[{"data":1,"prerenderedAt":4226},["ShallowReactive",2],{"locale-switcher-blog-es-null":3,"blog-series-page-es-probability-basics":4,"series-page-authors":4214},null,{"meta":5,"members":18},{"id":6,"title":7,"_locale":8,"date":9,"description":10,"extension":11,"meta":12,"slug":15,"stem":16,"__hash__":17},"series\u002Fseries\u002Fprobability-basics.es.yml","Fundamentos de probabilidad","es","2025-06-13T01:00:00Z","Empieza aquí: un recorrido guiado desde qué significa la probabilidad hasta cómo las distribuciones describen el comportamiento de una moneda.","yml",{"path":13,"body":14},"\u002Fseries\u002Fprobability-basics.es",{"slug":15,"title":7,"description":10,"date":9,"_locale":8},"probability-basics","series\u002Fprobability-basics.es","vFZjDWGo1oZ31sYiiJc6-PqYrPV0Xd0cVFyXw_mzs34",[19,2242,2675],{"id":20,"title":7,"_locale":8,"author":3,"body":21,"date":2219,"description":2220,"excerpt":2221,"extension":2233,"level":2234,"meta":2235,"navigation":2236,"path":2237,"published":2236,"seo":2238,"series":15,"seriesOrder":2239,"slug":27,"stem":2240,"tags":3,"__hash__":2241},"blog\u002Fblog\u002Fprobability-basics.es.md",{"type":22,"value":23,"toc":2204},"minimal",[24,28,41,44,49,200,211,215,218,245,248,252,259,277,288,292,303,439,442,884,895,899,902,907,918,1270,1374,1378,1392,1915,1922,2127,2134,2138,2144,2157,2161,2164,2192,2195],[25,26,7],"h1",{"id":27},"fundamentos-de-probabilidad",[29,30,31,32,36,37,40],"p",{},"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 ",[33,34,35],"strong",{},"cara"," o en ",[33,38,39],{},"cruz",".",[29,42,43],{},"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.",[45,46,48],"h2",{"id":47},"lo-básico","Lo básico",[29,50,51,52,191,192,195,196,199],{},"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 ",[53,54,57,86],"span",{"className":55},[56],"katex",[53,58,61],{"className":59},[60],"katex-mathml",[62,63,65],"math",{"xmlns":64},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[66,67,68,81],"semantics",{},[69,70,71],"mrow",{},[72,73,74,78],"mfrac",{},[75,76,77],"mn",{},"1",[75,79,80],{},"2",[82,83,85],"annotation",{"encoding":84},"application\u002Fx-tex","\\frac{1}{2}",[53,87,91],{"className":88,"ariaHidden":90},[89],"katex-html","true",[53,92,95,100],{"className":93},[94],"base",[53,96],{"className":97,"style":99},[98],"strut","height:1.1901em;vertical-align:-0.345em;",[53,101,104,109,187],{"className":102},[103],"mord",[53,105],{"className":106},[107,108],"mopen","nulldelimiter",[53,110,112],{"className":111},[72],[53,113,117,178],{"className":114},[115,116],"vlist-t","vlist-t2",[53,118,121,173],{"className":119},[120],"vlist-r",[53,122,126,147,158],{"className":123,"style":125},[124],"vlist","height:0.8451em;",[53,127,129,134],{"style":128},"top:-2.655em;",[53,130],{"className":131,"style":133},[132],"pstrut","height:3em;",[53,135,141],{"className":136},[137,138,139,140],"sizing","reset-size6","size3","mtight",[53,142,144],{"className":143},[103,140],[53,145,80],{"className":146},[103,140],[53,148,150,153],{"style":149},"top:-3.23em;",[53,151],{"className":152,"style":133},[132],[53,154],{"className":155,"style":157},[156],"frac-line","border-bottom-width:0.04em;",[53,159,161,164],{"style":160},"top:-3.394em;",[53,162],{"className":163,"style":133},[132],[53,165,167],{"className":166},[137,138,139,140],[53,168,170],{"className":169},[103,140],[53,171,77],{"className":172},[103,140],[53,174,177],{"className":175},[176],"vlist-s","​",[53,179,181],{"className":180},[120],[53,182,185],{"className":183,"style":184},[124],"height:0.345em;",[53,186],{},[53,188],{"className":189},[190,108],"mclose",", que es lo mismo que ",[33,193,194],{},"0,5"," o ",[33,197,198],{},"50%",". Las tres representaciones son equivalentes y se pueden usar indistintamente.",[29,201,202,203,210],{},"Como veremos en ",[204,205,209],"a",{"href":206,"rel":207},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fla-estadistica-inesperada-de-los-lanzamientos-de-moneda",[208],"nofollow","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.",[45,212,214],{"id":213},"resultados-y-sucesos","Resultados y sucesos",[29,216,217],{},"Antes de seguir, dos piezas pequeñas de vocabulario:",[219,220,221,239],"ul",{},[222,223,224,225,228,229,232,233,235,236,40],"li",{},"Un ",[33,226,227],{},"resultado"," es una de las cosas posibles que pueden ocurrir. Cuando lanzas una moneda, los resultados son ",[230,231,35],"em",{},", ",[230,234,39],{}," y (muy raramente) ",[230,237,238],{},"canto",[222,240,224,241,244],{},[33,242,243],{},"suceso"," es una pregunta que hacemos sobre los resultados, como \"¿cayó en cara?\" o \"¿cayó en alguna cara?\".",[29,246,247],{},"La probabilidad es la herramienta que usamos para ponerle un número a lo probable que es un suceso.",[45,249,251],{"id":250},"la-escala-de-probabilidad","La escala de probabilidad",[29,253,254,255,258],{},"Las probabilidades siempre viven en una escala de ",[33,256,257],{},"0 a 1",":",[219,260,261,267,272],{},[222,262,263,266],{},[33,264,265],{},"0"," significa que el suceso es imposible. Una moneda convirtiéndose en paloma en pleno vuelo tiene probabilidad 0.",[222,268,269,271],{},[33,270,77],{}," significa que el suceso es seguro. Una moneda lanzada al aire acabando por caer, gracias a la gravedad, tiene probabilidad 1.",[222,273,274,275,40],{},"Cualquier cosa entre medias es \"algo probable\". Una moneda justa cayendo en cara se sitúa justo en el medio, en ",[33,276,194],{},[29,278,279,280,283,284,287],{},"En términos de porcentaje, esa misma escala va del ",[33,281,282],{},"0%"," al ",[33,285,286],{},"100%",". La misma idea, expresada de otra forma.",[45,289,291],{"id":290},"la-regla-del-complemento","La regla del complemento",[29,293,294,295,298,299,302],{},"Cada suceso tiene una imagen espejo llamada su ",[33,296,297],{},"complemento",": el suceso de que ",[230,300,301],{},"no"," ocurra. Y como algo o bien ocurre o bien no, las dos probabilidades siempre suman 1. En símbolos:",[29,304,307],{"className":305},[306],"math-block",[53,308,310,357],{"className":309},[56],[53,311,313],{"className":312},[60],[62,314,315],{"xmlns":64},[66,316,317,354],{},[69,318,319,323,328,332,335,338,341,343,346,348,350,352],{},[320,321,322],"mi",{},"P",[324,325,327],"mo",{"stretchy":326},"false","(",[329,330,331],"mtext",{},"no ",[320,333,334],{},"A",[324,336,337],{"stretchy":326},")",[324,339,340],{},"=",[75,342,77],{},[324,344,345],{},"−",[320,347,322],{},[324,349,327],{"stretchy":326},[320,351,334],{},[324,353,337],{"stretchy":326},[82,355,356],{"encoding":84},"P(\\text{no } A) = 1 - P(A)",[53,358,360,400,421],{"className":359,"ariaHidden":90},[89],[53,361,363,367,372,375,382,385,388,393,397],{"className":362},[94],[53,364],{"className":365,"style":366},[98],"height:1em;vertical-align:-0.25em;",[53,368,322],{"className":369,"style":371},[103,370],"mathnormal","margin-right:0.1389em;",[53,373,327],{"className":374},[107],[53,376,379],{"className":377},[103,378],"text",[53,380,331],{"className":381},[103],[53,383,334],{"className":384},[103,370],[53,386,337],{"className":387},[190],[53,389],{"className":390,"style":392},[391],"mspace","margin-right:0.2778em;",[53,394,340],{"className":395},[396],"mrel",[53,398],{"className":399,"style":392},[391],[53,401,403,407,410,414,418],{"className":402},[94],[53,404],{"className":405,"style":406},[98],"height:0.7278em;vertical-align:-0.0833em;",[53,408,77],{"className":409},[103],[53,411],{"className":412,"style":413},[391],"margin-right:0.2222em;",[53,415,345],{"className":416},[417],"mbin",[53,419],{"className":420,"style":413},[391],[53,422,424,427,430,433,436],{"className":423},[94],[53,425],{"className":426,"style":366},[98],[53,428,322],{"className":429,"style":371},[103,370],[53,431,327],{"className":432},[107],[53,434,334],{"className":435},[103,370],[53,437,337],{"className":438},[190],[29,440,441],{},"Para una moneda justa:",[219,443,444,674],{},[222,445,446,533,534,40],{},[53,447,449,479],{"className":448},[56],[53,450,452],{"className":451},[60],[62,453,454],{"xmlns":64},[66,455,456,476],{},[69,457,458,460,462,464,466,468,470,473],{},[320,459,322],{},[324,461,327],{"stretchy":326},[329,463,35],{},[324,465,337],{"stretchy":326},[324,467,340],{},[75,469,265],{},[324,471,472],{"separator":90},",",[75,474,475],{},"5",[82,477,478],{"encoding":84},"P(\\text{cara}) = 0,5",[53,480,482,512],{"className":481,"ariaHidden":90},[89],[53,483,485,488,491,494,500,503,506,509],{"className":484},[94],[53,486],{"className":487,"style":366},[98],[53,489,322],{"className":490,"style":371},[103,370],[53,492,327],{"className":493},[107],[53,495,497],{"className":496},[103,378],[53,498,35],{"className":499},[103],[53,501,337],{"className":502},[190],[53,504],{"className":505,"style":392},[391],[53,507,340],{"className":508},[396],[53,510],{"className":511,"style":392},[391],[53,513,515,519,522,526,530],{"className":514},[94],[53,516],{"className":517,"style":518},[98],"height:0.8389em;vertical-align:-0.1944em;",[53,520,265],{"className":521},[103],[53,523,472],{"className":524},[525],"mpunct",[53,527],{"className":528,"style":529},[391],"margin-right:0.1667em;",[53,531,475],{"className":532},[103],", así que ",[53,535,537,578],{"className":536},[56],[53,538,540],{"className":539},[60],[62,541,542],{"xmlns":64},[66,543,544,575],{},[69,545,546,548,550,553,555,557,559,561,563,565,567,569,571,573],{},[320,547,322],{},[324,549,327],{"stretchy":326},[329,551,552],{},"no cara",[324,554,337],{"stretchy":326},[324,556,340],{},[75,558,77],{},[324,560,345],{},[75,562,265],{},[324,564,472],{"separator":90},[75,566,475],{},[324,568,340],{},[75,570,265],{},[324,572,472],{"separator":90},[75,574,475],{},[82,576,577],{"encoding":84},"P(\\text{no cara}) = 1 - 0,5 = 0,5",[53,579,581,611,629,656],{"className":580,"ariaHidden":90},[89],[53,582,584,587,590,593,599,602,605,608],{"className":583},[94],[53,585],{"className":586,"style":366},[98],[53,588,322],{"className":589,"style":371},[103,370],[53,591,327],{"className":592},[107],[53,594,596],{"className":595},[103,378],[53,597,552],{"className":598},[103],[53,600,337],{"className":601},[190],[53,603],{"className":604,"style":392},[391],[53,606,340],{"className":607},[396],[53,609],{"className":610,"style":392},[391],[53,612,614,617,620,623,626],{"className":613},[94],[53,615],{"className":616,"style":406},[98],[53,618,77],{"className":619},[103],[53,621],{"className":622,"style":413},[391],[53,624,345],{"className":625},[417],[53,627],{"className":628,"style":413},[391],[53,630,632,635,638,641,644,647,650,653],{"className":631},[94],[53,633],{"className":634,"style":518},[98],[53,636,265],{"className":637},[103],[53,639,472],{"className":640},[525],[53,642],{"className":643,"style":529},[391],[53,645,475],{"className":646},[103],[53,648],{"className":649,"style":392},[391],[53,651,340],{"className":652},[396],[53,654],{"className":655,"style":392},[391],[53,657,659,662,665,668,671],{"className":658},[94],[53,660],{"className":661,"style":518},[98],[53,663,265],{"className":664},[103],[53,666,472],{"className":667},[525],[53,669],{"className":670,"style":529},[391],[53,672,475],{"className":673},[103],[222,675,676,878,879,883],{},[53,677,679,726],{"className":678},[56],[53,680,682],{"className":681},[60],[62,683,684],{"xmlns":64},[66,685,686,723],{},[69,687,688,690,692,695,697,700,702,704,714,716,718,720],{},[320,689,322],{},[324,691,327],{"stretchy":326},[329,693,694],{},"cara o cruz",[324,696,337],{"stretchy":326},[324,698,699],{},"≈",[75,701,77],{},[324,703,345],{},[705,706,707],"mstyle",{"scriptlevel":265,"displaystyle":326},[72,708,709,711],{},[75,710,77],{},[75,712,713],{},"6000",[324,715,699],{},[75,717,265],{},[324,719,472],{"separator":90},[75,721,722],{},"99983",[82,724,725],{"encoding":84},"P(\\text{cara o cruz}) \\approx 1 - \\tfrac{1}{6000} \\approx 0,99983",[53,727,729,759,777,860],{"className":728,"ariaHidden":90},[89],[53,730,732,735,738,741,747,750,753,756],{"className":731},[94],[53,733],{"className":734,"style":366},[98],[53,736,322],{"className":737,"style":371},[103,370],[53,739,327],{"className":740},[107],[53,742,744],{"className":743},[103,378],[53,745,694],{"className":746},[103],[53,748,337],{"className":749},[190],[53,751],{"className":752,"style":392},[391],[53,754,699],{"className":755},[396],[53,757],{"className":758,"style":392},[391],[53,760,762,765,768,771,774],{"className":761},[94],[53,763],{"className":764,"style":406},[98],[53,766,77],{"className":767},[103],[53,769],{"className":770,"style":413},[391],[53,772,345],{"className":773},[417],[53,775],{"className":776,"style":413},[391],[53,778,780,783,851,854,857],{"className":779},[94],[53,781],{"className":782,"style":99},[98],[53,784,786,789,848],{"className":785},[103],[53,787],{"className":788},[107,108],[53,790,792],{"className":791},[72],[53,793,795,840],{"className":794},[115,116],[53,796,798,837],{"className":797},[120],[53,799,801,815,823],{"className":800,"style":125},[124],[53,802,803,806],{"style":128},[53,804],{"className":805,"style":133},[132],[53,807,809],{"className":808},[137,138,139,140],[53,810,812],{"className":811},[103,140],[53,813,713],{"className":814},[103,140],[53,816,817,820],{"style":149},[53,818],{"className":819,"style":133},[132],[53,821],{"className":822,"style":157},[156],[53,824,825,828],{"style":160},[53,826],{"className":827,"style":133},[132],[53,829,831],{"className":830},[137,138,139,140],[53,832,834],{"className":833},[103,140],[53,835,77],{"className":836},[103,140],[53,838,177],{"className":839},[176],[53,841,843],{"className":842},[120],[53,844,846],{"className":845,"style":184},[124],[53,847],{},[53,849],{"className":850},[190,108],[53,852],{"className":853,"style":392},[391],[53,855,699],{"className":856},[396],[53,858],{"className":859,"style":392},[391],[53,861,863,866,869,872,875],{"className":862},[94],[53,864],{"className":865,"style":518},[98],[53,867,265],{"className":868},[103],[53,870,472],{"className":871},[525],[53,873],{"className":874,"style":529},[391],[53,876,722],{"className":877},[103],", porque la única otra opción es el raro \"canto\". (Más sobre ese 1 entre 6000 en ",[204,880,882],{"href":206,"rel":881},[208],"La estadística inesperada de los lanzamientos de moneda",".)",[29,885,886,887,890,891,894],{},"La regla del complemento suele ser la forma más fácil de calcular una probabilidad: en lugar de contar todas las formas en que algo ",[230,888,889],{},"puede"," pasar, cuenta la única forma en que ",[230,892,893],{},"no puede"," y réstala a 1.",[45,896,898],{"id":897},"combinar-sucesos-y-vs-o","Combinar sucesos: \"y\" vs \"o\"",[29,900,901],{},"La cosa se pone interesante cuando preguntamos por más de un lanzamiento a la vez. Dos reglas cubren casi todo lo que necesitarás:",[903,904,906],"h3",{"id":905},"y-ambas-cosas-ocurren","\"Y\" — ambas cosas ocurren",[29,908,909,910,913,914,917],{},"Los lanzamientos de moneda no se influyen entre sí (son ",[33,911,912],{},"independientes","). En este caso, ",[33,915,916],{},"multiplicas"," sus probabilidades:",[219,919,920,1078],{},[222,921,922,1077],{},[53,923,925,972],{"className":924},[56],[53,926,928],{"className":927},[60],[62,929,930],{"xmlns":64},[66,931,932,969],{},[69,933,934,936,938,941,943,945,947,949,951,954,956,958,960,962,964,966],{},[320,935,322],{},[324,937,327],{"stretchy":326},[329,939,940],{},"OO",[324,942,337],{"stretchy":326},[324,944,340],{},[75,946,265],{},[324,948,472],{"separator":90},[75,950,475],{},[324,952,953],{},"×",[75,955,265],{},[324,957,472],{"separator":90},[75,959,475],{},[324,961,340],{},[75,963,265],{},[324,965,472],{"separator":90},[75,967,968],{},"25",[82,970,971],{"encoding":84},"P(\\text{OO}) = 0,5 \\times 0,5 = 0,25",[53,973,975,1005,1032,1059],{"className":974,"ariaHidden":90},[89],[53,976,978,981,984,987,993,996,999,1002],{"className":977},[94],[53,979],{"className":980,"style":366},[98],[53,982,322],{"className":983,"style":371},[103,370],[53,985,327],{"className":986},[107],[53,988,990],{"className":989},[103,378],[53,991,940],{"className":992},[103],[53,994,337],{"className":995},[190],[53,997],{"className":998,"style":392},[391],[53,1000,340],{"className":1001},[396],[53,1003],{"className":1004,"style":392},[391],[53,1006,1008,1011,1014,1017,1020,1023,1026,1029],{"className":1007},[94],[53,1009],{"className":1010,"style":518},[98],[53,1012,265],{"className":1013},[103],[53,1015,472],{"className":1016},[525],[53,1018],{"className":1019,"style":529},[391],[53,1021,475],{"className":1022},[103],[53,1024],{"className":1025,"style":413},[391],[53,1027,953],{"className":1028},[417],[53,1030],{"className":1031,"style":413},[391],[53,1033,1035,1038,1041,1044,1047,1050,1053,1056],{"className":1034},[94],[53,1036],{"className":1037,"style":518},[98],[53,1039,265],{"className":1040},[103],[53,1042,472],{"className":1043},[525],[53,1045],{"className":1046,"style":529},[391],[53,1048,475],{"className":1049},[103],[53,1051],{"className":1052,"style":392},[391],[53,1054,340],{"className":1055},[396],[53,1057],{"className":1058,"style":392},[391],[53,1060,1062,1065,1068,1071,1074],{"className":1061},[94],[53,1063],{"className":1064,"style":518},[98],[53,1066,265],{"className":1067},[103],[53,1069,472],{"className":1070},[525],[53,1072],{"className":1073,"style":529},[391],[53,1075,968],{"className":1076},[103],", o 25%.",[222,1079,1080,1269],{},[53,1081,1083,1137],{"className":1082},[56],[53,1084,1086],{"className":1085},[60],[62,1087,1088],{"xmlns":64},[66,1089,1090,1134],{},[69,1091,1092,1094,1096,1099,1101,1103,1105,1107,1109,1111,1113,1115,1117,1119,1121,1123,1125,1127,1129,1131],{},[320,1093,322],{},[324,1095,327],{"stretchy":326},[329,1097,1098],{},"OOO",[324,1100,337],{"stretchy":326},[324,1102,340],{},[75,1104,265],{},[324,1106,472],{"separator":90},[75,1108,475],{},[324,1110,953],{},[75,1112,265],{},[324,1114,472],{"separator":90},[75,1116,475],{},[324,1118,953],{},[75,1120,265],{},[324,1122,472],{"separator":90},[75,1124,475],{},[324,1126,340],{},[75,1128,265],{},[324,1130,472],{"separator":90},[75,1132,1133],{},"125",[82,1135,1136],{"encoding":84},"P(\\text{OOO}) = 0,5 \\times 0,5 \\times 0,5 = 