23923,23925,23927],{},[57,23917,16246],{},[276,23919,1028],{"stretchy":9816},[606,23921,23922],{},"cara o cruz",[276,23924,1037],{"stretchy":9816},[276,23926,278],{},[61,23928,272],{},[65,23930,23931],{"encoding":67},"P(\\text{cara o cruz}) = 1",[35,23933,23935,23965],{"className":23934,"ariaHidden":73},[72],[35,23936,23938,23941,23944,23947,23953,23956,23959,23962],{"className":23937},[77],[35,23939],{"className":23940,"style":9837},[81],[35,23942,16246],{"className":23943,"style":16272},[86,156],[35,23945,1028],{"className":23946},[90],[35,23948,23950],{"className":23949},[86,625],[35,23951,23922],{"className":23952},[86],[35,23954,1037],{"className":23955},[175],[35,23957],{"className":23958,"style":367},[366],[35,23960,278],{"className":23961},[371],[35,23963],{"className":23964,"style":367},[366],[35,23966,23968,23971],{"className":23967},[77],[35,23969],{"className":23970,"style":381},[81],[35,23972,272],{"className":23973},[86],[17,23975,23976,23977,23980,23981,176],{},"Esa independencia tiene una implicación famosa: si acabas de sacar diez caras seguidas, el undécimo lanzamiento sigue siendo 50\u002F50. La moneda no tiene memoria. Creer lo contrario es la ",[239,23978,23979],{},"Falacia del Jugador"," — ",[21,23982,23985],{"href":23983,"rel":23984},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fla-estadistica-inesperada-de-los-lanzamientos-de-moneda",[25],"explorada en profundidad aquí",[17,23987,23988,23989,23991,23992,23995,23996,530,23998,24000,24001,24003],{},"El número de caras esperado en ",[212,23990,2197],{}," lanzamientos sigue una ",[239,23993,23994],{},"distribución binomial"," con parámetros ",[212,23997,2197],{},[212,23999,17],{}," = 0,5. Para ",[212,24002,2197],{}," grande, esto se aproxima a una distribución normal — razón por la que los lanzamientos de moneda se usan para enseñar el Teorema Central del Límite, el cual explica que la suma de muchas variables aleatorias independientes sigue una distribución normal (forma de campana).",[561,24005,24007],{"id":24006},"es-realmente-justo-un-lanzamiento-de-moneda","¿Es realmente justo un lanzamiento de moneda?",[17,24009,24010,24011,24014],{},"No del todo. El estudio del matemático de Stanford Persi Diaconis mencionado en este blog demostró que una moneda lanzada de forma natural cae sobre la misma cara con la que empezó aproximadamente el ",[239,24012,24013],{},"51% de las veces",". El sesgo viene de:",[3549,24016,24017,24020,24023],{},[232,24018,24019],{},"La moneda pasa ligeramente más tiempo rotando en su orientación inicial antes de que la gravedad tome el control",[232,24021,24022],{},"La mecánica del pulgar que consistentemente favorece una dirección",[232,24024,24025],{},"La resistencia del aire que afecta la cara que empieza hacia arriba",[17,24027,24028],{},"Un sesgo del 1% suena pequeño. En un solo lanzamiento, es irrelevante. En miles de lanzamientos en un torneo o estudio estadístico, importa. Aunque parezca que los simuladores online basados en física no heredan la mecánica humana de lanzamiento, arrastran parte de ese sesgo y no pueden garantizar un resultado 50\u002F50, aunque se repita el lanzamiento muchas veces.",[561,24030,24032],{"id":24031},"lanzamientos-de-moneda-famosos-en-la-historia","Lanzamientos de moneda famosos en la historia",[17,24034,24035,24038],{},[239,24036,24037],{},"La Copa Davis",": El protocolo oficial de la ITF exige un lanzamiento de moneda antes de cada eliminatoria para decidir quién elige pista o servicio. Millones de partidos han comenzado así.",[17,24040,24041,24044],{},[239,24042,24043],{},"Los Hermanos Wright (1903)",": Orville y Wilbur Wright supuestamente lanzaron una moneda para decidir quién intentaría el primer vuelo motorizado en Kitty Hawk. Wilbur ganó el sorteo pero su primer intento falló. El segundo intento de Orville, tres días después, tuvo éxito.",[17,24046,24047,24050],{},[239,24048,24049],{},"El Caucus Demócrata de Iowa (2015)",": Varios distritos empataron entre Hillary Clinton y Bernie Sanders. ¿El desempate? Un lanzamiento de moneda. Clinton supuestamente ganó los seis lanzamientos — una secuencia con una probabilidad de aproximadamente el 1,6%, improbable pero completamente posible.",[17,24052,24053,24056],{},[239,24054,24055],{},"El fútbol",": La FIFA exige lanzamientos de moneda antes de los partidos internacionales. El equipo perdedor elige el extremo que defenderá; el capitán ganador decide si hace el saque inicial o lo cede al rival.",[561,24058,24060],{"id":24059},"por-qué-confiamos-en-los-lanzamientos-de-moneda","Por qué confiamos en los lanzamientos de moneda",[17,24062,24063,24064,24067],{},"Hay algo filosóficamente interesante en el lanzamiento de moneda: se acepta como justo precisamente porque ",[212,24065,24066],{},"nadie lo controla",". La moneda es un árbitro que no tiene interés en el resultado.",[17,24069,24070],{},"Por eso los lanzamientos de moneda se usan para resolver disputas en lugar de, por ejemplo, piedra-papel-tijera (que puede verse influenciado por la psicología y las señales). El elemento aleatorio elimina la agencia humana — y con ella, la posibilidad de queja.",[17,24072,24073],{},"Un lanzamiento de moneda online cumple la misma función. Cuando ambas partes pueden ver el lanzamiento en tiempo real, el resultado tiene el mismo peso social que un lanzamiento físico.",[561,24075,24077],{"id":24076},"conclusión","Conclusión",[17,24079,24080,24083],{},[239,24081,24082],{},"Cara o cruz"," es una de las herramientas de aleatorización más antiguas de la historia humana — simple, fiable y, como resulta, no perfectamente justa en el mundo real. Un simulador de física online te acerca más al ideal teórico de lo que podría hacerlo un lanzamiento humano.",[17,24085,24086,24087,24092],{},"¿Listo para lanzar? ",[21,24088,24091],{"href":24089,"rel":24090},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fplay",[25],"Prueba cara o cruz online"," — física real, resultado instantáneo.",{"title":1291,"searchDepth":12954,"depth":12954,"links":24094},[24095,24096,24097,24098,24099,24100],{"id":23691,"depth":12954,"text":23692},{"id":23722,"depth":12954,"text":23723},{"id":24006,"depth":12954,"text":24007},{"id":24031,"depth":12954,"text":24032},{"id":24059,"depth":12954,"text":24060},{"id":24076,"depth":12954,"text":24077},"2026-05-22T12:00:00Z","¿Por qué se llama cara o cruz? Historia, matemáticas y la sorprendente ciencia del lanzamiento de moneda — de Julio César a la física moderna.",{"excerpt":24104},{"type":10,"value":24105},[24106,24108],[13,24107,23680],{"id":23685},[17,24109,23688],{},"\u002Fblog\u002Fheads-or-tails.es",{"title":23680,"description":24102},"cara-o-cruz","blog\u002Fheads-or-tails.es","d01guGbrghT6q7EPXImYAgI3Mnc7bMm9ywHZ4KkDC2U",{"id":24116,"title":24117,"_locale":8,"body":24118,"date":24278,"description":24279,"extension":12971,"meta":24280,"navigation":13119,"path":24287,"published":13119,"seo":24288,"slug":24289,"stem":24290,"__hash__":24291},"blog\u002Fblog\u002Fhow-to-flip-a-coin-online.es.md","Cómo Lanzar una Moneda Online — Y Por Qué es Más Justo que el Lanzamiento Real",{"type":10,"value":24119,"toc":24270},[24120,24123,24126,24130,24133,24145,24148,24152,24159,24169,24172,24176,24208,24212,24215,24222,24226,24229,24255,24258,24260,24263],[13,24121,24117],{"id":24122},"cómo-lanzar-una-moneda-online-y-por-qué-es-más-justo-que-el-lanzamiento-real",[17,24124,24125],{},"¿Necesitas tomar una decisión rápida y justa? Lanzar una moneda online es la forma más rápida — sin moneda, sin mesa, sin discusión sobre si el lanzamiento fue justo. Esta guía explica cómo funciona, por qué importa, y cuándo es mejor usar un lanzador de moneda online en lugar de buscar monedas en los bolsillos.",[561,24127,24129],{"id":24128},"por-qué-lanzar-una-moneda-online","¿Por qué lanzar una moneda online?",[17,24131,24132],{},"La respuesta obvia: no tienes una moneda. Pero hay una razón más interesante.",[17,24134,24135,24136,24141,24142,24144],{},"Los lanzamientos de moneda reales no son perfectamente justos. Un ",[21,24137,24140],{"href":24138,"rel":24139},"https:\u002F\u002Fwww.stat.berkeley.edu\u002F~aldous\u002F157\u002FPapers\u002Fdiaconis_coinbias.pdf",[25],"estudio de 2007 de Diaconis, Holmes y Montgomery"," descubrió que una moneda cae sobre la misma cara con la que empezó aproximadamente el ",[239,24143,24013],{},". La mecánica de un lanzamiento humano — ángulo del brazo, fuerza del pulgar, resistencia del aire — introduce consistentemente un pequeño sesgo hacia la cara inicial.",[17,24146,24147],{},"Un lanzamiento de moneda online que usa un generador de números aleatorios adecuado elimina ese sesgo por completo.",[561,24149,24151],{"id":24150},"cómo-funciona-flipthecoinapp","Cómo funciona FlipTheCoin.app",[17,24153,24154,24155,24158],{},"La mayoría de lanzadores de moneda online usan una sola llamada a ",[1293,24156,24157],{},"Math.random()"," y animan algo bonito encima. Funciona, pero significa que la animación es puramente cosmética — el resultado ya está decidido antes de que la moneda empiece a moverse.",[17,24160,24161,24165,24166,24168],{},[21,24162,24164],{"href":24089,"rel":24163},[25],"FlipTheCoin.app"," lo hace de otra manera. La moneda es una simulación física real: un disco con masa, una fuerza de lanzamiento, un giro y gravedad. Los tres parámetros aleatorios — fuerza del empuje, deriva lateral y velocidad angular — se generan a partir de ",[1293,24167,24157],{}," en el momento en que lanzas. El motor de física ejecuta la simulación hasta que la moneda se detiene. La cara que queda mirando al techo es el resultado.",[17,24170,24171],{},"Eso significa que la animación es la física, no decoración. El resultado no se decide hasta que la moneda deja de moverse — igual que en la realidad.",[561,24173,24175],{"id":24174},"cuándo-deberías-lanzar-una-moneda-online","¿Cuándo deberías lanzar una moneda online?",[3549,24177,24178,24184,24190,24196,24202],{},[232,24179,24180,24183],{},[239,24181,24182],{},"Tomar una decisión",": El uso clásico. Si realmente estás indeciso entre dos opciones, lanzar una moneda externaliza la elección y — lo importante — tu reacción al resultado a menudo revela cuál opción querías realmente.",[232,24185,24186,24189],{},[239,24187,24188],{},"Deportes y juegos",": Decidir quién empieza en un juego de mesa, quién saca primero en tenis, quién elige el mapa en un videojuego.",[232,24191,24192,24195],{},[239,24193,24194],{},"Clase y trabajo remoto",": Hacer una lotería justa cuando todos están en pantalla. Comparte tu pantalla, lanza, listo.",[232,24197,24198,24201],{},[239,24199,24200],{},"Reemplazar una moneda perdida",": Antes de un evento deportivo cuando nadie tiene cambio. Los árbitros han usado lanzamientos de teléfono en partidos oficiales.",[232,24203,24204,24207],{},[239,24205,24206],{},"Experimentos de probabilidad",": Lanzar 100 veces y graficar la proporción cara\u002Fcruz — FlipTheCoin.app guarda un historial y estadísticas exactamente para esto.",[561,24209,24211],{"id":24210},"es-suficiente-con-un-solo-lanzamiento","¿Es suficiente con un solo lanzamiento?",[17,24213,24214],{},"Para decisiones simples: sí. Para resolver una disputa al mejor de varios, quizás quieras varios lanzamientos.",[17,24216,24217,24218,24221],{},"Algo a tener en cuenta: cada lanzamiento es estadísticamente independiente. Una racha de cinco caras seguidas no hace que cruz sea más probable en el sexto lanzamiento. Esa intuición — \"tiene que equilibrarse pronto\" — se llama la ",[21,24219,23979],{"href":23983,"rel":24220},[25]," y es incorrecta. La moneda no tiene memoria.",[561,24223,24225],{"id":24224},"importa-qué-aplicación-se-use","¿Importa qué aplicación se use?",[17,24227,24228],{},"Depende de cuánto te importe la imparcialidad.",[3549,24230,24231,24239,24249],{},[232,24232,24233,24238],{},[239,24234,24235,24236],{},"Envoltorios de ",[1293,24237,24157],{},": Bien para uso casual. La calidad de la aleatoriedad es suficiente para decisiones cotidianas.",[232,24240,24241,24244,24245,24248],{},[239,24242,24243],{},"Simuladores basados en física"," (como FlipTheCoin.app): Mejor para situaciones donde el ",[212,24246,24247],{},"proceso"," necesita sentirse tan justo como serlo — la animación es la fuente de verdad, no una cobertura.",[232,24250,24251,24254],{},[239,24252,24253],{},"Aleatorio criptográficamente seguro",": Excesivo para lanzamientos de moneda, apropiado para generación de claves criptográficas.",[17,24256,24257],{},"Para la gran mayoría de usos — decidir quién friega, elegir un capitán de equipo, desempatar — cualquier lanzador de moneda online de confianza es más que suficientemente justo.",[561,24259,24077],{"id":24076},[17,24261,24262],{},"Lanzar una moneda online es más rápido, más conveniente y — para los simuladores basados en física — más imparcial que un lanzamiento físico. Ya sea para tomar una decisión rápida o para un experimento estadístico, no necesitas buscar una moneda.",[17,24264,24265,24269],{},[21,24266,24268],{"href":24089,"rel":24267},[25],"Lanza una moneda online ahora"," — cara o cruz con un solo clic.",{"title":1291,"searchDepth":12954,"depth":12954,"links":24271},[24272,24273,24274,24275,24276,24277],{"id":24128,"depth":12954,"text":24129},{"id":24150,"depth":12954,"text":24151},{"id":24174,"depth":12954,"text":24175},{"id":24210,"depth":12954,"text":24211},{"id":24224,"depth":12954,"text":24225},{"id":24076,"depth":12954,"text":24077},"2026-05-22T10:00:00Z","Guía práctica para lanzar una moneda online, por qué un simulador basado en física es más imparcial que una moneda real, y cuándo usarlo.",{"excerpt":24281},{"type":10,"value":24282},[24283,24285],[13,24284,24117],{"id":24122},[17,24286,24125],{},"\u002Fblog\u002Fhow-to-flip-a-coin-online.es",{"title":24117,"description":24279},"como-lanzar-una-moneda-online","blog\u002Fhow-to-flip-a-coin-online.es","0PcX4s21_VPx2IrXnqHAnODvUBA5kFocwQ1gzzFJhhA",{"id":24293,"title":24294,"_locale":8,"body":24295,"date":24542,"description":24543,"extension":12971,"meta":24544,"navigation":13119,"path":24551,"published":13119,"seo":24552,"slug":24553,"stem":24554,"__hash__":24555},"blog\u002Fblog\u002Fhow-we-toss-a-coin.es.md","Cómo lanzamos una moneda en FlipTheCoin.app",{"type":10,"value":24296,"toc":24530},[24297,24300,24303,24315,24318,24322,24325,24343,24350,24354,24357,24361,24364,24368,24371,24374,24378,24388,24391,24394,24398,24405,24408,24412,24415,24422,24451,24464,24468,24471,24480,24497,24505,24507,24510,24522],[13,24298,24294],{"id":24299},"cómo-lanzamos-una-moneda-en-flipthecoinapp",[17,24301,24302],{},"Las matemáticas detrás de un lanzamiento de moneda son simples: una moneda tiene dos caras, cara y cruz, y cada cara tiene la misma probabilidad de quedar hacia arriba. Pero, ¿cómo lanzamos una moneda de verdad? En este artículo exploraremos la ciencia detrás de los lanzamientos y cómo nos aseguramos de que cada uno sea justo y aleatorio.",[17,24304,24305,24306,24308,24309,24314],{},"Se te perdonaría pensar que un lanzamiento de moneda en un navegador no es más que ",[1293,24307,24157],{}," disfrazado de animación bonita. Es una forma de hacerlo — y de hecho ofrecemos ese modo en el ",[21,24310,24313],{"href":24311,"rel":24312},"https:\u002F\u002Fflipthecoin.app\u002Ftoss-engine",[25],"motor de lanzamiento"," si quieres un 50\u002F50 limpio sin dramas.",[17,24316,24317],{},"Pero la moneda que ves en la portada y en nuestra extensión del navegador no es \"un número aleatorio disfrazado\". Es una pequeña simulación física real: un disco, un suelo, gravedad y un empujón. La cara que acaba mirando al techo gana. Vamos a ver qué significa eso.",[561,24319,24321],{"id":24320},"por-qué-simular-la-física","¿Por qué simular la física?",[17,24323,24324],{},"Dos razones:",[229,24326,24327,24333],{},[232,24328,24329,24332],{},[239,24330,24331],{},"Es más divertido de mirar."," Una moneda lanzada, girando y rebotando parece una moneda. Una escena que dice \"CARA\" no lo parece.",[232,24334,24335,24338,24339,24342],{},[239,24336,24337],{},"Refleja cómo funcionan los lanzamientos reales."," En el mundo real, un lanzamiento de moneda no es matemáticamente aleatorio — es un proceso físico determinista que es sensible a las diferencias minúsculas en cómo lo lanzas. Nuestra simulación tiene el mismo carácter: la aleatoriedad vive enteramente en ",[212,24340,24341],{},"cómo la lanzamos",", y el resto es física.",[17,24344,24345,24346,24349],{},"Si quieres un 50\u002F50 matemáticamente limpio (y nunca un \"canto\"), usa el ",[21,24347,24313],{"href":24311,"rel":24348},[25],". Si quieres la sensación real, sigue leyendo.",[561,24351,24353],{"id":24352},"los-tres-ingredientes-de-un-lanzamiento","Los tres ingredientes de un lanzamiento",[17,24355,24356],{},"Cada vez que haces clic en la moneda, le damos tres cosas: un empujón hacia arriba, una pequeña deriva lateral y un giro.",[582,24358,24360],{"id":24359},"_1-un-empujón-hacia-arriba","1. Un empujón hacia arriba",[17,24362,24363],{},"Primero la empujamos hacia el aire. En nuestra simulación ese empujón es un valor fijo — unas 9 unidades de velocidad vertical — idéntico cada vez. Esto marca cuánto tiempo la moneda permanece en el aire. No es aleatorio; solo la altura suficiente para un buen tumbo.",[582,24365,24367],{"id":24366},"_2-una-inclinación-lateral-consistente-con-algo-de-variación","2. Una inclinación lateral consistente con algo de variación",[17,24369,24370],{},"Si cada lanzamiento saliera del mismo punto y aterrizara en el mismo sitio, la moneda se apilaría como una pila de tortitas. Así que también le damos un empujón lateral en los dos ejes horizontales — mayormente un valor fijo, con una pequeña adición aleatoria cada vez. Es la misma dirección general en cada lanzamiento (para que la moneda no salga volando — al menos no demasiado), pero el punto exacto de aterrizaje varía.",[17,24372,24373],{},"Aquí es donde nace la aleatoriedad en nuestra simulación: introducimos un valor \"semilla\" que determina la dirección exacta del empujón lateral.",[582,24375,24377],{"id":24376},"_3-un-giro","3. Un giro",[17,24379,24380,24381,24384,24385,176],{},"Esta es la parte que realmente decide el resultado. En el modo puramente aleatorio aplicamos una ",[239,24382,24383],{},"velocidad angular aleatoria"," entre −10 y +10 en dos de los tres ejes de rotación — números aleatorios frescos en cada lanzamiento. En palabras llanas: le estamos diciendo a la moneda ",[212,24386,24387],{},"a qué velocidad y en qué dirección rodar",[17,24389,24390],{},"Dejamos el tercer eje de rotación a cero a propósito: mantiene el movimiento legible en lugar de dejar que la moneda se bambolee caóticamente en todas las direcciones posibles a la vez.",[17,24392,24393],{},"Una vez fijados esos tres valores, soltamos. A partir de ahí, todo es física.",[561,24395,24397],{"id":24396},"la-gravedad-hace-el-resto","La gravedad hace el resto",[17,24399,24400,24401,24404],{},"La moneda vive en un mundo simulado en miniatura donde la gravedad es ",[239,24402,24403],{},"9,82 m\u002Fs²"," — la misma atracción que sentirías en la Tierra. Sube, frena, cae, golpea el suelo, rebota un poco y finalmente se asienta.",[17,24406,24407],{},"Importante: no miramos el resultado por adelantado. No decidimos previamente \"esta saldrá cara\". La simulación corre, la moneda hace lo suyo, y lo que esté haciendo cuando se detiene es lo que llamamos el resultado. En ese sentido, es exactamente como un lanzamiento real: el momento en que la moneda deja tu pulgar, el resultado ya está decidido en principio, pero nadie lo sabe hasta que aterriza.",[561,24409,24411],{"id":24410},"cómo-leemos-el-resultado","Cómo leemos el resultado",[17,24413,24414],{},"Una vez que la moneda deja de moverse, necesitamos responder: \"¿qué cara está mirando al techo?\"",[17,24416,24417,24418,24421],{},"El truco es tomar una flecha virtual que apunte justo ",[212,24419,24420],{},"hacia arriba desde la cara superior de la moneda",", y ver hacia dónde acaba apuntando después de todo ese giro:",[3549,24423,24424,24433,24442],{},[232,24425,24426,24427,24430,24431,176],{},"Si la flecha apunta más o menos ",[239,24428,24429],{},"hacia arriba"," — la cara superior mira al cielo — es ",[239,24432,15293],{},[232,24434,24435,24436,24439,24440,176],{},"Si apunta más o menos ",[239,24437,24438],{},"hacia abajo"," — la cara superior ahora está boca abajo en el suelo — es ",[239,24441,15297],{},[232,24443,24435,24444,24447,24448,176],{},[239,24445,24446],{},"de lado"," — la moneda ha quedado apoyada sobre su canto — es ",[239,24449,24450],{},"canto",[17,24452,24453,24454,24458,24459,176],{},"Ese último es el raro \"canto\" mencionado en el ",[21,24455,24457],{"href":23983,"rel":24456},[25],"artículo de estadísticas"," — la moneda equilibrada sobre su borde — y ocurre en la simulación por la misma razón que ocurre en la vida real: a veces el tumbo simplemente sale así. Si tienes curiosidad por saber por qué llamarlo \"edge case\" es un gran juego de palabras, escribí sobre ello ",[21,24460,24463],{"href":24461,"rel":24462},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fde-donde-viene-el-termino-edge-case",[25],"aquí",[561,24465,24467],{"id":24466},"entonces-es-realmente-aleatorio","¿Entonces es realmente aleatorio?",[17,24469,24470],{},"Más o menos — con una nota al pie.",[17,24472,24473,24474,24476,24477,24479],{},"La aleatoriedad en nuestro lanzamiento viene de la deriva lateral aleatoria y del giro aleatorio que le damos a la moneda al lanzar. Bajo el capó, esos números vienen de ",[1293,24475,24157],{}," de JavaScript. La física posterior es completamente determinista: mismas entradas, mismo resultado, siempre. Así que la simulación hereda la calidad de aleatoriedad que proporcione ",[1293,24478,24157],{},", repartida entre tres valores con semilla.",[17,24481,24482,24483,24486,24487,24492,24493,24496],{},"¿Es eso aleatoriedad \"verdadera\"? No — es pseudoaleatoria, como la mayoría de la aleatoriedad en un ordenador. ¿Es aleatoria ",[212,24484,24485],{},"lo suficiente"," como para que no puedas trucarla haciendo clic de una forma concreta? Por supuesto. Y resulta que esto también es cierto para los lanzamientos reales: ",[21,24488,24491],{"href":24489,"rel":24490},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2310.04153",[25],"la investigación ha demostrado"," que los lanzamientos reales están ligeramente sesgados hacia la cara con la que empezaron, lo que significa que un lanzamiento simulado es, irónicamente, un poco ",[212,24494,24495],{},"más"," justo que el real.",[12832,24498,24499],{},[17,24500,24501,24502,24504],{},"Un lanzamiento simulado es, irónicamente, un poco ",[212,24503,24495],{}," justo que el real",[561,24506,24077],{"id":24076},[17,24508,24509],{},"Así que la próxima vez que lances una moneda en el sitio, esto es lo que pasa en realidad: un lanzamiento de tres números (empujón, deriva, giro), una breve simulación física y una lectura de hacia dónde apunta la cara superior de la moneda cuando se detiene. Sin trucos, sin ganador preseleccionado. Solo un mundo muy pequeño con una gravedad muy normal.",[17,24511,24512,24513,24517,24518,24521],{},"¿Con curiosidad por las matemáticas de fondo? Empieza por ",[21,24514,15769],{"href":24515,"rel":24516},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ffundamentos-de-probabilidad",[25],". ¿Con curiosidad por la rarísima realidad estadística de los lanzamientos reales? ",[21,24519,19270],{"href":23983,"rel":24520},[25]," es el que buscas.",[17,24523,24524,24525,24529],{},"¿Listo para ver la física en acción? ",[21,24526,24528],{"href":24089,"rel":24527},[25],"Lanza una moneda online"," ahora mismo.",{"title":1291,"searchDepth":12954,"depth":12954,"links":24531},[24532,24533,24538,24539,24540,24541],{"id":24320,"depth":12954,"text":24321},{"id":24352,"depth":12954,"text":24353,"children":24534},[24535,24536,24537],{"id":24359,"depth":12959,"text":24360},{"id":24366,"depth":12959,"text":24367},{"id":24376,"depth":12959,"text":24377},{"id":24396,"depth":12954,"text":24397},{"id":24410,"depth":12954,"text":24411},{"id":24466,"depth":12954,"text":24467},{"id":24076,"depth":12954,"text":24077},"2026-04-23T01:00:00Z","Un paseo en lenguaje llano por la física de cada lanzamiento de moneda en FlipTheCoin.app — el empujón, el giro y la gravedad que decide cara o cruz.",{"excerpt":24545},{"type":10,"value":24546},[24547,24549],[13,24548,24294],{"id":24299},[17,24550,24302],{},"\u002Fblog\u002Fhow-we-toss-a-coin.es",{"title":24294,"description":24543},"como-lanzamos-una-moneda","blog\u002Fhow-we-toss-a-coin.es","XX0PfMaOywxzNabodbwFufVii-yMIby27kSV2n0Gxho",{"id":24557,"title":15769,"_locale":8,"body":24558,"date":24542,"description":26647,"extension":12971,"meta":26648,"navigation":13119,"path":26661,"published":13119,"seo":26662,"slug":24562,"stem":26663,"__hash__":26664},"blog\u002Fblog\u002Fprobability-basics.es.md",{"type":10,"value":24559,"toc":26635},[24560,24563,24571,24574,24578,24686,24694,24698,24701,24723,24726,24730,24736,24753,24764,24768,24779,24895,24898,25330,25341,25345,25348,25352,25363,25712,25816,25820,25834,26352,26359,26564,26571,26575,26581,26594,26596,26599,26626,26629],[13,24561,15769],{"id":24562},"fundamentos-de-probabilidad",[17,24564,24565,24566,24568,24569,176],{},"La probabilidad es una rama de las matemáticas que se ocupa de la verosimilitud de que ocurra un suceso. Es una forma de cuantificar la incertidumbre y medir la probabilidad de distintos resultados. En el contexto de los lanzamientos de moneda, la probabilidad se usa para determinar la verosimilitud de que la moneda caiga en ",[239,24567,15293],{}," o en ",[239,24570,15297],{},[17,24572,24573],{},"Este artículo es una introducción al tema de la probabilidad. Nos quedaremos en lenguaje llano y usaremos una moneda como único ejemplo. Estadísticas más avanzadas llegarán en futuros artículos.",[561,24575,24577],{"id":24576},"lo-básico","Lo básico",[17,24579,24580,24581,24678,24679,24681,24682,24685],{},"Cuando hablamos de probabilidad, estamos hablando de un concepto matemático que se puede representar de varias formas. La manera más común es como una fracción, un decimal o un porcentaje. Por ejemplo, si lanzas una moneda justa, la probabilidad de que caiga en cara es ",[35,24582,24584,24601],{"className":24583},[38],[35,24585,24587],{"className":24586},[42],[44,24588,24589],{"xmlns":46},[48,24590,24591,24599],{},[51,24592,24593],{},[54,24594,24595,24597],{},[61,24596,272],{},[61,24598,404],{},[65,24600,407],{"encoding":67},[35,24602,24604],{"className":24603,"ariaHidden":73},[72],[35,24605,24607,24610],{"className":24606},[77],[35,24608],{"className":24609,"style":293},[81],[35,24611,24613,24616,24675],{"className":24612},[86],[35,24614],{"className":24615},[90,91],[35,24617,24619],{"className":24618},[54],[35,24620,24622,24667],{"className":24621},[98,99],[35,24623,24625,24664],{"className":24624},[103],[35,24626,24628,24642,24650],{"className":24627,"style":312},[107],[35,24629,24630,24633],{"style":111},[35,24631],{"className":24632,"style":116},[115],[35,24634,24636],{"className":24635},[120,121,122,123],[35,24637,24639],{"className":24638},[86,123],[35,24640,404],{"className":24641},[86,123],[35,24643,24644,24647],{"style":132},[35,24645],{"className":24646,"style":116},[115],[35,24648],{"className":24649,"style":140},[139],[35,24651,24652,24655],{"style":143},[35,24653],{"className":24654,"style":116},[115],[35,24656,24658],{"className":24657},[120,121,122,123],[35,24659,24661],{"className":24660},[86,123],[35,24662,272],{"className":24663},[86,123],[35,24665,162],{"className":24666},[161],[35,24668,24670],{"className":24669},[103],[35,24671,24673],{"className":24672,"style":169},[107],[35,24674],{},[35,24676],{"className":24677},[175,91],", que es lo mismo que ",[239,24680,500],{}," o ",[239,24683,24684],{},"50%",". Las tres representaciones son equivalentes y se pueden usar indistintamente.",[17,24687,24688,24689,24693],{},"Como veremos en ",[21,24690,24692],{"href":23983,"rel":24691},[25],"el artículo La estadística inesperada de los lanzamientos de moneda",", la probabilidad real de aterrizaje de una moneda no es exactamente 50-50 cada vez.",[561,24695,24697],{"id":24696},"resultados-y-sucesos","Resultados y sucesos",[17,24699,24700],{},"Antes de seguir, dos piezas pequeñas de vocabulario:",[3549,24702,24703,24717],{},[232,24704,19384,24705,24708,24709,24711,24712,24714,24715,176],{},[239,24706,24707],{},"resultado"," es una de las cosas posibles que pueden ocurrir. Cuando lanzas una moneda, los resultados son ",[212,24710,15293],{},", ",[212,24713,15297],{}," y (muy raramente) ",[212,24716,24450],{},[232,24718,19384,24719,24722],{},[239,24720,24721],{},"suceso"," es una pregunta que hacemos sobre los resultados, como \"¿cayó en cara?\" o \"¿cayó en alguna cara?\".",[17,24724,24725],{},"La probabilidad es la herramienta que usamos para ponerle un número a lo probable que es un suceso.",[561,24727,24729],{"id":24728},"la-escala-de-probabilidad","La escala de probabilidad",[17,24731,24732,24733,11815],{},"Las probabilidades siempre viven en una escala de ",[239,24734,24735],{},"0 a 1",[3549,24737,24738,24743,24748],{},[232,24739,24740,24742],{},[239,24741,517],{}," significa que el suceso es imposible. Una moneda convirtiéndose en paloma en pleno vuelo tiene probabilidad 0.",[232,24744,24745,24747],{},[239,24746,272],{}," significa que el suceso es seguro. Una moneda lanzada al aire acabando por caer, gracias a la gravedad, tiene probabilidad 1.",[232,24749,24750,24751,176],{},"Cualquier cosa entre medias es \"algo probable\". Una moneda justa cayendo en cara se sitúa justo en el medio, en ",[239,24752,500],{},[17,24754,24755,24756,24759,24760,24763],{},"En términos de porcentaje, esa misma escala va del ",[239,24757,24758],{},"0%"," al ",[239,24761,24762],{},"100%",". La misma idea, expresada de otra forma.",[561,24765,24767],{"id":24766},"la-regla-del-complemento","La regla del complemento",[17,24769,24770,24771,24774,24775,24778],{},"Cada suceso tiene una imagen espejo llamada su ",[239,24772,24773],{},"complemento",": el suceso de que ",[212,24776,24777],{},"no"," ocurra. Y como algo o bien ocurre o bien no, las dos probabilidades siempre suman 1. En símbolos:",[17,24780,24782],{"className":24781},[592],[35,24783,24785,24823],{"className":24784},[38],[35,24786,24788],{"className":24787},[42],[44,24789,24790],{"xmlns":46},[48,24791,24792,24820],{},[51,24793,24794,24796,24798,24801,24804,24806,24808,24810,24812,24814,24816,24818],{},[57,24795,16246],{},[276,24797,1028],{"stretchy":9816},[606,24799,24800],{},"no ",[57,24802,24803],{},"A",[276,24805,1037],{"stretchy":9816},[276,24807,278],{},[61,24809,272],{},[276,24811,1353],{},[57,24813,16246],{},[276,24815,1028],{"stretchy":9816},[57,24817,24803],{},[276,24819,1037],{"stretchy":9816},[65,24821,24822],{"encoding":67},"P(\\text{no } A) = 1 - P(A)",[35,24824,24826,24859,24877],{"className":24825,"ariaHidden":73},[72],[35,24827,24829,24832,24835,24838,24844,24847,24850,24853,24856],{"className":24828},[77],[35,24830],{"className":24831,"style":9837},[81],[35,24833,16246],{"className":24834,"style":16272},[86,156],[35,24836,1028],{"className":24837},[90],[35,24839,24841],{"className":24840},[86,625],[35,24842,24800],{"className":24843},[86],[35,24845,24803],{"className":24846},[86,156],[35,24848,1037],{"className":24849},[175],[35,24851],{"className":24852,"style":367},[366],[35,24854,278],{"className":24855},[371],[35,24857],{"className":24858,"style":367},[366],[35,24860,24862,24865,24868,24871,24874],{"className":24861},[77],[35,24863],{"className":24864,"style":1331},[81],[35,24866,272],{"className":24867},[86],[35,24869],{"className":24870,"style":2223},[366],[35,24872,1353],{"className":24873},[2227],[35,24875],{"className":24876,"style":2223},[366],[35,24878,24880,24883,24886,24889,24892],{"className":24879},[77],[35,24881],{"className":24882,"style":9837},[81],[35,24884,16246],{"className":24885,"style":16272},[86,156],[35,24887,1028],{"className":24888},[90],[35,24890,24803],{"className":24891},[86,156],[35,24893,1037],{"className":24894},[175],[17,24896,24897],{},"Para una moneda justa:",[3549,24899,24900,25124],{},[232,24901,24902,24983,24984,176],{},[35,24903,24905,24932],{"className":24904},[38],[35,24906,24908],{"className":24907},[42],[44,24909,24910],{"xmlns":46},[48,24911,24912,24930],{},[51,24913,24914,24916,24918,24920,24922,24924,24926,24928],{},[57,24915,16246],{},[276,24917,1028],{"stretchy":9816},[606,24919,15293],{},[276,24921,1037],{"stretchy":9816},[276,24923,278],{},[61,24925,517],{},[276,24927,525],{"separator":73},[61,24929,529],{},[65,24931,23764],{"encoding":67},[35,24933,24935,24965],{"className":24934,"ariaHidden":73},[72],[35,24936,24938,24941,24944,24947,24953,24956,24959,24962],{"className":24937},[77],[35,24939],{"className":24940,"style":9837},[81],[35,24942,16246],{"className":24943,"style":16272},[86,156],[35,24945,1028],{"className":24946},[90],[35,24948,24950],{"className":24949},[86,625],[35,24951,15293],{"className":24952},[86],[35,24954,1037],{"className":24955},[175],[35,24957],{"className":24958,"style":367},[366],[35,24960,278],{"className":24961},[371],[35,24963],{"className":24964,"style":367},[366],[35,24966,24968,24971,24974,24977,24980],{"className":24967},[77],[35,24969],{"className":24970,"style":513},[81],[35,24972,517],{"className":24973},[86],[35,24975,525],{"className":24976},[524],[35,24978],{"className":24979,"style":1452},[366],[35,24981,529],{"className":24982},[86],", así