0,125",[53,1138,1140,1170,1197,1224,1251],{"className":1139,"ariaHidden":90},[89],[53,1141,1143,1146,1149,1152,1158,1161,1164,1167],{"className":1142},[94],[53,1144],{"className":1145,"style":366},[98],[53,1147,322],{"className":1148,"style":371},[103,370],[53,1150,327],{"className":1151},[107],[53,1153,1155],{"className":1154},[103,378],[53,1156,1098],{"className":1157},[103],[53,1159,337],{"className":1160},[190],[53,1162],{"className":1163,"style":392},[391],[53,1165,340],{"className":1166},[396],[53,1168],{"className":1169,"style":392},[391],[53,1171,1173,1176,1179,1182,1185,1188,1191,1194],{"className":1172},[94],[53,1174],{"className":1175,"style":518},[98],[53,1177,265],{"className":1178},[103],[53,1180,472],{"className":1181},[525],[53,1183],{"className":1184,"style":529},[391],[53,1186,475],{"className":1187},[103],[53,1189],{"className":1190,"style":413},[391],[53,1192,953],{"className":1193},[417],[53,1195],{"className":1196,"style":413},[391],[53,1198,1200,1203,1206,1209,1212,1215,1218,1221],{"className":1199},[94],[53,1201],{"className":1202,"style":518},[98],[53,1204,265],{"className":1205},[103],[53,1207,472],{"className":1208},[525],[53,1210],{"className":1211,"style":529},[391],[53,1213,475],{"className":1214},[103],[53,1216],{"className":1217,"style":413},[391],[53,1219,953],{"className":1220},[417],[53,1222],{"className":1223,"style":413},[391],[53,1225,1227,1230,1233,1236,1239,1242,1245,1248],{"className":1226},[94],[53,1228],{"className":1229,"style":518},[98],[53,1231,265],{"className":1232},[103],[53,1234,472],{"className":1235},[525],[53,1237],{"className":1238,"style":529},[391],[53,1240,475],{"className":1241},[103],[53,1243],{"className":1244,"style":392},[391],[53,1246,340],{"className":1247},[396],[53,1249],{"className":1250,"style":392},[391],[53,1252,1254,1257,1260,1263,1266],{"className":1253},[94],[53,1255],{"className":1256,"style":518},[98],[53,1258,265],{"className":1259},[103],[53,1261,472],{"className":1262},[525],[53,1264],{"className":1265,"style":529},[391],[53,1267,1133],{"className":1268},[103],", o 12,5%.",[29,1271,1272,1273,1373],{},"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 ",[53,1274,1276,1296],{"className":1275},[56],[53,1277,1279],{"className":1278},[60],[62,1280,1281],{"xmlns":64},[66,1282,1283,1293],{},[69,1284,1285],{},[705,1286,1287],{"scriptlevel":265,"displaystyle":326},[72,1288,1289,1291],{},[75,1290,77],{},[75,1292,80],{},[82,1294,1295],{"encoding":84},"\\tfrac{1}{2}",[53,1297,1299],{"className":1298,"ariaHidden":90},[89],[53,1300,1302,1305],{"className":1301},[94],[53,1303],{"className":1304,"style":99},[98],[53,1306,1308,1311,1370],{"className":1307},[103],[53,1309],{"className":1310},[107,108],[53,1312,1314],{"className":1313},[72],[53,1315,1317,1362],{"className":1316},[115,116],[53,1318,1320,1359],{"className":1319},[120],[53,1321,1323,1337,1345],{"className":1322,"style":125},[124],[53,1324,1325,1328],{"style":128},[53,1326],{"className":1327,"style":133},[132],[53,1329,1331],{"className":1330},[137,138,139,140],[53,1332,1334],{"className":1333},[103,140],[53,1335,80],{"className":1336},[103,140],[53,1338,1339,1342],{"style":149},[53,1340],{"className":1341,"style":133},[132],[53,1343],{"className":1344,"style":157},[156],[53,1346,1347,1350],{"style":160},[53,1348],{"className":1349,"style":133},[132],[53,1351,1353],{"className":1352},[137,138,139,140],[53,1354,1356],{"className":1355},[103,140],[53,1357,77],{"className":1358},[103,140],[53,1360,177],{"className":1361},[176],[53,1363,1365],{"className":1364},[120],[53,1366,1368],{"className":1367,"style":184},[124],[53,1369],{},[53,1371],{"className":1372},[190,108]," al producto.",[903,1375,1377],{"id":1376},"o-al-menos-una-de-varias-cosas-ocurre","\"O\" — al menos una de varias cosas ocurre",[29,1379,1380,1381,1384,1385,1388,1389,917],{},"Si dos resultados ",[33,1382,1383],{},"no pueden ocurrir al mismo tiempo"," (son ",[33,1386,1387],{},"mutuamente excluyentes","), ",[33,1390,1391],{},"sumas",[219,1393,1394,1566],{},[222,1395,1396,1565],{},[53,1397,1399,1447],{"className":1398},[56],[53,1400,1402],{"className":1401},[60],[62,1403,1404],{"xmlns":64},[66,1405,1406,1444],{},[69,1407,1408,1410,1412,1415,1418,1421,1423,1425,1427,1429,1431,1434,1436,1438,1440,1442],{},[320,1409,322],{},[324,1411,327],{"stretchy":326},[329,1413,1414],{},"O",[324,1416,1417],{},"∪",[329,1419,1420],{},"X",[324,1422,337],{"stretchy":326},[324,1424,340],{},[75,1426,265],{},[324,1428,472],{"separator":90},[75,1430,475],{},[324,1432,1433],{},"+",[75,1435,265],{},[324,1437,472],{"separator":90},[75,1439,475],{},[324,1441,340],{},[75,1443,77],{},[82,1445,1446],{"encoding":84},"P(\\text{O} \\cup \\text{X}) = 0,5 + 0,5 = 1",[53,1448,1450,1477,1501,1528,1555],{"className":1449,"ariaHidden":90},[89],[53,1451,1453,1456,1459,1462,1468,1471,1474],{"className":1452},[94],[53,1454],{"className":1455,"style":366},[98],[53,1457,322],{"className":1458,"style":371},[103,370],[53,1460,327],{"className":1461},[107],[53,1463,1465],{"className":1464},[103,378],[53,1466,1414],{"className":1467},[103],[53,1469],{"className":1470,"style":413},[391],[53,1472,1417],{"className":1473},[417],[53,1475],{"className":1476,"style":413},[391],[53,1478,1480,1483,1489,1492,1495,1498],{"className":1479},[94],[53,1481],{"className":1482,"style":366},[98],[53,1484,1486],{"className":1485},[103,378],[53,1487,1420],{"className":1488},[103],[53,1490,337],{"className":1491},[190],[53,1493],{"className":1494,"style":392},[391],[53,1496,340],{"className":1497},[396],[53,1499],{"className":1500,"style":392},[391],[53,1502,1504,1507,1510,1513,1516,1519,1522,1525],{"className":1503},[94],[53,1505],{"className":1506,"style":518},[98],[53,1508,265],{"className":1509},[103],[53,1511,472],{"className":1512},[525],[53,1514],{"className":1515,"style":529},[391],[53,1517,475],{"className":1518},[103],[53,1520],{"className":1521,"style":413},[391],[53,1523,1433],{"className":1524},[417],[53,1526],{"className":1527,"style":413},[391],[53,1529,1531,1534,1537,1540,1543,1546,1549,1552],{"className":1530},[94],[53,1532],{"className":1533,"style":518},[98],[53,1535,265],{"className":1536},[103],[53,1538,472],{"className":1539},[525],[53,1541],{"className":1542,"style":529},[391],[53,1544,475],{"className":1545},[103],[53,1547],{"className":1548,"style":392},[391],[53,1550,340],{"className":1551},[396],[53,1553],{"className":1554,"style":392},[391],[53,1556,1558,1562],{"className":1557},[94],[53,1559],{"className":1560,"style":1561},[98],"height:0.6444em;",[53,1563,77],{"className":1564},[103],". Nada sorprendente — eso es cualquier resultado no-canto.",[222,1567,1568,1914],{},[53,1569,1571,1626],{"className":1570},[56],[53,1572,1574],{"className":1573},[60],[62,1575,1576],{"xmlns":64},[66,1577,1578,1623],{},[69,1579,1580,1582,1584,1586,1588,1590,1592,1594,1603,1605,1613,1615],{},[320,1581,322],{},[324,1583,327],{"stretchy":326},[75,1585,77],{},[324,1587,1417],{},[75,1589,80],{},[324,1591,337],{"stretchy":326},[324,1593,340],{},[705,1595,1596],{"scriptlevel":265,"displaystyle":326},[72,1597,1598,1600],{},[75,1599,77],{},[75,1601,1602],{},"6",[324,1604,1433],{},[705,1606,1607],{"scriptlevel":265,"displaystyle":326},[72,1608,1609,1611],{},[75,1610,77],{},[75,1612,1602],{},[324,1614,340],{},[705,1616,1617],{"scriptlevel":265,"displaystyle":326},[72,1618,1619,1621],{},[75,1620,80],{},[75,1622,1602],{},[82,1624,1625],{"encoding":84},"P(1 \\cup 2) = \\tfrac{1}{6} + \\tfrac{1}{6} = \\tfrac{2}{6}",[53,1627,1629,1653,1674,1757,1840],{"className":1628,"ariaHidden":90},[89],[53,1630,1632,1635,1638,1641,1644,1647,1650],{"className":1631},[94],[53,1633],{"className":1634,"style":366},[98],[53,1636,322],{"className":1637,"style":371},[103,370],[53,1639,327],{"className":1640},[107],[53,1642,77],{"className":1643},[103],[53,1645],{"className":1646,"style":413},[391],[53,1648,1417],{"className":1649},[417],[53,1651],{"className":1652,"style":413},[391],[53,1654,1656,1659,1662,1665,1668,1671],{"className":1655},[94],[53,1657],{"className":1658,"style":366},[98],[53,1660,80],{"className":1661},[103],[53,1663,337],{"className":1664},[190],[53,1666],{"className":1667,"style":392},[391],[53,1669,340],{"className":1670},[396],[53,1672],{"className":1673,"style":392},[391],[53,1675,1677,1680,1748,1751,1754],{"className":1676},[94],[53,1678],{"className":1679,"style":99},[98],[53,1681,1683,1686,1745],{"className":1682},[103],[53,1684],{"className":1685},[107,108],[53,1687,1689],{"className":1688},[72],[53,1690,1692,1737],{"className":1691},[115,116],[53,1693,1695,1734],{"className":1694},[120],[53,1696,1698,1712,1720],{"className":1697,"style":125},[124],[53,1699,1700,1703],{"style":128},[53,1701],{"className":1702,"style":133},[132],[53,1704,1706],{"className":1705},[137,138,139,140],[53,1707,1709],{"className":1708},[103,140],[53,1710,1602],{"className":1711},[103,140],[53,1713,1714,1717],{"style":149},[53,1715],{"className":1716,"style":133},[132],[53,1718],{"className":1719,"style":157},[156],[53,1721,1722,1725],{"style":160},[53,1723],{"className":1724,"style":133},[132],[53,1726,1728],{"className":1727},[137,138,139,140],[53,1729,1731],{"className":1730},[103,140],[53,1732,77],{"className":1733},[103,140],[53,1735,177],{"className":1736},[176],[53,1738,1740],{"className":1739},[120],[53,1741,1743],{"className":1742,"style":184},[124],[53,1744],{},[53,1746],{"className":1747},[190,108],[53,1749],{"className":1750,"style":413},[391],[53,1752,1433],{"className":1753},[417],[53,1755],{"className":1756,"style":413},[391],[53,1758,1760,1763,1831,1834,1837],{"className":1759},[94],[53,1761],{"className":1762,"style":99},[98],[53,1764,1766,1769,1828],{"className":1765},[103],[53,1767],{"className":1768},[107,108],[53,1770,1772],{"className":1771},[72],[53,1773,1775,1820],{"className":1774},[115,116],[53,1776,1778,1817],{"className":1777},[120],[53,1779,1781,1795,1803],{"className":1780,"style":125},[124],[53,1782,1783,1786],{"style":128},[53,1784],{"className":1785,"style":133},[132],[53,1787,1789],{"className":1788},[137,138,139,140],[53,1790,1792],{"className":1791},[103,140],[53,1793,1602],{"className":1794},[103,140],[53,1796,1797,1800],{"style":149},[53,1798],{"className":1799,"style":133},[132],[53,1801],{"className":1802,"style":157},[156],[53,1804,1805,1808],{"style":160},[53,1806],{"className":1807,"style":133},[132],[53,1809,1811],{"className":1810},[137,138,139,140],[53,1812,1814],{"className":1813},[103,140],[53,1815,77],{"className":1816},[103,140],[53,1818,177],{"className":1819},[176],[53,1821,1823],{"className":1822},[120],[53,1824,1826],{"className":1825,"style":184},[124],[53,1827],{},[53,1829],{"className":1830},[190,108],[53,1832],{"className":1833,"style":392},[391],[53,1835,340],{"className":1836},[396],[53,1838],{"className":1839,"style":392},[391],[53,1841,1843,1846],{"className":1842},[94],[53,1844],{"className":1845,"style":99},[98],[53,1847,1849,1852,1911],{"className":1848},[103],[53,1850],{"className":1851},[107,108],[53,1853,1855],{"className":1854},[72],[53,1856,1858,1903],{"className":1857},[115,116],[53,1859,1861,1900],{"className":1860},[120],[53,1862,1864,1878,1886],{"className":1863,"style":125},[124],[53,1865,1866,1869],{"style":128},[53,1867],{"className":1868,"style":133},[132],[53,1870,1872],{"className":1871},[137,138,139,140],[53,1873,1875],{"className":1874},[103,140],[53,1876,1602],{"className":1877},[103,140],[53,1879,1880,1883],{"style":149},[53,1881],{"className":1882,"style":133},[132],[53,1884],{"className":1885,"style":157},[156],[53,1887,1888,1891],{"style":160},[53,1889],{"className":1890,"style":133},[132],[53,1892,1894],{"className":1893},[137,138,139,140],[53,1895,1897],{"className":1896},[103,140],[53,1898,80],{"className":1899},[103,140],[53,1901,177],{"className":1902},[176],[53,1904,1906],{"className":1905},[120],[53,1907,1909],{"className":1908,"style":184},[124],[53,1910],{},[53,1912],{"className":1913},[190,108]," (sacar un 1 o un 2 en un dado de seis caras).",[29,1916,1917,1918,1921],{},"Si los sucesos ",[230,1919,1920],{},"pueden"," solaparse, la simple suma cuenta doble el solape y tienes que restarlo:",[29,1923,1925],{"className":1924},[306],[53,1926,1928,1988],{"className":1927},[56],[53,1929,1931],{"className":1930},[60],[62,1932,1933],{"xmlns":64},[66,1934,1935,1985],{},[69,1936,1937,1939,1941,1943,1945,1948,1950,1952,1954,1956,1958,1960,1962,1964,1966,1968,1970,1972,1974,1976,1978,1981,1983],{},[320,1938,322],{},[324,1940,327],{"stretchy":326},[320,1942,334],{},[324,1944,1417],{},[320,1946,1947],{},"B",[324,1949,337],{"stretchy":326},[324,1951,340],{},[320,1953,322],{},[324,1955,327],{"stretchy":326},[320,1957,334],{},[324,1959,337],{"stretchy":326},[324,1961,1433],{},[320,1963,322],{},[324,1965,327],{"stretchy":326},[320,1967,1947],{},[324,1969,337],{"stretchy":326},[324,1971,345],{},[320,1973,322],{},[324,1975,327],{"stretchy":326},[320,1977,334],{},[324,1979,1980],{},"∩",[320,1982,1947],{},[324,1984,337],{"stretchy":326},[82,1986,1987],{"encoding":84},"P(A \\cup B) = P(A) + P(B) - P(A \\cap B)",[53,1989,1991,2015,2037,2064,2091,2115],{"className":1990,"ariaHidden":90},[89],[53,1992,1994,1997,2000,2003,2006,2009,2012],{"className":1993},[94],[53,1995],{"className":1996,"style":366},[98],[53,1998,322],{"className":1999,"style":371},[103,370],[53,2001,327],{"className":2002},[107],[53,2004,334],{"className":2005},[103,370],[53,2007],{"className":2008,"style":413},[391],[53,2010,1417],{"className":2011},[417],[53,2013],{"className":2014,"style":413},[391],[53,2016,2018,2021,2025,2028,2031,2034],{"className":2017},[94],[53,2019],{"className":2020,"style":366},[98],[53,2022,1947],{"className":2023,"style":2024},[103,370],"margin-right:0.0502em;",[53,2026,337],{"className":2027},[190],[53,2029],{"className":2030,"style":392},[391],[53,2032,340],{"className":2033},[396],[53,2035],{"className":2036,"style":392},[391],[53,2038,2040,2043,2046,2049,2052,2055,2058,2061],{"className":2039},[94],[53,2041],{"className":2042,"style":366},[98],[53,2044,322],{"className":2045,"style":371},[103,370],[53,2047,327],{"className":2048},[107],[53,2050,334],{"className":2051},[103,370],[53,2053,337],{"className":2054},[190],[53,2056],{"className":2057,"style":413},[391],[53,2059,1433],{"className":2060},[417],[53,2062],{"className":2063,"style":413},[391],[53,2065,2067,2070,2073,2076,2079,2082,2085,2088],{"className":2066},[94],[53,2068],{"className":2069,"style":366},[98],[53,2071,322],{"className":2072,"style":371},[103,370],[53,2074,327],{"className":2075},[107],[53,2077,1947],{"className":2078,"style":2024},[103,370],[53,2080,337],{"className":2081},[190],[53,2083],{"className":2084,"style":413},[391],[53,2086,345],{"className":2087},[417],[53,2089],{"className":2090,"style":413},[391],[53,2092,2094,2097,2100,2103,2106,2109,2112],{"className":2093},[94],[53,2095],{"className":2096,"style":366},[98],[53,2098,322],{"className":2099,"style":371},[103,370],[53,2101,327],{"className":2102},[107],[53,2104,334],{"className":2105},[103,370],[53,2107],{"className":2108,"style":413},[391],[53,2110,1980],{"className":2111},[417],[53,2113],{"className":2114,"style":413},[391],[53,2116,2118,2121,2124],{"className":2117},[94],[53,2119],{"className":2120,"style":366},[98],[53,2122,1947],{"className":2123,"style":2024},[103,370],[53,2125,337],{"className":2126},[190],[29,2128,2129,2130,2133],{},"Eso es el ",[33,2131,2132],{},"principio de inclusión–exclusión",", y su versión más profunda es tema para un futuro artículo.",[45,2135,2137],{"id":2136},"una-nota-rápida-sobre-independencia","Una nota rápida sobre independencia",[29,2139,2140,2141,2143],{},"Dos sucesos son ",[33,2142,912],{}," 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.",[29,2145,2146,2147,2150,2151,2156],{},"Este es el montaje detrás de la famosa ",[33,2148,2149],{},"Falacia del jugador",", que desgrano en ",[204,2152,2155],{"href":2153,"rel":2154},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fla-estadistica-inesperada-de-los-lanzamientos-de-moneda#the-gamblers-fallacy",[208],"el artículo de estadísticas",". Por ahora, la versión corta: cada lanzamiento es un nuevo comienzo.",[45,2158,2160],{"id":2159},"conclusión","Conclusión",[29,2162,2163],{},"Eso es realmente todo para lo básico:",[2165,2166,2167,2170,2179,2189],"ol",{},[222,2168,2169],{},"Las probabilidades viven entre 0 y 1.",[222,2171,2172,2173,2175,2176,2178],{},"Usa el ",[33,2174,297],{}," (1 − P) cuando sea más fácil contar lo que ",[230,2177,301],{}," ocurre.",[222,2180,2181,2184,2185,2188],{},[33,2182,2183],{},"Multiplica"," para el \"y\", ",[33,2186,2187],{},"suma"," para el \"o\" mutuamente excluyente.",[222,2190,2191],{},"Los sucesos independientes no se preocupan por la historia del otro.",[29,2193,2194],{},"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.",[29,2196,2197,2198,2203],{},"Pon la teoría en práctica: ",[204,2199,2202],{"href":2200,"rel":2201},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fplay",[208],"lanza una moneda online"," y sigue tu propia racha.",{"title":2205,"searchDepth":2206,"depth":2206,"links":2207},"",2,[2208,2209,2210,2211,2212,2217,2218],{"id":47,"depth":2206,"text":48},{"id":213,"depth":2206,"text":214},{"id":250,"depth":2206,"text":251},{"id":290,"depth":2206,"text":291},{"id":897,"depth":2206,"text":898,"children":2213},[2214,2216],{"id":905,"depth":2215,"text":906},3,{"id":1376,"depth":2215,"text":1377},{"id":2136,"depth":2206,"text":2137},{"id":2159,"depth":2206,"text":2160},"2026-04-23T01:00:00Z","Aprende los fundamentos de la probabilidad y cómo se relacionan con los lanzamientos de moneda.",{"type":22,"value":2222},[2223,2225,2231],[25,2224,7],{"id":27},[29,2226,31,2227,36,2229,40],{},[33,2228,35],{},[33,2230,39],{},[29,2232,43],{},"md","beginner",{},true,"\u002Fblog\u002Fprobability-basics.es",{"title":7,"description":2220},1,"blog\u002Fprobability-basics.es","UDMtCPfKihC8ZPXit0j2o6pIQuJsJXiMaiR4DOfQzfw",{"id":2243,"title":2244,"_locale":8,"author":3,"body":2245,"date":2661,"description":2662,"excerpt":2663,"extension":2233,"level":2234,"meta":2669,"navigation":2236,"path":2670,"published":2236,"seo":2671,"series":15,"seriesOrder":2206,"slug":2672,"stem":2673,"tags":3,"__hash__":2674},"blog\u002Fblog\u002Fheads-or-tails.es.md","Cara o Cruz — La Guía Completa sobre el Lanzamiento de Moneda",{"type":22,"value":2246,"toc":2653},[2247,2250,2253,2257,2263,2277,2284,2288,2295,2379,2464,2536,2547,2567,2571,2578,2589,2592,2596,2602,2608,2614,2620,2624,2631,2634,2637,2639,2645],[25,2248,2244],{"id":2249},"cara-o-cruz-la-guía-completa-sobre-el-lanzamiento-de-moneda",[29,2251,2252],{},"\"¿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?",[45,2254,2256],{"id":2255},"de-dónde-vienen-cara-y-cruz","¿De