que ",[35,24985,24987,25028],{"className":24986},[38],[35,24988,24990],{"className":24989},[42],[44,24991,24992],{"xmlns":46},[48,24993,24994,25025],{},[51,24995,24996,24998,25000,25003,25005,25007,25009,25011,25013,25015,25017,25019,25021,25023],{},[57,24997,16246],{},[276,24999,1028],{"stretchy":9816},[606,25001,25002],{},"no cara",[276,25004,1037],{"stretchy":9816},[276,25006,278],{},[61,25008,272],{},[276,25010,1353],{},[61,25012,517],{},[276,25014,525],{"separator":73},[61,25016,529],{},[276,25018,278],{},[61,25020,517],{},[276,25022,525],{"separator":73},[61,25024,529],{},[65,25026,25027],{"encoding":67},"P(\\text{no cara}) = 1 - 0,5 = 0,5",[35,25029,25031,25061,25079,25106],{"className":25030,"ariaHidden":73},[72],[35,25032,25034,25037,25040,25043,25049,25052,25055,25058],{"className":25033},[77],[35,25035],{"className":25036,"style":9837},[81],[35,25038,16246],{"className":25039,"style":16272},[86,156],[35,25041,1028],{"className":25042},[90],[35,25044,25046],{"className":25045},[86,625],[35,25047,25002],{"className":25048},[86],[35,25050,1037],{"className":25051},[175],[35,25053],{"className":25054,"style":367},[366],[35,25056,278],{"className":25057},[371],[35,25059],{"className":25060,"style":367},[366],[35,25062,25064,25067,25070,25073,25076],{"className":25063},[77],[35,25065],{"className":25066,"style":1331},[81],[35,25068,272],{"className":25069},[86],[35,25071],{"className":25072,"style":2223},[366],[35,25074,1353],{"className":25075},[2227],[35,25077],{"className":25078,"style":2223},[366],[35,25080,25082,25085,25088,25091,25094,25097,25100,25103],{"className":25081},[77],[35,25083],{"className":25084,"style":513},[81],[35,25086,517],{"className":25087},[86],[35,25089,525],{"className":25090},[524],[35,25092],{"className":25093,"style":1452},[366],[35,25095,529],{"className":25096},[86],[35,25098],{"className":25099,"style":367},[366],[35,25101,278],{"className":25102},[371],[35,25104],{"className":25105,"style":367},[366],[35,25107,25109,25112,25115,25118,25121],{"className":25108},[77],[35,25110],{"className":25111,"style":513},[81],[35,25113,517],{"className":25114},[86],[35,25116,525],{"className":25117},[524],[35,25119],{"className":25120,"style":1452},[366],[35,25122,529],{"className":25123},[86],[232,25125,25126,25326,25327,4213],{},[35,25127,25129,25174],{"className":25128},[38],[35,25130,25132],{"className":25131},[42],[44,25133,25134],{"xmlns":46},[48,25135,25136,25171],{},[51,25137,25138,25140,25142,25144,25146,25148,25150,25152,25162,25164,25166,25168],{},[57,25139,16246],{},[276,25141,1028],{"stretchy":9816},[606,25143,23922],{},[276,25145,1037],{"stretchy":9816},[276,25147,8030],{},[61,25149,272],{},[276,25151,1353],{},[25153,25154,25155],"mstyle",{"scriptlevel":517,"displaystyle":9816},[54,25156,25157,25159],{},[61,25158,272],{},[61,25160,25161],{},"6000",[276,25163,8030],{},[61,25165,517],{},[276,25167,525],{"separator":73},[61,25169,25170],{},"99983",[65,25172,25173],{"encoding":67},"P(\\text{cara o cruz}) \\approx 1 - \\tfrac{1}{6000} \\approx 0,99983",[35,25175,25177,25207,25225,25308],{"className":25176,"ariaHidden":73},[72],[35,25178,25180,25183,25186,25189,25195,25198,25201,25204],{"className":25179},[77],[35,25181],{"className":25182,"style":9837},[81],[35,25184,16246],{"className":25185,"style":16272},[86,156],[35,25187,1028],{"className":25188},[90],[35,25190,25192],{"className":25191},[86,625],[35,25193,23922],{"className":25194},[86],[35,25196,1037],{"className":25197},[175],[35,25199],{"className":25200,"style":367},[366],[35,25202,8030],{"className":25203},[371],[35,25205],{"className":25206,"style":367},[366],[35,25208,25210,25213,25216,25219,25222],{"className":25209},[77],[35,25211],{"className":25212,"style":1331},[81],[35,25214,272],{"className":25215},[86],[35,25217],{"className":25218,"style":2223},[366],[35,25220,1353],{"className":25221},[2227],[35,25223],{"className":25224,"style":2223},[366],[35,25226,25228,25231,25299,25302,25305],{"className":25227},[77],[35,25229],{"className":25230,"style":293},[81],[35,25232,25234,25237,25296],{"className":25233},[86],[35,25235],{"className":25236},[90,91],[35,25238,25240],{"className":25239},[54],[35,25241,25243,25288],{"className":25242},[98,99],[35,25244,25246,25285],{"className":25245},[103],[35,25247,25249,25263,25271],{"className":25248,"style":312},[107],[35,25250,25251,25254],{"style":111},[35,25252],{"className":25253,"style":116},[115],[35,25255,25257],{"className":25256},[120,121,122,123],[35,25258,25260],{"className":25259},[86,123],[35,25261,25161],{"className":25262},[86,123],[35,25264,25265,25268],{"style":132},[35,25266],{"className":25267,"style":116},[115],[35,25269],{"className":25270,"style":140},[139],[35,25272,25273,25276],{"style":143},[35,25274],{"className":25275,"style":116},[115],[35,25277,25279],{"className":25278},[120,121,122,123],[35,25280,25282],{"className":25281},[86,123],[35,25283,272],{"className":25284},[86,123],[35,25286,162],{"className":25287},[161],[35,25289,25291],{"className":25290},[103],[35,25292,25294],{"className":25293,"style":169},[107],[35,25295],{},[35,25297],{"className":25298},[175,91],[35,25300],{"className":25301,"style":367},[366],[35,25303,8030],{"className":25304},[371],[35,25306],{"className":25307,"style":367},[366],[35,25309,25311,25314,25317,25320,25323],{"className":25310},[77],[35,25312],{"className":25313,"style":513},[81],[35,25315,517],{"className":25316},[86],[35,25318,525],{"className":25319},[524],[35,25321],{"className":25322,"style":1452},[366],[35,25324,25170],{"className":25325},[86],", porque la única otra opción es el raro \"canto\". (Más sobre ese 1 entre 6000 en ",[21,25328,19270],{"href":23983,"rel":25329},[25],[17,25331,25332,25333,25336,25337,25340],{},"La regla del complemento suele ser la forma más fácil de calcular una probabilidad: en lugar de contar todas las formas en que algo ",[212,25334,25335],{},"puede"," pasar, cuenta la única forma en que ",[212,25338,25339],{},"no puede"," y réstala a 1.",[561,25342,25344],{"id":25343},"combinar-sucesos-y-vs-o","Combinar sucesos: \"y\" vs \"o\"",[17,25346,25347],{},"La cosa se pone interesante cuando preguntamos por más de un lanzamiento a la vez. Dos reglas cubren casi todo lo que necesitarás:",[582,25349,25351],{"id":25350},"y-ambas-cosas-ocurren","\"Y\" — ambas cosas ocurren",[17,25353,25354,25355,25358,25359,25362],{},"Los lanzamientos de moneda no se influyen entre sí (son ",[239,25356,25357],{},"independientes","). En este caso, ",[239,25360,25361],{},"multiplicas"," sus probabilidades:",[3549,25364,25365,25522],{},[232,25366,25367,25521],{},[35,25368,25370,25416],{"className":25369},[38],[35,25371,25373],{"className":25372},[42],[44,25374,25375],{"xmlns":46},[48,25376,25377,25413],{},[51,25378,25379,25381,25383,25386,25388,25390,25392,25394,25396,25398,25400,25402,25404,25406,25408,25410],{},[57,25380,16246],{},[276,25382,1028],{"stretchy":9816},[606,25384,25385],{},"OO",[276,25387,1037],{"stretchy":9816},[276,25389,278],{},[61,25391,517],{},[276,25393,525],{"separator":73},[61,25395,529],{},[276,25397,6472],{},[61,25399,517],{},[276,25401,525],{"separator":73},[61,25403,529],{},[276,25405,278],{},[61,25407,517],{},[276,25409,525],{"separator":73},[61,25411,25412],{},"25",[65,25414,25415],{"encoding":67},"P(\\text{OO}) = 0,5 \\times 0,5 = 0,25",[35,25417,25419,25449,25476,25503],{"className":25418,"ariaHidden":73},[72],[35,25420,25422,25425,25428,25431,25437,25440,25443,25446],{"className":25421},[77],[35,25423],{"className":25424,"style":9837},[81],[35,25426,16246],{"className":25427,"style":16272},[86,156],[35,25429,1028],{"className":25430},[90],[35,25432,25434],{"className":25433},[86,625],[35,25435,25385],{"className":25436},[86],[35,25438,1037],{"className":25439},[175],[35,25441],{"className":25442,"style":367},[366],[35,25444,278],{"className":25445},[371],[35,25447],{"className":25448,"style":367},[366],[35,25450,25452,25455,25458,25461,25464,25467,25470,25473],{"className":25451},[77],[35,25453],{"className":25454,"style":513},[81],[35,25456,517],{"className":25457},[86],[35,25459,525],{"className":25460},[524],[35,25462],{"className":25463,"style":1452},[366],[35,25465,529],{"className":25466},[86],[35,25468],{"className":25469,"style":2223},[366],[35,25471,6472],{"className":25472},[2227],[35,25474],{"className":25475,"style":2223},[366],[35,25477,25479,25482,25485,25488,25491,25494,25497,25500],{"className":25478},[77],[35,25480],{"className":25481,"style":513},[81],[35,25483,517],{"className":25484},[86],[35,25486,525],{"className":25487},[524],[35,25489],{"className":25490,"style":1452},[366],[35,25492,529],{"className":25493},[86],[35,25495],{"className":25496,"style":367},[366],[35,25498,278],{"className":25499},[371],[35,25501],{"className":25502,"style":367},[366],[35,25504,25506,25509,25512,25515,25518],{"className":25505},[77],[35,25507],{"className":25508,"style":513},[81],[35,25510,517],{"className":25511},[86],[35,25513,525],{"className":25514},[524],[35,25516],{"className":25517,"style":1452},[366],[35,25519,25412],{"className":25520},[86],", o 25%.",[232,25523,25524,25711],{},[35,25525,25527,25579],{"className":25526},[38],[35,25528,25530],{"className":25529},[42],[44,25531,25532],{"xmlns":46},[48,25533,25534,25576],{},[51,25535,25536,25538,25540,25542,25544,25546,25548,25550,25552,25554,25556,25558,25560,25562,25564,25566,25568,25570,25572,25574],{},[57,25537,16246],{},[276,25539,1028],{"stretchy":9816},[606,25541,15923],{},[276,25543,1037],{"stretchy":9816},[276,25545,278],{},[61,25547,517],{},[276,25549,525],{"separator":73},[61,25551,529],{},[276,25553,6472],{},[61,25555,517],{},[276,25557,525],{"separator":73},[61,25559,529],{},[276,25561,6472],{},[61,25563,517],{},[276,25565,525],{"separator":73},[61,25567,529],{},[276,25569,278],{},[61,25571,517],{},[276,25573,525],{"separator":73},[61,25575,17110],{},[65,25577,25578],{"encoding":67},"P(\\text{OOO}) = 0,5 \\times 0,5 \\times 0,5 = 