dónde vienen \"cara\" y \"cruz\"?",[29,2258,2259,2262],{},[33,2260,2261],{},"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.",[29,2264,2265,2268,2269,2272,2273,2276],{},[33,2266,2267],{},"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 ",[230,2270,2271],{},"caput aut navis"," (\"cabeza o barco\"), porque el reverso mostraba la proa de una nave. En inglés moderno se dice ",[230,2274,2275],{},"heads or tails"," (\"cabeza o cola\"), refiriéndose a los animales representados en el reverso de las monedas británicas.",[29,2278,2279,2280,2283],{},"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 ",[230,2281,2282],{},"capita aut navia"," (\"cabeza o barco\") en su honor.",[45,2285,2287],{"id":2286},"las-matemáticas-del-lanzamiento-de-moneda","Las matemáticas del lanzamiento de moneda",[29,2289,2290,2291,2294],{},"En teoría de la probabilidad, el lanzamiento de una moneda justa es el ejemplo por excelencia de un ",[33,2292,2293],{},"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:",[29,2296,2298],{"className":2297},[306],[53,2299,2301,2328],{"className":2300},[56],[53,2302,2304],{"className":2303},[60],[62,2305,2306],{"xmlns":64},[66,2307,2308,2326],{},[69,2309,2310,2312,2314,2316,2318,2320,2322,2324],{},[320,2311,322],{},[324,2313,327],{"stretchy":326},[329,2315,35],{},[324,2317,337],{"stretchy":326},[324,2319,340],{},[75,2321,265],{},[324,2323,472],{"separator":90},[75,2325,475],{},[82,2327,478],{"encoding":84},[53,2329,2331,2361],{"className":2330,"ariaHidden":90},[89],[53,2332,2334,2337,2340,2343,2349,2352,2355,2358],{"className":2333},[94],[53,2335],{"className":2336,"style":366},[98],[53,2338,322],{"className":2339,"style":371},[103,370],[53,2341,327],{"className":2342},[107],[53,2344,2346],{"className":2345},[103,378],[53,2347,35],{"className":2348},[103],[53,2350,337],{"className":2351},[190],[53,2353],{"className":2354,"style":392},[391],[53,2356,340],{"className":2357},[396],[53,2359],{"className":2360,"style":392},[391],[53,2362,2364,2367,2370,2373,2376],{"className":2363},[94],[53,2365],{"className":2366,"style":518},[98],[53,2368,265],{"className":2369},[103],[53,2371,472],{"className":2372},[525],[53,2374],{"className":2375,"style":529},[391],[53,2377,475],{"className":2378},[103],[29,2380,2382],{"className":2381},[306],[53,2383,2385,2413],{"className":2384},[56],[53,2386,2388],{"className":2387},[60],[62,2389,2390],{"xmlns":64},[66,2391,2392,2410],{},[69,2393,2394,2396,2398,2400,2402,2404,2406,2408],{},[320,2395,322],{},[324,2397,327],{"stretchy":326},[329,2399,39],{},[324,2401,337],{"stretchy":326},[324,2403,340],{},[75,2405,265],{},[324,2407,472],{"separator":90},[75,2409,475],{},[82,2411,2412],{"encoding":84},"P(\\text{cruz}) = 0,5",[53,2414,2416,2446],{"className":2415,"ariaHidden":90},[89],[53,2417,2419,2422,2425,2428,2434,2437,2440,2443],{"className":2418},[94],[53,2420],{"className":2421,"style":366},[98],[53,2423,322],{"className":2424,"style":371},[103,370],[53,2426,327],{"className":2427},[107],[53,2429,2431],{"className":2430},[103,378],[53,2432,39],{"className":2433},[103],[53,2435,337],{"className":2436},[190],[53,2438],{"className":2439,"style":392},[391],[53,2441,340],{"className":2442},[396],[53,2444],{"className":2445,"style":392},[391],[53,2447,2449,2452,2455,2458,2461],{"className":2448},[94],[53,2450],{"className":2451,"style":518},[98],[53,2453,265],{"className":2454},[103],[53,2456,472],{"className":2457},[525],[53,2459],{"className":2460,"style":529},[391],[53,2462,475],{"className":2463},[103],[29,2465,2467],{"className":2466},[306],[53,2468,2470,2494],{"className":2469},[56],[53,2471,2473],{"className":2472},[60],[62,2474,2475],{"xmlns":64},[66,2476,2477,2491],{},[69,2478,2479,2481,2483,2485,2487,2489],{},[320,2480,322],{},[324,2482,327],{"stretchy":326},[329,2484,694],{},[324,2486,337],{"stretchy":326},[324,2488,340],{},[75,2490,77],{},[82,2492,2493],{"encoding":84},"P(\\text{cara o cruz}) = 1",[53,2495,2497,2527],{"className":2496,"ariaHidden":90},[89],[53,2498,2500,2503,2506,2509,2515,2518,2521,2524],{"className":2499},[94],[53,2501],{"className":2502,"style":366},[98],[53,2504,322],{"className":2505,"style":371},[103,370],[53,2507,327],{"className":2508},[107],[53,2510,2512],{"className":2511},[103,378],[53,2513,694],{"className":2514},[103],[53,2516,337],{"className":2517},[190],[53,2519],{"className":2520,"style":392},[391],[53,2522,340],{"className":2523},[396],[53,2525],{"className":2526,"style":392},[391],[53,2528,2530,2533],{"className":2529},[94],[53,2531],{"className":2532,"style":1561},[98],[53,2534,77],{"className":2535},[103],[29,2537,2538,2539,2542,2543,40],{},"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 ",[33,2540,2541],{},"Falacia del Jugador"," — ",[204,2544,2546],{"href":206,"rel":2545},[208],"explorada en profundidad aquí",[29,2548,2549,2550,2553,2554,2557,2558,2560,2561,2563,2564,2566],{},"El número de caras esperado en ",[230,2551,2552],{},"n"," lanzamientos sigue una ",[33,2555,2556],{},"distribución binomial"," con parámetros ",[230,2559,2552],{}," y ",[230,2562,29],{}," = 0,5. Para ",[230,2565,2552],{}," 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).",[45,2568,2570],{"id":2569},"es-realmente-justo-un-lanzamiento-de-moneda","¿Es realmente justo un lanzamiento de moneda?",[29,2572,2573,2574,2577],{},"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 ",[33,2575,2576],{},"51% de las veces",". El sesgo viene de:",[219,2579,2580,2583,2586],{},[222,2581,2582],{},"La moneda pasa ligeramente más tiempo rotando en su orientación inicial antes de que la gravedad tome el control",[222,2584,2585],{},"La mecánica del pulgar que consistentemente favorece una dirección",[222,2587,2588],{},"La resistencia del aire que afecta la cara que empieza hacia arriba",[29,2590,2591],{},"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.",[45,2593,2595],{"id":2594},"lanzamientos-de-moneda-famosos-en-la-historia","Lanzamientos de moneda famosos en la historia",[29,2597,2598,2601],{},[33,2599,2600],{},"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.",[29,2603,2604,2607],{},[33,2605,2606],{},"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.",[29,2609,2610,2613],{},[33,2611,2612],{},"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.",[29,2615,2616,2619],{},[33,2617,2618],{},"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.",[45,2621,2623],{"id":2622},"por-qué-confiamos-en-los-lanzamientos-de-moneda","Por qué confiamos en los lanzamientos de moneda",[29,2625,2626,2627,2630],{},"Hay algo filosóficamente interesante en el lanzamiento de moneda: se acepta como justo precisamente porque ",[230,2628,2629],{},"nadie lo controla",". La moneda es un árbitro que no tiene interés en el resultado.",[29,2632,2633],{},"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.",[29,2635,2636],{},"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.",[45,2638,2160],{"id":2159},[29,2640,2641,2644],{},[33,2642,2643],{},"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.",[29,2646,2647,2648,2652],{},"¿Listo para lanzar? ",[204,2649,2651],{"href":2200,"rel":2650},[208],"Prueba cara o cruz online"," — física real, resultado instantáneo.",{"title":2205,"searchDepth":2206,"depth":2206,"links":2654},[2655,2656,2657,2658,2659,2660],{"id":2255,"depth":2206,"text":2256},{"id":2286,"depth":2206,"text":2287},{"id":2569,"depth":2206,"text":2570},{"id":2594,"depth":2206,"text":2595},{"id":2622,"depth":2206,"text":2623},{"id":2159,"depth":2206,"text":2160},"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.",{"type":22,"value":2664},[2665,2667],[25,2666,2244],{"id":2249},[29,2668,2252],{},{},"\u002Fblog\u002Fheads-or-tails.es",{"title":2244,"description":2662},"cara-o-cruz","blog\u002Fheads-or-tails.es","PV9X9hHBkFhGG9qdj47oYZTMSeQehLZovxsGbsI1HLk",{"id":2676,"title":2677,"_locale":8,"author":3,"body":2678,"date":4190,"description":4191,"excerpt":4192,"extension":2233,"level":2234,"meta":4202,"navigation":2236,"path":4203,"published":2236,"seo":4204,"series":15,"seriesOrder":2215,"slug":4205,"stem":4206,"tags":4207,"__hash__":4213},"blog\u002Fblog\u002Fwhat-is-a-probability-distribution.es.md","¿Qué es una distribución de probabilidad?",{"type":22,"value":2679,"toc":4184},[2680,2683,2694,2698,2708,2711,2985,2996,3214,3228,3232,3354,3357,3672,3705,3977,3994,4107,4111,4129,4140,4144,4155,4158,4165,4176],[25,2681,2677],{"id":2682},"qué-es-una-distribución-de-probabilidad",[29,2684,2685,2686,2689,2690,2693],{},"Lanza una moneda una vez y el resultado es solo un lanzamiento. Lánzala cien veces y aparece algo más interesante: una ",[230,2687,2688],{},"forma",". Cuenta con qué frecuencia ocurre cada resultado y ya no estarás mirando la suerte, sino una ",[33,2691,2692],{},"distribución de probabilidad",". Este breve artículo explica qué significa eso, presenta la distribución más famosa nacida de los lanzamientos de moneda y te deja verla construirse, bola a bola, en un tablero de Galton interactivo.",[45,2695,2697],{"id":2696},"una-distribución-es-solo-un-mapa-de-qué-tan-probable","Una distribución es solo un mapa de \"qué tan probable\"",[29,2699,2700,2701,2703,2704,2707],{},"Una ",[33,2702,2692],{}," es la respuesta completa a la pregunta ",[230,2705,2706],{},"\"¿qué puede pasar y qué tan probable es cada cosa?