0,125",[35,25580,25582,25612,25639,25666,25693],{"className":25581,"ariaHidden":73},[72],[35,25583,25585,25588,25591,25594,25600,25603,25606,25609],{"className":25584},[77],[35,25586],{"className":25587,"style":9837},[81],[35,25589,16246],{"className":25590,"style":16272},[86,156],[35,25592,1028],{"className":25593},[90],[35,25595,25597],{"className":25596},[86,625],[35,25598,15923],{"className":25599},[86],[35,25601,1037],{"className":25602},[175],[35,25604],{"className":25605,"style":367},[366],[35,25607,278],{"className":25608},[371],[35,25610],{"className":25611,"style":367},[366],[35,25613,25615,25618,25621,25624,25627,25630,25633,25636],{"className":25614},[77],[35,25616],{"className":25617,"style":513},[81],[35,25619,517],{"className":25620},[86],[35,25622,525],{"className":25623},[524],[35,25625],{"className":25626,"style":1452},[366],[35,25628,529],{"className":25629},[86],[35,25631],{"className":25632,"style":2223},[366],[35,25634,6472],{"className":25635},[2227],[35,25637],{"className":25638,"style":2223},[366],[35,25640,25642,25645,25648,25651,25654,25657,25660,25663],{"className":25641},[77],[35,25643],{"className":25644,"style":513},[81],[35,25646,517],{"className":25647},[86],[35,25649,525],{"className":25650},[524],[35,25652],{"className":25653,"style":1452},[366],[35,25655,529],{"className":25656},[86],[35,25658],{"className":25659,"style":2223},[366],[35,25661,6472],{"className":25662},[2227],[35,25664],{"className":25665,"style":2223},[366],[35,25667,25669,25672,25675,25678,25681,25684,25687,25690],{"className":25668},[77],[35,25670],{"className":25671,"style":513},[81],[35,25673,517],{"className":25674},[86],[35,25676,525],{"className":25677},[524],[35,25679],{"className":25680,"style":1452},[366],[35,25682,529],{"className":25683},[86],[35,25685],{"className":25686,"style":367},[366],[35,25688,278],{"className":25689},[371],[35,25691],{"className":25692,"style":367},[366],[35,25694,25696,25699,25702,25705,25708],{"className":25695},[77],[35,25697],{"className":25698,"style":513},[81],[35,25700,517],{"className":25701},[86],[35,25703,525],{"className":25704},[524],[35,25706],{"className":25707,"style":1452},[366],[35,25709,17110],{"className":25710},[86],", o 12,5%.",[17,25713,25714,25715,25815],{},"Cada lanzamiento extra reduce la probabilidad a la mitad. Por eso las rachas largas resultan sorprendentes — no porque la moneda esté \"tocada\" para el otro lado, sino porque cada nuevo lanzamiento añade otro ",[35,25716,25718,25738],{"className":25717},[38],[35,25719,25721],{"className":25720},[42],[44,25722,25723],{"xmlns":46},[48,25724,25725,25735],{},[51,25726,25727],{},[25153,25728,25729],{"scriptlevel":517,"displaystyle":9816},[54,25730,25731,25733],{},[61,25732,272],{},[61,25734,404],{},[65,25736,25737],{"encoding":67},"\\tfrac{1}{2}",[35,25739,25741],{"className":25740,"ariaHidden":73},[72],[35,25742,25744,25747],{"className":25743},[77],[35,25745],{"className":25746,"style":293},[81],[35,25748,25750,25753,25812],{"className":25749},[86],[35,25751],{"className":25752},[90,91],[35,25754,25756],{"className":25755},[54],[35,25757,25759,25804],{"className":25758},[98,99],[35,25760,25762,25801],{"className":25761},[103],[35,25763,25765,25779,25787],{"className":25764,"style":312},[107],[35,25766,25767,25770],{"style":111},[35,25768],{"className":25769,"style":116},[115],[35,25771,25773],{"className":25772},[120,121,122,123],[35,25774,25776],{"className":25775},[86,123],[35,25777,404],{"className":25778},[86,123],[35,25780,25781,25784],{"style":132},[35,25782],{"className":25783,"style":116},[115],[35,25785],{"className":25786,"style":140},[139],[35,25788,25789,25792],{"style":143},[35,25790],{"className":25791,"style":116},[115],[35,25793,25795],{"className":25794},[120,121,122,123],[35,25796,25798],{"className":25797},[86,123],[35,25799,272],{"className":25800},[86,123],[35,25802,162],{"className":25803},[161],[35,25805,25807],{"className":25806},[103],[35,25808,25810],{"className":25809,"style":169},[107],[35,25811],{},[35,25813],{"className":25814},[175,91]," al producto.",[582,25817,25819],{"id":25818},"o-al-menos-una-de-varias-cosas-ocurre","\"O\" — al menos una de varias cosas ocurre",[17,25821,25822,25823,25826,25827,25830,25831,25362],{},"Si dos resultados ",[239,25824,25825],{},"no pueden ocurrir al mismo tiempo"," (son ",[239,25828,25829],{},"mutuamente excluyentes","), ",[239,25832,25833],{},"sumas",[3549,25835,25836,26004],{},[232,25837,25838,26003],{},[35,25839,25841,25886],{"className":25840},[38],[35,25842,25844],{"className":25843},[42],[44,25845,25846],{"xmlns":46},[48,25847,25848,25883],{},[51,25849,25850,25852,25854,25856,25859,25861,25863,25865,25867,25869,25871,25873,25875,25877,25879,25881],{},[57,25851,16246],{},[276,25853,1028],{"stretchy":9816},[606,25855,570],{},[276,25857,25858],{},"∪",[606,25860,574],{},[276,25862,1037],{"stretchy":9816},[276,25864,278],{},[61,25866,517],{},[276,25868,525],{"separator":73},[61,25870,529],{},[276,25872,1316],{},[61,25874,517],{},[276,25876,525],{"separator":73},[61,25878,529],{},[276,25880,278],{},[61,25882,272],{},[65,25884,25885],{"encoding":67},"P(\\text{O} \\cup \\text{X}) = 0,5 + 0,5 = 1",[35,25887,25889,25916,25940,25967,25994],{"className":25888,"ariaHidden":73},[72],[35,25890,25892,25895,25898,25901,25907,25910,25913],{"className":25891},[77],[35,25893],{"className":25894,"style":9837},[81],[35,25896,16246],{"className":25897,"style":16272},[86,156],[35,25899,1028],{"className":25900},[90],[35,25902,25904],{"className":25903},[86,625],[35,25905,570],{"className":25906},[86],[35,25908],{"className":25909,"style":2223},[366],[35,25911,25858],{"className":25912},[2227],[35,25914],{"className":25915,"style":2223},[366],[35,25917,25919,25922,25928,25931,25934,25937],{"className":25918},[77],[35,25920],{"className":25921,"style":9837},[81],[35,25923,25925],{"className":25924},[86,625],[35,25926,574],{"className":25927},[86],[35,25929,1037],{"className":25930},[175],[35,25932],{"className":25933,"style":367},[366],[35,25935,278],{"className":25936},[371],[35,25938],{"className":25939,"style":367},[366],[35,25941,25943,25946,25949,25952,25955,25958,25961,25964],{"className":25942},[77],[35,25944],{"className":25945,"style":513},[81],[35,25947,517],{"className":25948},[86],[35,25950,525],{"className":25951},[524],[35,25953],{"className":25954,"style":1452},[366],[35,25956,529],{"className":25957},[86],[35,25959],{"className":25960,"style":2223},[366],[35,25962,1316],{"className":25963},[2227],[35,25965],{"className":25966,"style":2223},[366],[35,25968,25970,25973,25976,25979,25982,25985,25988,25991],{"className":25969},[77],[35,25971],{"className":25972,"style":513},[81],[35,25974,517],{"className":25975},[86],[35,25977,525],{"className":25978},[524],[35,25980],{"className":25981,"style":1452},[366],[35,25983,529],{"className":25984},[86],[35,25986],{"className":25987,"style":367},[366],[35,25989,278],{"className":25990},[371],[35,25992],{"className":25993,"style":367},[366],[35,25995,25997,26000],{"className":25996},[77],[35,25998],{"className":25999,"style":381},[81],[35,26001,272],{"className":26002},[86],". Nada sorprendente — eso es cualquier resultado no-canto.",[232,26005,26006,26351],{},[35,26007,26009,26063],{"className":26008},[38],[35,26010,26012],{"className":26011},[42],[44,26013,26014],{"xmlns":46},[48,26015,26016,26060],{},[51,26017,26018,26020,26022,26024,26026,26028,26030,26032,26040,26042,26050,26052],{},[57,26019,16246],{},[276,26021,1028],{"stretchy":9816},[61,26023,272],{},[276,26025,25858],{},[61,26027,404],{},[276,26029,1037],{"stretchy":9816},[276,26031,278],{},[25153,26033,26034],{"scriptlevel":517,"displaystyle":9816},[54,26035,26036,26038],{},[61,26037,272],{},[61,26039,5442],{},[276,26041,1316],{},[25153,26043,26044],{"scriptlevel":517,"displaystyle":9816},[54,26045,26046,26048],{},[61,26047,272],{},[61,26049,5442],{},[276,26051,278],{},[25153,26053,26054],{"scriptlevel":517,"displaystyle":9816},[54,26055,26056,26058],{},[61,26057,404],{},[61,26059,5442],{},[65,26061,26062],{"encoding":67},"P(1 \\cup 2) = \\tfrac{1}{6} + \\tfrac{1}{6} = \\tfrac{2}{6}",[35,26064,26066,26090,26111,26194,26277],{"className":26065,"ariaHidden":73},[72],[35,26067,26069,26072,26075,26078,26081,26084,26087],{"className":26068},[77],[35,26070],{"className":26071,"style":9837},[81],[35,26073,16246],{"className":26074,"style":16272},[86,156],[35,26076,1028],{"className":26077},[90],[35,26079,272],{"className":26080},[86],[35,26082],{"className":26083,"style":2223},[366],[35,26085,25858],{"className":26086},[2227],[35,26088],{"className":26089,"style":2223},[366],[35,26091,26093,26096,26099,26102,26105,26108],{"className":26092},[77],[35,26094],{"className":26095,"style":9837},[81],[35,26097,404],{"className":26098},[86],[35,26100,1037],{"className":26101},[175],[35,26103],{"className":26104,"style":367},[366],[35,26106,278],{"className":26107},[371],[35,26109],{"className":26110,"style":367},[366],[35,26112,26114,26117,26185,26188,26191],{"className":26113},[77],[35,26115],{"className":26116,"style":293},[81],[35,26118,26120,26123,26182],{"className":26119},[86],[35,26121],{"className":26122},[90,91],[35,26124,26126],{"className":26125},[54],[35,26127,26129,26174],{"className":26128},[98,99],[35,26130,26132,26171],{"className":26131},[103],[35,26133,26135,26149,26157],{"className":26134,"style":312},[107],[35,26136,26137,26140],{"style":111},[35,26138],{"className":26139,"style":116},[115],[35,26141,26143],{"className":26142},[120,121,122,123],[35,26144,26146],{"className":26145},[86,123],[35,26147,5442],{"className":26148},[86,123],[35,26150,26151,26154],{"style":132},[35,26152],{"className":26153,"style":116},[115],[35,26155],{"className":26156,"style":140},[139],[35,26158,26159,26162],{"style":143},[35,26160],{"className":26161,"style":116},[115],[35,26163,26165],{"className":26164},[120,121,122,123],[35,26166,26168],{"className":26167},[86,123],[35,26169,272],{"className":26170},[86,123],[35,26172,162],{"className":26173},[161],[35,26175,26177],{"className":26176},[103],[35,26178,26180],{"className":26179,"style":169},[107],[35,26181],{},[35,26183],{"className":26184},[175,91],[35,26186],{"className":26187,"style":2223},[366],[35,26189,1316],{"className":26190},[2227],[35,26192],{"className":26193,"style":2223},[366],[35,26195,26197,26200,26268,26271,26274],{"className":26196},[77],[35,26198],{"className":26199,"style":293},[81],[35,26201,26203,26206,26265],{"className":26202},[86],[35,26204],{"className":26205},[90,91],[35,26207,26209],{"className":26208},[54],[35,26210,26212,26257],{"className":26211},[98,99],[35,26213,26215,26254],{"className":26214},[103],[35,26216,26218,26232,26240],{"className":26217,"style":312},[107],[35,26219,26220,26223],{"style":111},[35,26221],{"className":26222,"style":116},[115],[35,26224,26226],{"className":26225},[120,121,122,123],[35,26227,26229],{"className":26228},[86,123],[35,26230,5442],{"className":26231},[86,123],[35,26233,26234,26237],{"style":132},[35,26235],{"className":26236,"style":116},[115],[35,26238],{"className":26239,"style":140},[139],[35,26241,26242,26245],{"style":143},[35,26243],{"className":26244,"style":116},[115],[35,26246,26248],{"className":26247},[120,121,122,123],[35,26249,26251],{"className":26250},[86,123],[35,26252,272],{"className":26253},[86,123],[35,26255,162],{"className":26256},[161],[35,26258,26260],{"className":26259},[103],[35,26261,26263],{"className":26262,"style":169},[107],[35,26264],{},[35,26266],{"className":26267},[175,91],[35,26269],{"className":26270,"style":367},[366],[35,26272,278],{"className":26273},[371],[35,26275],{"className":26276,"style":367},[366],[35,26278,26280,26283],{"className":26279},[77],[35,26281],{"className":26282,"style":293},[81],[35,26284,26286,26289,26348],{"className":26285},[86],[35,26287],{"className":26288},[90,91],[35,26290,26292],{"className":26291},[54],[35,26293,26295,26340],{"className":26294},[98,99],[35,26296,26298,26337],{"className":26297},[103],[35,26299,26301,26315,26323],{"className":26300,"style":312},[107],[35,26302,26303,26306],{"style":111},[35,26304],{"className":26305,"style":116},[115],[35,26307,26309],{"className":26308},[120,121,122,123],[35,26310,26312],{"className":26311},[86,123],[35,26313,5442],{"className":26314},[86,123],[35,26316,26317,26320],{"style":132},[35,26318],{"className":26319,"style":116},[115],[35,26321],{"className":26322,"style":140},[139],[35,26324,26325,26328],{"style":143},[35,26326],{"className":26327,"style":116},[115],[35,26329,26331],{"className":26330},[120,121,122,123],[35,26332,26334],{"className":26333},[86,123],[35,26335,404],{"className":26336},[86,123],[35,26338,162],{"className":26339},[161],[35,26341,26343],{"className":26342},[103],[35,26344,26346],{"className":26345,"style":169},[107],[35,26347],{},[35,26349],{"className":26350},[175,91]," (sacar