\"",". Empareja cada resultado posible con su probabilidad.",[29,2709,2710],{},"Para un único lanzamiento de una moneda justa, la distribución es diminuta:",[29,2712,2714],{"className":2713},[306],[53,2715,2717,2770],{"className":2716},[56],[53,2718,2720],{"className":2719},[60],[62,2721,2722],{"xmlns":64},[66,2723,2724,2767],{},[69,2725,2726,2728,2730,2732,2734,2736,2744,2746,2749,2751,2753,2755,2757,2759],{},[320,2727,322],{},[324,2729,327],{"stretchy":326},[329,2731,35],{},[324,2733,337],{"stretchy":326},[324,2735,340],{},[705,2737,2738],{"scriptlevel":265,"displaystyle":326},[72,2739,2740,2742],{},[75,2741,77],{},[75,2743,80],{},[324,2745,472],{"separator":90},[391,2747],{"width":2748},"2em",[320,2750,322],{},[324,2752,327],{"stretchy":326},[329,2754,39],{},[324,2756,337],{"stretchy":326},[324,2758,340],{},[705,2760,2761],{"scriptlevel":265,"displaystyle":326},[72,2762,2763,2765],{},[75,2764,77],{},[75,2766,80],{},[82,2768,2769],{"encoding":84},"P(\\text{cara}) = \\tfrac{1}{2}, \\qquad P(\\text{cruz}) = \\tfrac{1}{2}",[53,2771,2773,2803,2911],{"className":2772,"ariaHidden":90},[89],[53,2774,2776,2779,2782,2785,2791,2794,2797,2800],{"className":2775},[94],[53,2777],{"className":2778,"style":366},[98],[53,2780,322],{"className":2781,"style":371},[103,370],[53,2783,327],{"className":2784},[107],[53,2786,2788],{"className":2787},[103,378],[53,2789,35],{"className":2790},[103],[53,2792,337],{"className":2793},[190],[53,2795],{"className":2796,"style":392},[391],[53,2798,340],{"className":2799},[396],[53,2801],{"className":2802,"style":392},[391],[53,2804,2806,2809,2877,2880,2884,2887,2890,2893,2899,2902,2905,2908],{"className":2805},[94],[53,2807],{"className":2808,"style":99},[98],[53,2810,2812,2815,2874],{"className":2811},[103],[53,2813],{"className":2814},[107,108],[53,2816,2818],{"className":2817},[72],[53,2819,2821,2866],{"className":2820},[115,116],[53,2822,2824,2863],{"className":2823},[120],[53,2825,2827,2841,2849],{"className":2826,"style":125},[124],[53,2828,2829,2832],{"style":128},[53,2830],{"className":2831,"style":133},[132],[53,2833,2835],{"className":2834},[137,138,139,140],[53,2836,2838],{"className":2837},[103,140],[53,2839,80],{"className":2840},[103,140],[53,2842,2843,2846],{"style":149},[53,2844],{"className":2845,"style":133},[132],[53,2847],{"className":2848,"style":157},[156],[53,2850,2851,2854],{"style":160},[53,2852],{"className":2853,"style":133},[132],[53,2855,2857],{"className":2856},[137,138,139,140],[53,2858,2860],{"className":2859},[103,140],[53,2861,77],{"className":2862},[103,140],[53,2864,177],{"className":2865},[176],[53,2867,2869],{"className":2868},[120],[53,2870,2872],{"className":2871,"style":184},[124],[53,2873],{},[53,2875],{"className":2876},[190,108],[53,2878,472],{"className":2879},[525],[53,2881],{"className":2882,"style":2883},[391],"margin-right:2em;",[53,2885],{"className":2886,"style":529},[391],[53,2888,322],{"className":2889,"style":371},[103,370],[53,2891,327],{"className":2892},[107],[53,2894,2896],{"className":2895},[103,378],[53,2897,39],{"className":2898},[103],[53,2900,337],{"className":2901},[190],[53,2903],{"className":2904,"style":392},[391],[53,2906,340],{"className":2907},[396],[53,2909],{"className":2910,"style":392},[391],[53,2912,2914,2917],{"className":2913},[94],[53,2915],{"className":2916,"style":99},[98],[53,2918,2920,2923,2982],{"className":2919},[103],[53,2921],{"className":2922},[107,108],[53,2924,2926],{"className":2925},[72],[53,2927,2929,2974],{"className":2928},[115,116],[53,2930,2932,2971],{"className":2931},[120],[53,2933,2935,2949,2957],{"className":2934,"style":125},[124],[53,2936,2937,2940],{"style":128},[53,2938],{"className":2939,"style":133},[132],[53,2941,2943],{"className":2942},[137,138,139,140],[53,2944,2946],{"className":2945},[103,140],[53,2947,80],{"className":2948},[103,140],[53,2950,2951,2954],{"style":149},[53,2952],{"className":2953,"style":133},[132],[53,2955],{"className":2956,"style":157},[156],[53,2958,2959,2962],{"style":160},[53,2960],{"className":2961,"style":133},[132],[53,2963,2965],{"className":2964},[137,138,139,140],[53,2966,2968],{"className":2967},[103,140],[53,2969,77],{"className":2970},[103,140],[53,2972,177],{"className":2973},[176],[53,2975,2977],{"className":2976},[120],[53,2978,2980],{"className":2979,"style":184},[124],[53,2981],{},[53,2983],{"className":2984},[190,108],[29,2986,2987,2988,2991,2992,2995],{},"Dos resultados, cada uno con probabilidad un medio. Hay una regla que una distribución nunca puede romper: ",[33,2989,2990],{},"todas las probabilidades tienen que sumar 1",", porque ",[230,2993,2994],{},"algo"," siempre ocurre.",[29,2997,2999],{"className":2998},[306],[53,3000,3002,3036],{"className":3001},[56],[53,3003,3005],{"className":3004},[60],[62,3006,3007],{"xmlns":64},[66,3008,3009,3033],{},[69,3010,3011,3019,3021,3029,3031],{},[705,3012,3013],{"scriptlevel":265,"displaystyle":326},[72,3014,3015,3017],{},[75,3016,77],{},[75,3018,80],{},[324,3020,1433],{},[705,3022,3023],{"scriptlevel":265,"displaystyle":326},[72,3024,3025,3027],{},[75,3026,77],{},[75,3028,80],{},[324,3030,340],{},[75,3032,77],{},[82,3034,3035],{"encoding":84},"\\tfrac{1}{2} + \\tfrac{1}{2} = 1",[53,3037,3039,3122,3205],{"className":3038,"ariaHidden":90},[89],[53,3040,3042,3045,3113,3116,3119],{"className":3041},[94],[53,3043],{"className":3044,"style":99},[98],[53,3046,3048,3051,3110],{"className":3047},[103],[53,3049],{"className":3050},[107,108],[53,3052,3054],{"className":3053},[72],[53,3055,3057,3102],{"className":3056},[115,116],[53,3058,3060,3099],{"className":3059},[120],[53,3061,3063,3077,3085],{"className":3062,"style":125},[124],[53,3064,3065,3068],{"style":128},[53,3066],{"className":3067,"style":133},[132],[53,3069,3071],{"className":3070},[137,138,139,140],[53,3072,3074],{"className":3073},[103,140],[53,3075,80],{"className":3076},[103,140],[53,3078,3079,3082],{"style":149},[53,3080],{"className":3081,"style":133},[132],[53,3083],{"className":3084,"style":157},[156],[53,3086,3087,3090],{"style":160},[53,3088],{"className":3089,"style":133},[132],[53,3091,3093],{"className":3092},[137,138,139,140],[53,3094,3096],{"className":3095},[103,140],[53,3097,77],{"className":3098},[103,140],[53,3100,177],{"className":3101},[176],[53,3103,3105],{"className":3104},[120],[53,3106,3108],{"className":3107,"style":184},[124],[53,3109],{},[53,3111],{"className":3112},[190,108],[53,3114],{"className":3115,"style":413},[391],[53,3117,1433],{"className":3118},[417],[53,3120],{"className":3121,"style":413},[391],[53,3123,3125,3128,3196,3199,3202],{"className":3124},[94],[53,3126],{"className":3127,"style":99},[98],[53,3129,3131,3134,3193],{"className":3130},[103],[53,3132],{"className":3133},[107,108],[53,3135,3137],{"className":3136},[72],[53,3138,3140,3185],{"className":3139},[115,116],[53,3141,3143,3182],{"className":3142},[120],[53,3144,3146,3160,3168],{"className":3145,"style":125},[124],[53,3147,3148,3151],{"style":128},[53,3149],{"className":3150,"style":133},[132],[53,3152,3154],{"className":3153},[137,138,139,140],[53,3155,3157],{"className":3156},[103,140],[53,3158,80],{"className":3159},[103,140],[53,3161,3162,3165],{"style":149},[53,3163],{"className":3164,"style":133},[132],[53,3166],{"className":3167,"style":157},[156],[53,3169,3170,3173],{"style":160},[53,3171],{"className":3172,"style":133},[132],[53,3174,3176],{"className":3175},[137,138,139,140],[53,3177,3179],{"className":3178},[103,140],[53,3180,77],{"className":3181},[103,140],[53,3183,177],{"className":3184},[176],[53,3186,3188],{"className":3187},[120],[53,3189,3191],{"className":3190,"style":184},[124],[53,3192],{},[53,3194],{"className":3195},[190,108],[53,3197],{"className":3198,"style":392},[391],[53,3200,340],{"className":3201},[396],[53,3203],{"className":3204,"style":392},[391],[53,3206,3208,3211],{"className":3207},[94],[53,3209],{"className":3210,"style":1561},[98],[53,3212,77],{"className":3213},[103],[29,3215,3216,3217,2560,3220,3223,3224,883],{},"Esa es toda la idea. Las distribuciones se vuelven interesantes no cuando ocurre un solo suceso, sino cuando repetimos un experimento muchas veces y nos preguntamos cómo se reparten los resultados. (Si las palabras ",[230,3218,3219],{},"probabilidad",[230,3221,3222],{},"verosimilitud"," todavía te resultan resbaladizas, empieza por ",[204,3225,7],{"href":3226,"rel":3227},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ffundamentos-de-probabilidad\u002F",[208],[45,3229,3231],{"id":3230},"la-distribución-binomial","La distribución binomial",[29,3233,3234,3235,3264,3265,2560,3293,3321,3322,3324,3325,3353],{},"Ahora lanza una moneda justa ",[53,3236,3238,3251],{"className":3237},[56],[53,3239,3241],{"className":3240},[60],[62,3242,3243],{"xmlns":64},[66,3244,3245,3249],{},[69,3246,3247],{},[320,3248,2552],{},[82,3250,2552],{"encoding":84},[53,3252,3254],{"className":3253,"ariaHidden":90},[89],[53,3255,3257,3261],{"className":3256},[94],[53,3258],{"className":3259,"style":3260},[98],"height:0.4306em;",[53,3262,2552],{"className":3263},[103,370]," veces y cuenta las caras. El número de caras puede ser cualquier valor