un 1 o un 2 en un dado de seis caras).",[17,26353,26354,26355,26358],{},"Si los sucesos ",[212,26356,26357],{},"pueden"," solaparse, la simple suma cuenta doble el solape y tienes que restarlo:",[17,26360,26362],{"className":26361},[592],[35,26363,26365,26425],{"className":26364},[38],[35,26366,26368],{"className":26367},[42],[44,26369,26370],{"xmlns":46},[48,26371,26372,26422],{},[51,26373,26374,26376,26378,26380,26382,26385,26387,26389,26391,26393,26395,26397,26399,26401,26403,26405,26407,26409,26411,26413,26415,26418,26420],{},[57,26375,16246],{},[276,26377,1028],{"stretchy":9816},[57,26379,24803],{},[276,26381,25858],{},[57,26383,26384],{},"B",[276,26386,1037],{"stretchy":9816},[276,26388,278],{},[57,26390,16246],{},[276,26392,1028],{"stretchy":9816},[57,26394,24803],{},[276,26396,1037],{"stretchy":9816},[276,26398,1316],{},[57,26400,16246],{},[276,26402,1028],{"stretchy":9816},[57,26404,26384],{},[276,26406,1037],{"stretchy":9816},[276,26408,1353],{},[57,26410,16246],{},[276,26412,1028],{"stretchy":9816},[57,26414,24803],{},[276,26416,26417],{},"∩",[57,26419,26384],{},[276,26421,1037],{"stretchy":9816},[65,26423,26424],{"encoding":67},"P(A \\cup B) = P(A) + P(B) - P(A \\cap B)",[35,26426,26428,26452,26474,26501,26528,26552],{"className":26427,"ariaHidden":73},[72],[35,26429,26431,26434,26437,26440,26443,26446,26449],{"className":26430},[77],[35,26432],{"className":26433,"style":9837},[81],[35,26435,16246],{"className":26436,"style":16272},[86,156],[35,26438,1028],{"className":26439},[90],[35,26441,24803],{"className":26442},[86,156],[35,26444],{"className":26445,"style":2223},[366],[35,26447,25858],{"className":26448},[2227],[35,26450],{"className":26451,"style":2223},[366],[35,26453,26455,26458,26462,26465,26468,26471],{"className":26454},[77],[35,26456],{"className":26457,"style":9837},[81],[35,26459,26384],{"className":26460,"style":26461},[86,156],"margin-right:0.0502em;",[35,26463,1037],{"className":26464},[175],[35,26466],{"className":26467,"style":367},[366],[35,26469,278],{"className":26470},[371],[35,26472],{"className":26473,"style":367},[366],[35,26475,26477,26480,26483,26486,26489,26492,26495,26498],{"className":26476},[77],[35,26478],{"className":26479,"style":9837},[81],[35,26481,16246],{"className":26482,"style":16272},[86,156],[35,26484,1028],{"className":26485},[90],[35,26487,24803],{"className":26488},[86,156],[35,26490,1037],{"className":26491},[175],[35,26493],{"className":26494,"style":2223},[366],[35,26496,1316],{"className":26497},[2227],[35,26499],{"className":26500,"style":2223},[366],[35,26502,26504,26507,26510,26513,26516,26519,26522,26525],{"className":26503},[77],[35,26505],{"className":26506,"style":9837},[81],[35,26508,16246],{"className":26509,"style":16272},[86,156],[35,26511,1028],{"className":26512},[90],[35,26514,26384],{"className":26515,"style":26461},[86,156],[35,26517,1037],{"className":26518},[175],[35,26520],{"className":26521,"style":2223},[366],[35,26523,1353],{"className":26524},[2227],[35,26526],{"className":26527,"style":2223},[366],[35,26529,26531,26534,26537,26540,26543,26546,26549],{"className":26530},[77],[35,26532],{"className":26533,"style":9837},[81],[35,26535,16246],{"className":26536,"style":16272},[86,156],[35,26538,1028],{"className":26539},[90],[35,26541,24803],{"className":26542},[86,156],[35,26544],{"className":26545,"style":2223},[366],[35,26547,26417],{"className":26548},[2227],[35,26550],{"className":26551,"style":2223},[366],[35,26553,26555,26558,26561],{"className":26554},[77],[35,26556],{"className":26557,"style":9837},[81],[35,26559,26384],{"className":26560,"style":26461},[86,156],[35,26562,1037],{"className":26563},[175],[17,26565,26566,26567,26570],{},"Eso es el ",[239,26568,26569],{},"principio de inclusión–exclusión",", y su versión más profunda es tema para un futuro artículo.",[561,26572,26574],{"id":26573},"una-nota-rápida-sobre-independencia","Una nota rápida sobre independencia",[17,26576,26577,26578,26580],{},"Dos sucesos son ",[239,26579,25357],{}," cuando el resultado de uno no cambia la probabilidad del otro. Los lanzamientos de moneda son el ejemplo de libro: la moneda no tiene memoria, así que una racha de diez caras seguidas no hace que la cruz sea más probable en el undécimo lanzamiento.",[17,26582,26583,26584,26587,26588,26593],{},"Este es el montaje detrás de la famosa ",[239,26585,26586],{},"Falacia del jugador",", que desgrano en ",[21,26589,26592],{"href":26590,"rel":26591},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fla-estadistica-inesperada-de-los-lanzamientos-de-moneda#the-gamblers-fallacy",[25],"el artículo de estadísticas",". Por ahora, la versión corta: cada lanzamiento es un nuevo comienzo.",[561,26595,24077],{"id":24076},[17,26597,26598],{},"Eso es realmente todo para lo básico:",[229,26600,26601,26604,26613,26623],{},[232,26602,26603],{},"Las probabilidades viven entre 0 y 1.",[232,26605,26606,26607,26609,26610,26612],{},"Usa el ",[239,26608,24773],{}," (1 − P) cuando sea más fácil contar lo que ",[212,26611,24777],{}," ocurre.",[232,26614,26615,26618,26619,26622],{},[239,26616,26617],{},"Multiplica"," para el \"y\", ",[239,26620,26621],{},"suma"," para el \"o\" mutuamente excluyente.",[232,26624,26625],{},"Los sucesos independientes no se preocupan por la historia del otro.",[17,26627,26628],{},"Con solo esas cuatro ideas puedes razonar sobre casi cualquier pregunta de lanzamiento de moneda — y sobre la mayoría de problemas tipo dados, cartas y lotería también.",[17,26630,12947,26631,26634],{},[21,26632,12951],{"href":24089,"rel":26633},[25]," y sigue tu propia racha.",{"title":1291,"searchDepth":12954,"depth":12954,"links":26636},[26637,26638,26639,26640,26641,26645,26646],{"id":24576,"depth":12954,"text":24577},{"id":24696,"depth":12954,"text":24697},{"id":24728,"depth":12954,"text":24729},{"id":24766,"depth":12954,"text":24767},{"id":25343,"depth":12954,"text":25344,"children":26642},[26643,26644],{"id":25350,"depth":12959,"text":25351},{"id":25818,"depth":12959,"text":25819},{"id":26573,"depth":12954,"text":26574},{"id":24076,"depth":12954,"text":24077},"Aprende los fundamentos de la probabilidad y cómo se relacionan con los lanzamientos de moneda.",{"excerpt":26649},{"type":10,"value":26650},[26651,26653,26659],[13,26652,15769],{"id":24562},[17,26654,24565,26655,24568,26657,176],{},[239,26656,15293],{},[239,26658,15297],{},[17,26660,24573],{},"\u002Fblog\u002Fprobability-basics.es",{"title":15769,"description":26647},"blog\u002Fprobability-basics.es","6yxNewZ-8-Ifp0r0_XaURHiahdISyPwS20XdYbgwuLY",{"id":26666,"title":26667,"_locale":8,"body":26668,"date":24542,"description":26890,"extension":12971,"meta":26891,"navigation":13119,"path":26915,"published":13119,"seo":26916,"slug":26917,"stem":26918,"__hash__":26919},"blog\u002Fblog\u002Fwhere-does-the-term-edge-case-come-from.es.md","¿De dónde viene el término \"edge case\"?",{"type":10,"value":26669,"toc":26883},[26670,26673,26676,26692,26695,26705,26709,26715,26732,26735,26739,26748,26785,26788,26792,26799,26809,26818,26821,26825,26836,26854,26861,26864,26871,26873,26876],[13,26671,26667],{"id":26672},"de-dónde-viene-el-término-edge-case",[17,26674,26675],{},"Te ha caído una moneda sobre su canto? Has tenido una moneda en tus manos y te has preguntado por qué se llama \"canto\" y no \"borde\"? ¿O has oído el término \"edge case\" y te has preguntado de dónde viene?",[17,26677,26678,26679,26682,26683,26687,26688,26691],{},"Está el ",[239,26680,26681],{},"borde literal"," de una moneda: el fino canto entre cara y cruz. Que una moneda real caiga sobre ese canto es un suceso raro, alrededor de ",[21,26684,26686],{"href":23983,"rel":26685},[25],"1 entre 6000 lanzamientos",", y sí, nuestra simulación también lo contempla. Y luego está el ",[239,26689,26690],{},"\"edge case\" de la ingeniería de software",": el término que usan los desarrolladores para entradas raras, extremas, en los límites, que las pruebas cotidianas no detectan.",[17,26693,26694],{},"¿Están relacionados los dos?",[17,26696,26697,26700,26701,26704],{},[239,26698,26699],{},"Respuesta corta",": no. El \"edge\" en \"edge case\" no tiene nada que ver con el canto de las monedas. Pero el hecho de que una app de lanzar monedas tenga que manejar ambos — un canto literal y un ",[212,26702,26703],{},"edge case"," — es una coincidencia que no puedo dejar pasar. Así que aquí va la historia.",[561,26706,26708],{"id":26707},"qué-quieren-decir-los-ingenieros-con-edge-case","Qué quieren decir los ingenieros con \"edge case\"",[17,26710,26711,26712,26714],{},"En software (y más ampliamente, en ingeniería), un ",[239,26713,26703],{}," es una entrada o estado que está en el extremo del límite de lo que un sistema está diseñado para manejar. Cosas como:",[3549,26716,26717,26720,26723,26726],{},[232,26718,26719],{},"Una lista vacía, cuando la mayoría de las listas tienen elementos.",[232,26721,26722],{},"Un número en el valor máximo permitido — el entero más grande, el año más alto que tu selector de fechas acepta.",[232,26724,26725],{},"Exactamente medianoche en un día bisiesto.",[232,26727,26728,26729],{},"Un usuario sin nombre, o un nombre hecho enteramente de emoji. ",[212,26730,26731],{},"¡Hola, 🪙🐬!",[17,26733,26734],{},"Estos casos rara vez aparecen en el uso normal, pero es donde los sistemas fallan silenciosamente. Algunos trabajos te entrenan a buscarlos de la misma forma en que los humanos somos muy buenos detectando patrones en general.",[561,26736,26738],{"id":26737},"el-borde-la-esquina-y-el-límite","El borde, la esquina y el límite",[17,26740,26741,26742,26747],{},"Tres términos muy emparentados se usan para ideas similares, y la gente los intercambia sin pensar. El ",[21,26743,26746],{"href":26744,"rel":26745},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCorner_case",[25],"artículo de Wikipedia sobre corner cases"," tiene la distinción más limpia:",[3549,26749,26750,26759,26779],{},[232,26751,19384,26752,26754,26755,26758],{},[239,26753,26703],{}," empuja ",[239,26756,26757],{},"una"," variable a su mínimo o máximo. Unidimensional.",[232,26760,19384,26761,26754,26764,26767,26768,26771,26772,26775,26776,26778],{},[239,26762,26763],{},"corner case",[239,26765,26766],{},"varias"," variables a sus extremos ",[212,26769,26770],{},"al mismo tiempo",". Piensa en una caja n-dimensional: las esquinas son donde varios bordes se encuentran. El ejemplo de Wikipedia es bueno — un altavoz que solo falla cuando el volumen está al máximo ",[212,26773,26774],{},"y"," los graves están al máximo ",[212,26777,26774],{}," la humedad es alta. Cada mando está bien por sí solo; la esquina es donde se cruzan.",[232,26780,19384,26781,26784],{},[239,26782,26783],{},"boundary case"," está justo en el umbral entre dos comportamientos — como 17 vs 18 en una comprobación de mayoría de edad.",[17,26786,26787],{},"En conversaciones informales nadie se molesta con la distinción. Verás \"edge case\" usado para los tres, y está bien. El lenguaje va a la deriva, y todo el mundo sigue entendiendo lo que quieres decir.",[561,26789,26791],{"id":26790},"de-dónde-viene-realmente-la-palabra","De dónde viene realmente la palabra",[17,26793,26794,26795,26798],{},"El término no viene del software en absoluto. Viene de ",[239,26796,26797],{},"la ingeniería hardware"," — en concreto, de las pruebas de componentes electrónicos y sistemas complejos.",[17,26800,26801,26802,26805,26806,176],{},"Cuando diseñas un circuito, un chip, o en realidad cualquier sistema físico, el componente está calificado para funcionar en un ",[239,26803,26804],{},"rango"," de condiciones operativas: un rango de voltaje, un rango de temperatura, un rango de frecuencia, un rango de presión. Probar significa que no solo compruebas el medio de cada rango; compruebas los extremos, porque ahí es donde los componentes se rompen primero. Esos extremos son los ",[239,26807,26808],{},"bordes de la envolvente operativa",[17,26810,26811,26812,26817],{},"La metáfora misma toma prestado de la ",[21,26813,26816],{"href":26814,"rel":26815},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FFlight_envelope",[25],"envolvente de vuelo"," de la aeronáutica — la región de velocidades y altitudes en la que un avión puede operar con seguridad. El \"borde de la envolvente\" es jerga de pilotos e ingenieros para los límites exteriores de esa región. \"Edge case\" es la versión de pruebas de la misma idea: condiciones justo contra la pared donde las cosas empiezan a romperse.",[17,26819,26820],{},"El software heredó el lenguaje cuando la ingeniería informática creció a partir de la ingeniería eléctrica. La metáfora se quedó porque seguía siendo útil: cada sistema tiene una forma, y cada forma tiene bordes.",[561,26822,26824],{"id":26823},"y-qué-pasa-con-la-moneda","¿Y qué pasa con la moneda?",[17,26826,26827,26828,26831,26832,26835],{},"Aquí está la parte divertida. En ",[239,26829,26830],{},"nuestra"," app, el \"canto\" literal de una moneda — el fino borde sobre el que a veces cae — es ",[212,26833,26834],{},"también"," un edge case en el sentido software:",[3549,26837,26838,26845],{},[232,26839,26840,26841,26844],{},"Está en el ",[239,26842,26843],{},"extremo"," del espacio de resultados posibles. La mayoría de lanzamientos caen sólidamente en cara o cruz; solo una fracción mínima de todos los estados físicos posibles resulta en \"canto\".",[232,26846,26847,26848,26853],{},"Nuestro código tiene que manejarlo específicamente. Una vez que la física se estabiliza, comprobamos cómo de cerca está la parte superior de la moneda de apuntar de lado, y si está dentro de un pequeño umbral lo llamamos \"Canto\". (Si tienes curiosidad por esa comprobación, escribí ",[21,26849,26852],{"href":26850,"rel":26851},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fcomo-lanzamos-una-moneda",[25],"cómo lanzamos una moneda",", que incluye los detalles.)",[17,26855,26856,26857,26860],{},"Así que aunque la ",[212,26858,26859],{},"palabra"," \"edge\" en \"edge case\" no tiene nada que ver con cantos de moneda, en nuestro pequeño rincón del universo los dos significados se han fundido calladamente. El raro canto físico de la moneda es el raro edge case que tenemos que programar.",[17,26862,26863],{},"Este es el tipo de etimología accidental que me parece deliciosa. Dos términos no relacionados nacidos en campos diferentes, ambos significando \"el extremo raro en el límite\", chocando perfectamente dentro de una única app.",[17,26865,26866,26867,26870],{},"¿Habría sido más ordenado si \"edge case\" ",[212,26868,26869],{},"viniera"," de las monedas? Sí. Pero entonces no tendríamos la coincidencia, y yo no habría escrito este artículo.",[561,26872,24077],{"id":24076},[17,26874,26875],{},"\"Edge case\" viene de algún otro campo de la ingeniería, no de la estadística. Pero en una app de lanzar monedas, el borde es las dos cosas.",[17,26877,26878,26879,26882],{},"Hablando de bordes — ",[21,26880,12951],{"href":24089,"rel":26881},[25]," y a ver si consigues que caiga en el canto.",{"title":1291,"searchDepth":12954,"depth":12954,"links":26884},[26885,26886,26887,26888,26889],{"id":26707,"depth":12954,"text":26708},{"id":26737,"depth":12954,"text":26738},{"id":26790,"depth":12954,"text":26791},{"id":26823,"depth":12954,"text":26824},{"id":24076,"depth":12954,"text":24077},"\"Edge case\" en software no tiene nada que ver con una moneda cayendo sobre su canto — pero en una app de lanzar monedas, los dos significados se cruzan.",{"excerpt":26892},{"type":10,"value":26893},[26894,26896,26898,26907,26909],[13,26895,26667],{"id":26672},[17,26897,26675],{},[17,26899,26678,26900,26682,26902,26687,26905,26691],{},[239,26901,26681],{},[21,26903,26686],{"href":23983,"rel":26904},[25],[239,26906,26690],{},[17,26908,26694],{},[17,26910,26911,26700,26913,26704],{},[239,26912,26699],{},[212,26914,26703],{},"\u002Fblog\u002Fwhere-does-the-term-edge-case-come-from.es",{"title":26667,"description":26890},"de-donde-viene-el-termino-edge-case","blog\u002Fwhere-does-the-term-edge-case-come-from.es","KYQwX9YaTCJC05t27Q7DS-bU1tMhkhYs3eV1xDscWxE",{"id":26921,"title":19270,"_locale":8,"body":26922,"date":27502,"description":27503,"extension":12971,"meta":27504,"navigation":13119,"path":27511,"published":13119,"seo":27512,"slug":27513,"stem":27514,"__hash__":27515},"blog\u002Fblog\u002Fthe-unexpected-statistics-of-coin-flips.es.md",{"type":10,"value":26923,"toc":27492},[26924,26927,26930,26932,26935,26944,26951,26957,26961,26964,26971,26982,26986,26994,27003,27007,27010,27016,27224,27429,27433,27436,27443,27447,27450,27457,27474,27476,27479,27482,27485],[13,26925,19270],{"id":26926},"la-estadística-inesperada-de-los-lanzamientos-de-moneda",[17,26928,26929],{},"Probablemente hayas lanzado una moneda en algún momento de tu vida. Quizás la lanzaste para decidir quién empezaba en un juego, o para decidir dónde ir a cenar. Pero, ¿te has parado alguna vez a pensar en la estadística de los lanzamientos de moneda?",[561,26931,24577],{"id":24576},[17,26933,26934],{},"Una moneda tiene dos caras: cara y cruz. Cuando la lanzas, en teoría hay un 50% de probabilidad de que caiga en cara y un 50% de que caiga en cruz. Esto es porque la moneda tiene dos caras y cada una tiene la misma probabilidad de quedar arriba.",[17,26936,26937,26938,26943],{},"Pero hay otro caso en el que la moneda puede caer sobre su canto. Es un suceso raro, pero es posible. La probabilidad de que una moneda caiga sobre su canto es muy baja, pero no es cero. Se estima que la probabilidad de que una moneda caiga sobre su canto es de aproximadamente 1 entre 6000 lanzamientos dependiendo de las condiciones (",[21,26939,26942],{"href":26940,"rel":26941},"https:\u002F\u002Fui.adsabs.harvard.edu\u002Fabs\u002F1993PhRvE..48.2547M\u002Fabstract",[25],"fuente",").",[17,26945,26946,26947,176],{},"Este es el verdadero \"caso borde\" en los lanzamientos de moneda. Pero no te confundas, no está relacionado con el origen de la expresión. Lee más sobre de dónde viene el término ",[21,26948,26950],{"href":24461,"rel":26949},[25],"\"edge case\"",[17,26952,26953,26954,176],{},"Aprende más sobre cómo lanzamos una moneda en FlipTheCoin.app ",[21,26955,24463],{"href":26850,"rel":26956},[25],[561,26958,26960],{"id":26959},"la-realidad","La realidad",[17,26962,26963],{},"En realidad, la estadística de los lanzamientos de moneda es un poco más complicada. Por ejemplo, si lanzas una moneda 100 veces, quizás no obtengas exactamente 50 caras y 50 cruces. De hecho, podrías obtener 60 caras y 40 cruces, o 55 caras y 45 cruces. Esto es porque hay un elemento de aleatoriedad en los lanzamientos, y el resultado de cada lanzamiento no es del todo predecible.",[17,26965,26966,26967,26970],{},"Podrías culpar de esto a que la moneda no está perfectamente equilibrada, o a que la persona que la lanza no es perfectamente consistente en su técnica. Pero la verdad es que incluso con una moneda perfectamente equilibrada y un lanzador perfectamente consistente, seguirá habiendo algo de variación en los resultados. Ver la sección ",[21,26968,26586],{"href":26969},"#the-gamblers-fallacy"," más abajo.",[17,26972,26973,26974,26978,26979,26981],{},"💡 Si quieres ver una simulación ideal de lanzamiento de moneda, mira la ",[21,26975,26977],{"href":24311,"rel":26976},[25],"herramienta del motor de lanzamiento",". Está diseñada para ser lo más justa matemáticamente posible (y excluye los casos de \"canto\") usando el método ",[1293,26980,24157],{}," de JavaScript para simular el lanzamiento.",[561,26983,26985],{"id":26984},"la-injusticia-de-las-monedas-justas","La injusticia de las monedas justas",[17,26987,26988,26989,26993],{},"Recientemente, un grupo de investigadores holandeses publicó un ",[21,26990,26992],{"href":24489,"rel":26991},[25],"artículo"," que exploraba la estadística de los lanzamientos de moneda con más detalle. Descubrieron que el resultado de un lanzamiento está influido por una variedad de factores, incluidas las condiciones iniciales del lanzamiento. También descubrieron que el resultado no es del todo aleatorio, sino que está influido por esas condiciones iniciales: cuando la gente lanza una moneda corriente, tiende a caer sobre la misma cara con la que empezó.",[17,26995,26996,26997,27002],{},"Por este estudio, fueron condecorados con el ",[21,26998,27001],{"href":26999,"rel":27000},"https:\u002F\u002Fimprobable.com\u002Fig\u002Fwinners\u002F#ig2024",[25],"Premio Ig Nobel"," en 2024.",[561,27004,27006],{"id":27005},"la-falacia-del-jugador","La falacia del jugador",[17,27008,27009],{},"Una idea equivocada común sobre los lanzamientos de moneda se conoce como la Falacia del Jugador. Es la creencia errónea de que, si una moneda ha caído en cara varias veces seguidas, es más probable que caiga en cruz en el siguiente lanzamiento. En realidad, cada lanzamiento es independiente de los anteriores, y el resultado de uno no afecta al resultado del siguiente.",[17,27011,27012,27013,27015],{},"Está relacionado con un tema estadístico llamado \"independencia\". En estadística, dos sucesos se consideran ",[239,27014,25357],{}," si la ocurrencia de uno no afecta a la probabilidad del otro. En el caso de los lanzamientos de moneda, cada lanzamiento es un suceso independiente, y el resultado de uno no afecta al resultado del siguiente.",[17,27017,27018,27019,27121,27122,27223],{},"Un ejemplo de sucesos dependientes sería sacar cartas de una baraja sin