entre ",[53,3266,3268,3281],{"className":3267},[56],[53,3269,3271],{"className":3270},[60],[62,3272,3273],{"xmlns":64},[66,3274,3275,3279],{},[69,3276,3277],{},[75,3278,265],{},[82,3280,265],{"encoding":84},[53,3282,3284],{"className":3283,"ariaHidden":90},[89],[53,3285,3287,3290],{"className":3286},[94],[53,3288],{"className":3289,"style":1561},[98],[53,3291,265],{"className":3292},[103],[53,3294,3296,3309],{"className":3295},[56],[53,3297,3299],{"className":3298},[60],[62,3300,3301],{"xmlns":64},[66,3302,3303,3307],{},[69,3304,3305],{},[320,3306,2552],{},[82,3308,2552],{"encoding":84},[53,3310,3312],{"className":3311,"ariaHidden":90},[89],[53,3313,3315,3318],{"className":3314},[94],[53,3316],{"className":3317,"style":3260},[98],[53,3319,2552],{"className":3320},[103,370],", y las probabilidades de cada conteo forman la ",[33,3323,2556],{},": la distribución de \"cuántos éxitos hay en ",[53,3326,3328,3341],{"className":3327},[56],[53,3329,3331],{"className":3330},[60],[62,3332,3333],{"xmlns":64},[66,3334,3335,3339],{},[69,3336,3337],{},[320,3338,2552],{},[82,3340,2552],{"encoding":84},[53,3342,3344],{"className":3343,"ariaHidden":90},[89],[53,3345,3347,3350],{"className":3346},[94],[53,3348],{"className":3349,"style":3260},[98],[53,3351,2552],{"className":3352},[103,370]," pruebas independientes de sí\u002Fno\".",[29,3355,3356],{},"Para una moneda justa, su fórmula es:",[29,3358,3360],{"className":3359},[306],[53,3361,3363,3426],{"className":3362},[56],[53,3364,3366],{"className":3365},[60],[62,3367,3368],{"xmlns":64},[66,3369,3370,3423],{},[69,3371,3372,3374,3376,3379,3382,3384,3387,3389,3391,3404],{},[320,3373,322],{},[324,3375,327],{"stretchy":326},[320,3377,3378],{},"k",[329,3380,3381],{}," caras en ",[320,3383,2552],{},[329,3385,3386],{}," lanzamientos",[324,3388,337],{"stretchy":326},[324,3390,340],{},[69,3392,3393,3395,3402],{},[324,3394,327],{"fence":90},[72,3396,3398,3400],{"linethickness":3397},"0px",[320,3399,2552],{},[320,3401,3378],{},[324,3403,337],{"fence":90},[3405,3406,3407,3421],"msup",{},[69,3408,3409,3411,3419],{},[324,3410,327],{"fence":90},[705,3412,3413],{"scriptlevel":265,"displaystyle":326},[72,3414,3415,3417],{},[75,3416,77],{},[75,3418,80],{},[324,3420,337],{"fence":90},[320,3422,2552],{},[82,3424,3425],{"encoding":84},"P(k \\text{ caras en } n \\text{ lanzamientos}) = \\binom{n}{k}\\left(\\tfrac{1}{2}\\right)^{n}",[53,3427,3429,3472],{"className":3428,"ariaHidden":90},[89],[53,3430,3432,3435,3438,3441,3445,3451,3454,3460,3463,3466,3469],{"className":3431},[94],[53,3433],{"className":3434,"style":366},[98],[53,3436,322],{"className":3437,"style":371},[103,370],[53,3439,327],{"className":3440},[107],[53,3442,3378],{"className":3443,"style":3444},[103,370],"margin-right:0.0315em;",[53,3446,3448],{"className":3447},[103,378],[53,3449,3381],{"className":3450},[103],[53,3452,2552],{"className":3453},[103,370],[53,3455,3457],{"className":3456},[103,378],[53,3458,3386],{"className":3459},[103],[53,3461,337],{"className":3462},[190],[53,3464],{"className":3465,"style":392},[391],[53,3467,340],{"className":3468},[396],[53,3470],{"className":3471,"style":392},[391],[53,3473,3475,3479,3553,3556],{"className":3474},[94],[53,3476],{"className":3477,"style":3478},[98],"height:1.2543em;vertical-align:-0.35em;",[53,3480,3482,3492,3547],{"className":3481},[103],[53,3483,3487],{"className":3484,"style":3486},[107,3485],"delimcenter","top:0em;",[53,3488,327],{"className":3489},[3490,3491],"delimsizing","size1",[53,3493,3495],{"className":3494},[72],[53,3496,3498,3539],{"className":3497},[115,116],[53,3499,3501,3536],{"className":3500},[120],[53,3502,3505,3521],{"className":3503,"style":3504},[124],"height:0.7454em;",[53,3506,3508,3512],{"style":3507},"top:-2.355em;",[53,3509],{"className":3510,"style":3511},[132],"height:2.7em;",[53,3513,3515],{"className":3514},[137,138,139,140],[53,3516,3518],{"className":3517},[103,140],[53,3519,3378],{"className":3520,"style":3444},[103,370,140],[53,3522,3524,3527],{"style":3523},"top:-3.144em;",[53,3525],{"className":3526,"style":3511},[132],[53,3528,3530],{"className":3529},[137,138,139,140],[53,3531,3533],{"className":3532},[103,140],[53,3534,2552],{"className":3535},[103,370,140],[53,3537,177],{"className":3538},[176],[53,3540,3542],{"className":3541},[120],[53,3543,3545],{"className":3544,"style":184},[124],[53,3546],{},[53,3548,3550],{"className":3549,"style":3486},[190,3485],[53,3551,337],{"className":3552},[3490,3491],[53,3554],{"className":3555,"style":529},[391],[53,3557,3560,3643],{"className":3558},[3559],"minner",[53,3561,3563,3569,3637],{"className":3562},[3559],[53,3564,3566],{"className":3565,"style":3486},[107,3485],[53,3567,327],{"className":3568},[3490,3491],[53,3570,3572,3575,3634],{"className":3571},[103],[53,3573],{"className":3574},[107,108],[53,3576,3578],{"className":3577},[72],[53,3579,3581,3626],{"className":3580},[115,116],[53,3582,3584,3623],{"className":3583},[120],[53,3585,3587,3601,3609],{"className":3586,"style":125},[124],[53,3588,3589,3592],{"style":128},[53,3590],{"className":3591,"style":133},[132],[53,3593,3595],{"className":3594},[137,138,139,140],[53,3596,3598],{"className":3597},[103,140],[53,3599,80],{"className":3600},[103,140],[53,3602,3603,3606],{"style":149},[53,3604],{"className":3605,"style":133},[132],[53,3607],{"className":3608,"style":157},[156],[53,3610,3611,3614],{"style":160},[53,3612],{"className":3613,"style":133},[132],[53,3615,3617],{"className":3616},[137,138,139,140],[53,3618,3620],{"className":3619},[103,140],[53,3621,77],{"className":3622},[103,140],[53,3624,177],{"className":3625},[176],[53,3627,3629],{"className":3628},[120],[53,3630,3632],{"className":3631,"style":184},[124],[53,3633],{},[53,3635],{"className":3636},[190,108],[53,3638,3640],{"className":3639,"style":3486},[190,3485],[53,3641,337],{"className":3642},[3490,3491],[53,3644,3647],{"className":3645},[3646],"msupsub",[53,3648,3650],{"className":3649},[115],[53,3651,3653],{"className":3652},[120],[53,3654,3657],{"className":3655,"style":3656},[124],"height:0.9043em;",[53,3658,3660,3663],{"style":3659},"top:-3.3029em;margin-right:0.05em;",[53,3661],{"className":3662,"style":3511},[132],[53,3664,3666],{"className":3665},[137,138,139,140],[53,3667,3669],{"className":3668},[103,140],[53,3670,2552],{"className":3671},[103,370,140],[29,3673,3674,3675,3704],{},"y, de forma más general, cuando \"cara\" tiene probabilidad ",[53,3676,3678,3691],{"className":3677},[56],[53,3679,3681],{"className":3680},[60],[62,3682,3683],{"xmlns":64},[66,3684,3685,3689],{},[69,3686,3687],{},[320,3688,29],{},[82,3690,29],{"encoding":84},[53,3692,3694],{"className":3693,"ariaHidden":90},[89],[53,3695,3697,3701],{"className":3696},[94],[53,3698],{"className":3699,"style":3700},[98],"height:0.625em;vertical-align:-0.1944em;",[53,3702,29],{"className":3703},[103,370]," (no necesariamente un medio):",[29,3706,3708],{"className":3707},[306],[53,3709,3711,3774],{"className":3710},[56],[53,3712,3714],{"className":3713},[60],[62,3715,3716],{"xmlns":64},[66,3717,3718,3771],{},[69,3719,3720,3722,3724,3726,3728,3730,3742,3745,3751,3753,3755,3757,3759],{},[320,3721,322],{},[324,3723,327],{"stretchy":326},[320,3725,3378],{},[324,3727,337],{"stretchy":326},[324,3729,340],{},[69,3731,3732,3734,3740],{},[324,3733,327],{"fence":90},[72,3735,3736,3738],{"linethickness":3397},[320,3737,2552],{},[320,3739,3378],{},[324,3741,337],{"fence":90},[329,3743,3744],{}," ",[3405,3746,3747,3749],{},[320,3748,29],{},[320,3750,3378],{},[324,3752,327],{"stretchy":326},[75,3754,77],{},[324,3756,345],{},[320,3758,29],{},[3405,3760,3761,3763],{},[324,3762,337],{"stretchy":326},[69,3764,3765,3767,3769],{},[320,3766,2552],{},[324,3768,345],{},[320,3770,3378],{},[82,3772,3773],{"encoding":84},"P(k) = \\binom{n}{k}\\, p^{k} (1-p)^{n-k}",[53,3775,3777,3804,3929],{"className":3776,"ariaHidden":90},[89],[53,3778,3780,3783,3786,3789,3792,3795,3798,3801],{"className":3779},[94],[53,3781],{"className":3782,"style":366},[98],[53,3784,322],{"className":3785,"style":371},[103,370],[53,3787,327],{"className":3788},[107],[53,3790,3378],{"className":3791,"style":3444},[103,370],[53,3793,337],{"className":3794},[190],[53,3796],{"className":3797,"style":392},[391],[53,3799,340],{"className":3800},[396],[53,3802],{"className":3803,"style":392},[391],[53,3805,3807,3811,3877,3880,3914,3917,3920,3923,3926],{"className":3806},[94],[53,3808],{"className":3809,"style":3810},[98],"height:1.2em;vertical-align:-0.35em;",[53,3812,3814,3820,3871],{"className":3813},[103],[53,3815,3817],{"className":3816,"style":3486},[107,3485],[53,3818,327],{"className":3819},[3490,3491],[53,3821,3823],{"className":3822},[72],[53,3824,3826,3863],{"className":3825},[115,116],[53,3827,3829,3860],{"className":3828},[120],[53,3830,3832,3846],{"className":3831,"style":3504},[124],[53,3833,3834,3837],{"style":3507},[53,3835],{"className":3836,"style":3511},[132],[53,3838,3840],{"className":3839},[137,138,139,140],[53,3841,3843],{"className":3842},[103,140],[53,3844,3378],{"className":3845,"style":3444},[103,370,140],[53,3847,3848,3851],{"style":3523},[53,3849],{"className":3850,"style":3511},[132],[53,3852,3854],{"className":3853},[137,138,139,140],[53,3855,3857],{"className":3856},[103,140],[53,3858,2552],{"className":3859},[103,370,140],[53,3861,177],{"className":3862},[176],[53,3864,3866],{"className":3865},[120],[53,3867,3869],{"className":3868,"style":184},[124],[53,3870],{},[53,3872,3874],{"className":3873,"style":3486},[190,3485],[53,3875,337],{"className":3876},[3490,3491],[53,3878],{"className":3879,"style":529},[391],[53,3881,3883,3886],{"className":3882},[103],[53,3884,29],{"className":3885},[103,370],[53,3887,3889],{"className":3888},[3646],[53,3890,3892],{"className":3891},[115],[53,3893,3895],{"className":3894},[120],[53,3896,3899],{"className":3897,"style":3898},[124],"height:0.8491em;",[53,3900,3902,3905],{"style":3901},"top:-3.063em;margin-right:0.05em;",[53,3903],{"className":3904,"style":3511},[132],[53,3906,3908],{"className":3907},[137,138,139,140],[53,3909,3911],{"className":3910},[103,140],[53,3912,3378],{"className":3913,"style":3444},[103,370,140],[53,3915,327],{"className":3916},[107],[53,3918,77],{"className":3919},[103],[53,3921],{"className":3922,"style":413},[391],[53,3924,345],{"className":3925},[417],[53,3927],{"className":3928,"style":413},[391],[53,3930,3932,3936,3939],{"className":3931},[94],[53,3933],{"className":3934,"style":3935},[98],"height:1.0991em;vertical-align:-0.25em;",[53,3937,29],{"className":3938},[103,370],[53,3940,3942,3945],{"className":3941},[190],[53,3943,337],{"className":3944},[190],[53,3946,3948],{"className":3947},[3646],[53,3949,3951],{"className":3950},[115],[53,3952,3954],{"className":3953},[120],[53,3955,3957],{"className":3956,"style":3898},[124],[53,3958,3959,3962],{"style":3901},[53,3960],{"className":3961,"style":3511},[132],[53,3963,3965],{"className":3964},[137,138,139,140],[53,3966,3968,3971,3974],{"className":3967},[103,140],[53,3969,2552],{"className":3970},[103,370,140],[53,3972,345],{"className":3973},[417,140],[53,3975,3378],{"className":3976,"style":3444},[103,370,140],[29,3978,3979,3980,3982,3983,3986,3987,3990,3991,40],{},"Lo interesante es la ",[33,3981,2688],{},". Los resultados extremos —todo caras o todo cruces— son raros, porque solo hay ",[230,3984,3985],{},"una"," manera de obtener cada uno. Los resultados intermedios son frecuentes, porque hay ",[230,3988,3989],{},"muchas"," ordenaciones distintas que caen ahí. Si dibujas las probabilidades obtienes una joroba alta en el centro y fina en los bordes: el comienzo de la famosa ",[33,3992,3993],{},"campana de Gauss",[29,3995,3996,3997,4097,4098,4103,4104],{},"Por qué aparecen exactamente esos conteos ",[53,3998,4000,4022],{"className":3999},[56],[53,4001,4003],{"className":4002},[60],[62,4004,4005],{"xmlns":64},[66,4006,4007,4019],{},[69,4008,4009,4011,4017],{},[324,4010,327],{"fence":90},[72,4012,4013,4015],{"linethickness":3397},[320,4014,2552],{},[320,4016,3378],{},[324,4018,337],{"fence":90},[82,4020,4021],{"encoding":84},"\\binom{n}{k}",[53,4023,4025],{"className":4024,"ariaHidden":90},[89],[53,4026,4028,4031],{"className":4027},[94],[53,4029],{"className":4030,"style":3810},[98],[53,4032,4034,4040,4091],{"className":4033},[103],[53,4035,4037],{"className":4036,"style":3486},[107,3485],[53,4038,327],{"className":4039},[3490,3491],[53,4041,4043],{"className":4042},[72],[53,4044,4046,4083],{"className":4045},[115,116],[53,4047,4049,4080],{"className":4048},[120],[53,4050,4052,4066],{"className":4051,"style":3504},[124],[53,4053,4054,4057],{"style":3507},[53,4055],{"className":4056,"style":3511},[132],[53,4058,4060],{"className":4059},[137,138,139,140],[53,4061,4063],{"className":4062},[103,140],[53,4064,3378],{"className":4065,"style":3444},[103,370,140],[53,4067,4068,4071],{"style":3523},[53,4069],{"className":4070,"style":3511},[132],[53,4072,4074],{"className":4073},[137,138,139,140],[53,4075,4077],{"className":4076},[103,140],[53,4078,2552],{"className":4079},[103,370,140],[53,4081,177],{"className":4082},[176],[53,4084,4086],{"className":4085},[120],[53,4087,4089],{"className":4088,"style":184},[124],[53,4090],{},[53,4092,4094],{"className":4093,"style":3486},[190,3485],[53,4095,337],{"className":4096},[3490,3491]," —y por qué son las filas del triángulo de Tartaglia— es una historia en sí misma, contada en ",[204,4099,4102],{"href":4100,"rel":4101},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ftriangulo-de-tartaglia-combinatoria-y-probabilidad\u002F",[208],"¿Qué tienen en común el triángulo de Tartaglia y una moneda?",". Aquí solo necesitamos la conclusión: ",[33,4105,4106],{},"suma suficientes elecciones justas e independientes y los resultados se acumulan formando una campana.",[45,4108,4110],{"id":4109},"quién-fue-francis-galton","¿Quién fue Francis Galton?",[29,4112,4113,4114,4117,4118,4121,4122,4125,4126,40],{},"La forma más encantadora de ",[230,4115,4116],{},"ver"," una distribución binomial la ideó ",[33,4119,4120],{},"Sir Francis Galton"," (1822–1911), un inquieto erudito victoriano y pariente de Charles Darwin (compartían abuelo, Erasmus Darwin). De joven exploró el suroeste de África y luego dedicó el resto de su vida a medir y contar casi todo lo que podía: el tiempo atmosférico (dibujó el primer mapa meteorológico de un periódico), las huellas dactilares e incluso la supuesta \"eficacia\" de la oración. Por el camino fue pionero de ideas estadísticas que seguimos usando a diario, como la ",[33,4123,4124],{},"correlación"," y la ",[33,4127,4128],{},"regresión a la media",[29,4130,4131,4132,4135,4136,4139],{},"Para demostrar cómo el orden surge del azar, Galton construyó un aparato que llamó ",[33,4133,4134],{},"quincunce","; hoy lo llamamos ",[33,4137,4138],{},"tablero de Galton",". Deja caer una bola en una red triangular de clavijas. En cada clavija rebota a izquierda o derecha, más o menos al azar, exactamente como un lanzamiento de moneda. Tras muchas filas de clavijas, cada bola cae en una casilla del fondo. Una sola bola es impredecible. Pero vierte cientos y las casillas se llenan formando la inconfundible campana de la distribución binomial. Galton había convertido una fórmula abstracta en algo que puedes observar con tus propios ojos.",[45,4141,4143],{"id":4142},"míralo-suceder","Míralo suceder",[29,4145,4146,4147,4150,4151,4154],{},"Pulsa ",[33,4148,4149],{},"Soltar bolas"," y observa cómo crece el montón. Cada bola es un pequeño experimento; las pilas que se elevan son la distribución dibujándose a sí misma. Pulsa ",[33,4152,4153],{},"Reiniciar"," para empezar de nuevo.",[4156,4157],"galton-board",{},[29,4159,4160,4161,4164],{},"Fíjate en que ninguna bola \"sabe\" a dónde debería ir y, sin embargo, el conjunto traza con fiabilidad la misma campana una y otra vez. Esa es la tranquila magia de las distribuciones de probabilidad: los sucesos individuales siguen siendo impredecibles, pero su comportamiento ",[230,4162,4163],{},"colectivo"," es notablemente ordenado.",[29,4166,4167,4168,4171,4172,4175],{},"Entonces, ¿cómo se conecta esto con una sola moneda? ",[33,4169,4170],{},"Cada rebote a izquierda o derecha del tablero es un lanzamiento de moneda",", y cada bola que llega al fondo es un experimento completo de varios lanzamientos, que cae en la casilla correspondiente a cuántas veces salió \"cara\". Por eso un único lanzamiento nunca puede revelar una curva: una curva es un patrón que vive en ",[230,4173,4174],{},"muchos"," resultados, no en uno. Lanza una vez y simplemente obtienes cara o cruz. Lanza una y otra vez, lleva la cuenta, y esos recuentos van trazando poco a poco la misma campana que ves arriba.",[29,4177,4178,4183],{},[204,4179,4182],{"href":4180,"rel":4181},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fplay\u002F",[208],"Lanza una moneda online"," y observa cómo crece tu propio historial de lanzamientos: la distribución solo toma forma cuando los lanzamientos se acumulan, igual que las bolas.",{"title":2205,"searchDepth":2206,"depth":2206,"links":4185},[4186,4187,4188,4189],{"id":2696,"depth":2206,"text":2697},{"id":3230,"depth":2206,"text":3231},{"id":4109,"depth":2206,"text":4110},{"id":4142,"depth":2206,"text":4143},"2026-06-12T10:00:00Z","Una introducción amable a las distribuciones de probabilidad y a la distribución binomial, con un tablero de Galton interactivo para experimentar.",{"type":22,"value":4193},[4194,4196],[25,4195,2677],{"id":2682},[29,4197,2685,4198,2689,4200,2693],{},[230,4199,2688],{},[33,4201,2692],{},{},"\u002Fblog\u002Fwhat-is-a-probability-distribution.es",{"title":2677,"description":4191},"que-es-una-distribucion-de-probabilidad","blog\u002Fwhat-is-a-probability-distribution.es",[4208,4209,4210,4211,4212],"Distribución de probabilidad","Distribución binomial","Francis Galton","Campana de Gauss","Tablero de Galton","C_n7VFrlCfp0gd03VXZ85bXYyDMjwtYaCPh3sc6Hijw",[4215],{"id":4216,"avatar":4217,"extension":11,"meta":4218,"name":4221,"slug":4222,"stem":4224,"url":4223,"__hash__":4225},"authors\u002Fauthors\u002Fpaul-melero.yml","\u002Fimg\u002Fuploads\u002Fauthors\u002Fpaul-melero.webp",{"path":4219,"body":4220,"title":4221},"\u002Fauthors\u002Fpaul-melero",{"name":4221,"slug":4222,"url":4223,"avatar":4217},"Paul Melero","paul-melero","https:\u002F\u002Fgraficos.net","authors\u002Fpaul-melero","LTEA07PCVRlrCg3-G1fVOu2kWXF5EjQ1BxEF3zgZT6k",1781475125322]