reemplazo. Si sacas una carta de una baraja de 52, la probabilidad de sacar una carta roja es ",[35,27020,27022,27044],{"className":27021},[38],[35,27023,27025],{"className":27024},[42],[44,27026,27027],{"xmlns":46},[48,27028,27029,27041],{},[51,27030,27031],{},[25153,27032,27033],{"scriptlevel":517,"displaystyle":9816},[54,27034,27035,27038],{},[61,27036,27037],{},"26",[61,27039,27040],{},"52",[65,27042,27043],{"encoding":67},"\\tfrac{26}{52}",[35,27045,27047],{"className":27046,"ariaHidden":73},[72],[35,27048,27050,27053],{"className":27049},[77],[35,27051],{"className":27052,"style":293},[81],[35,27054,27056,27059,27118],{"className":27055},[86],[35,27057],{"className":27058},[90,91],[35,27060,27062],{"className":27061},[54],[35,27063,27065,27110],{"className":27064},[98,99],[35,27066,27068,27107],{"className":27067},[103],[35,27069,27071,27085,27093],{"className":27070,"style":312},[107],[35,27072,27073,27076],{"style":111},[35,27074],{"className":27075,"style":116},[115],[35,27077,27079],{"className":27078},[120,121,122,123],[35,27080,27082],{"className":27081},[86,123],[35,27083,27040],{"className":27084},[86,123],[35,27086,27087,27090],{"style":132},[35,27088],{"className":27089,"style":116},[115],[35,27091],{"className":27092,"style":140},[139],[35,27094,27095,27098],{"style":143},[35,27096],{"className":27097,"style":116},[115],[35,27099,27101],{"className":27100},[120,121,122,123],[35,27102,27104],{"className":27103},[86,123],[35,27105,27037],{"className":27106},[86,123],[35,27108,162],{"className":27109},[161],[35,27111,27113],{"className":27112},[103],[35,27114,27116],{"className":27115,"style":169},[107],[35,27117],{},[35,27119],{"className":27120},[175,91],". Pero si sacas una carta roja y no la repones, la probabilidad de sacar otra roja ahora es ",[35,27123,27125,27146],{"className":27124},[38],[35,27126,27128],{"className":27127},[42],[44,27129,27130],{"xmlns":46},[48,27131,27132,27143],{},[51,27133,27134],{},[25153,27135,27136],{"scriptlevel":517,"displaystyle":9816},[54,27137,27138,27140],{},[61,27139,25412],{},[61,27141,27142],{},"51",[65,27144,27145],{"encoding":67},"\\tfrac{25}{51}",[35,27147,27149],{"className":27148,"ariaHidden":73},[72],[35,27150,27152,27155],{"className":27151},[77],[35,27153],{"className":27154,"style":293},[81],[35,27156,27158,27161,27220],{"className":27157},[86],[35,27159],{"className":27160},[90,91],[35,27162,27164],{"className":27163},[54],[35,27165,27167,27212],{"className":27166},[98,99],[35,27168,27170,27209],{"className":27169},[103],[35,27171,27173,27187,27195],{"className":27172,"style":312},[107],[35,27174,27175,27178],{"style":111},[35,27176],{"className":27177,"style":116},[115],[35,27179,27181],{"className":27180},[120,121,122,123],[35,27182,27184],{"className":27183},[86,123],[35,27185,27142],{"className":27186},[86,123],[35,27188,27189,27192],{"style":132},[35,27190],{"className":27191,"style":116},[115],[35,27193],{"className":27194,"style":140},[139],[35,27196,27197,27200],{"style":143},[35,27198],{"className":27199,"style":116},[115],[35,27201,27203],{"className":27202},[120,121,122,123],[35,27204,27206],{"className":27205},[86,123],[35,27207,25412],{"className":27208},[86,123],[35,27210,162],{"className":27211},[161],[35,27213,27215],{"className":27214},[103],[35,27216,27218],{"className":27217,"style":169},[107],[35,27219],{},[35,27221],{"className":27222},[175,91],". El primer suceso ha afectado a la probabilidad del segundo.",[17,27225,27226,27227,24681,27326,27425,27426,176],{},"💡 Si quieres saber más sobre qué significa tener una probabilidad de ",[35,27228,27230,27249],{"className":27229},[38],[35,27231,27233],{"className":27232},[42],[44,27234,27235],{"xmlns":46},[48,27236,27237,27247],{},[51,27238,27239],{},[25153,27240,27241],{"scriptlevel":517,"displaystyle":9816},[54,27242,27243,27245],{},[61,27244,27037],{},[61,27246,27040],{},[65,27248,27043],{"encoding":67},[35,27250,27252],{"className":27251,"ariaHidden":73},[72],[35,27253,27255,27258],{"className":27254},[77],[35,27256],{"className":27257,"style":293},[81],[35,27259,27261,27264,27323],{"className":27260},[86],[35,27262],{"className":27263},[90,91],[35,27265,27267],{"className":27266},[54],[35,27268,27270,27315],{"className":27269},[98,99],[35,27271,27273,27312],{"className":27272},[103],[35,27274,27276,27290,27298],{"className":27275,"style":312},[107],[35,27277,27278,27281],{"style":111},[35,27279],{"className":27280,"style":116},[115],[35,27282,27284],{"className":27283},[120,121,122,123],[35,27285,27287],{"className":27286},[86,123],[35,27288,27040],{"className":27289},[86,123],[35,27291,27292,27295],{"style":132},[35,27293],{"className":27294,"style":116},[115],[35,27296],{"className":27297,"style":140},[139],[35,27299,27300,27303],{"style":143},[35,27301],{"className":27302,"style":116},[115],[35,27304,27306],{"className":27305},[120,121,122,123],[35,27307,27309],{"className":27308},[86,123],[35,27310,27037],{"className":27311},[86,123],[35,27313,162],{"className":27314},[161],[35,27316,27318],{"className":27317},[103],[35,27319,27321],{"className":27320,"style":169},[107],[35,27322],{},[35,27324],{"className":27325},[175,91],[35,27327,27329,27348],{"className":27328},[38],[35,27330,27332],{"className":27331},[42],[44,27333,27334],{"xmlns":46},[48,27335,27336,27346],{},[51,27337,27338],{},[25153,27339,27340],{"scriptlevel":517,"displaystyle":9816},[54,27341,27342,27344],{},[61,27343,25412],{},[61,27345,27142],{},[65,27347,27145],{"encoding":67},[35,27349,27351],{"className":27350,"ariaHidden":73},[72],[35,27352,27354,27357],{"className":27353},[77],[35,27355],{"className":27356,"style":293},[81],[35,27358,27360,27363,27422],{"className":27359},[86],[35,27361],{"className":27362},[90,91],[35,27364,27366],{"className":27365},[54],[35,27367,27369,27414],{"className":27368},[98,99],[35,27370,27372,27411],{"className":27371},[103],[35,27373,27375,27389,27397],{"className":27374,"style":312},[107],[35,27376,27377,27380],{"style":111},[35,27378],{"className":27379,"style":116},[115],[35,27381,27383],{"className":27382},[120,121,122,123],[35,27384,27386],{"className":27385},[86,123],[35,27387,27142],{"className":27388},[86,123],[35,27390,27391,27394],{"style":132},[35,27392],{"className":27393,"style":116},[115],[35,27395],{"className":27396,"style":140},[139],[35,27398,27399,27402],{"style":143},[35,27400],{"className":27401,"style":116},[115],[35,27403,27405],{"className":27404},[120,121,122,123],[35,27406,27408],{"className":27407},[86,123],[35,27409,25412],{"className":27410},[86,123],[35,27412,162],{"className":27413},[161],[35,27415,27417],{"className":27416},[103],[35,27418,27420],{"className":27419,"style":169},[107],[35,27421],{},[35,27423],{"className":27424},[175,91],", mira el artículo ",[21,27427,15769],{"href":24515,"rel":27428},[25],[561,27430,27432],{"id":27431},"la-ley-de-los-grandes-números","La ley de los grandes números",[17,27434,27435],{},"La ley de los grandes números dice que, cuantas más veces lances una moneda, más se acercarán los resultados a la partición teórica 50\u002F50. Por ejemplo, si lanzas una moneda 10 veces, podrías obtener 6 caras y 4 cruces. Pero si la lanzas 1 millón de veces, es más probable que te acerques al 50\u002F50.",[17,27437,27438,27439,27442],{},"En la práctica, la ley de los grandes números no es siempre perfecta. Seguirá habiendo algo de variación en los resultados, incluso con un ",[239,27440,27441],{},"número muy grande"," de lanzamientos. Pero cuantas más veces lances la moneda, más se acercarán los resultados a la partición teórica 50\u002F50.",[582,27444,27446],{"id":27445},"cuántas-veces-deberías-lanzar-una-moneda-para-obtener-un-5050","¿Cuántas veces deberías lanzar una moneda para obtener un 50\u002F50?",[17,27448,27449],{},"En teoría, cuantas más veces la lances, más se acercarán los resultados a un 50\u002F50. Pero, ¿llegas a un punto en que los resultados son exactamente 50\u002F50?",[17,27451,27452,27453,27456],{},"La respuesta es no. Incluso si lanzas una moneda un número infinito de veces, nunca obtendrás exactamente un 50\u002F50. Esto está relacionado con el concepto de ",[239,27454,27455],{},"desviación estándar"," en estadística. La desviación estándar mide cómo de dispersos están los valores en un conjunto de datos. En el caso de los lanzamientos, mide cuánto se alejan los resultados de la partición teórica 50\u002F50.",[17,27458,27459,27460,27462,27463,27467,27468,27470,27471,27473],{},"Imagina que lanzamos la moneda 1000 veces. La probabilidad de obtener ",[239,27461,12790],{}," 500 caras y 500 cruces es muy baja. De hecho, está alrededor del 2,5% (",[21,27464,26942],{"href":27465,"rel":27466},"https:\u002F\u002Fwww.wolframalpha.com\u002Finput?i=probability+of+getting+exactly+500+heads+and+500+tails+in+1000+flips+of+a+fair+coin",[25],"). Los rangos, como que el 95% de las veces caiga entre 469 y 530 caras, o el 99% entre 450 y 550, se derivan usando la ",[239,27469,23994],{}," y la ",[239,27472,27455],{},". Estas herramientas estadísticas ayudan a describir la dispersión de resultados en ensayos repetidos. Para aclarar: esto significa que el 99% de las veces que lanzas una moneda 1000 veces, el resultado cae en ese rango.",[561,27475,24077],{"id":24076},[17,27477,27478],{},"La estadística de los lanzamientos de moneda es un tema fascinante que puede enseñarnos mucho sobre probabilidad y aleatoriedad.",[17,27480,27481],{},"Se usa comúnmente como uno de los primeros ejemplos al enseñar estadística, pero también es un tema que puede explorarse en más profundidad. Es un concepto simple que puede llevar a discusiones complejas sobre independencia, aleatoriedad y la ley de los grandes números. Para algunas personas es solo una forma divertida de tomar decisiones, para otras es una forma de explorar el mundo de la estadística.",[17,27483,27484],{},"Así que la próxima vez que lances una moneda, tómate un momento para apreciar la estadística inesperada detrás de este simple acto.",[17,27486,27487,27491],{},[21,27488,27490],{"href":24089,"rel":27489},[25],"Prueba nuestro lanzador de moneda online gratuito"," — cara o cruz, decidido por física 3D real.",{"title":1291,"searchDepth":12954,"depth":12954,"links":27493},[27494,27495,27496,27497,27498,27501],{"id":24576,"depth":12954,"text":24577},{"id":26959,"depth":12954,"text":26960},{"id":26984,"depth":12954,"text":26985},{"id":27005,"depth":12954,"text":27006},{"id":27431,"depth":12954,"text":27432,"children":27499},[27500],{"id":27445,"depth":12959,"text":27446},{"id":24076,"depth":12954,"text":24077},"2025-03-22T01:00:00Z","Un vistazo a lo que realmente ocurre cuando lanzas una moneda — los casos extremos, la falacia del jugador y por qué las monedas justas no lo son tanto.",{"excerpt":27505},{"type":10,"value":27506},[27507,27509],[13,27508,19270],{"id":26926},[17,27510,26929],{},"\u002Fblog\u002Fthe-unexpected-statistics-of-coin-flips.es",{"title":19270,"description":27503},"la-estadistica-inesperada-de-los-lanzamientos-de-moneda","blog\u002Fthe-unexpected-statistics-of-coin-flips.es","FXFZGF1QBIxAknshdpYtuXd3UHyUiKcuSvjB8B0kPZQ",1779926067380]