[{"data":1,"prerenderedAt":26788},["ShallowReactive",2],{"locale-switcher-blog-en-null":3,"blog-series-page-en-random-walks":4,"series-page-authors":26776},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\u002Frandom-walks.en.yml","Random Walks","en","2026-06-06T01:00:00Z","Where coin flips meet deeper math: analyzing paths with Pascal's triangle and Catalan numbers, and why a fair coin can look unfair for so long, stochastic processes and the arcsine law.","yml",{"path":13,"body":14},"\u002Fseries\u002Frandom-walks.en",{"slug":15,"title":7,"description":10,"date":9,"_locale":8},"random-walks","series\u002Frandom-walks.en","eDMXmyx7RR9ieBpI6WoRGyauo7uPbttCGSy0AOVnW7Q",[19,10667,14013],{"id":20,"title":21,"_locale":8,"author":3,"body":22,"date":10644,"description":10645,"excerpt":10646,"extension":10652,"level":10653,"meta":10654,"navigation":10655,"path":10656,"published":10655,"seo":10657,"series":15,"seriesOrder":10658,"slug":10659,"stem":10660,"tags":10661,"__hash__":10666},"blog\u002Fblog\u002Fpascals-triangle-and-probability.en.md","What Do Pascal's Triangle and a Coin Have in Common?",{"type":23,"value":24,"toc":10627},"minimal",[25,29,33,38,55,58,64,68,71,95,98,109,119,394,399,406,542,665,828,831,1088,1392,1839,1871,2274,2278,2287,2497,2504,2570,2631,2772,2775,2785,2789,2875,2881,2887,2941,3017,3236,3240,3246,3380,3412,3852,3855,4523,4673,4676,4948,4952,4955,5039,5123,5255,5261,5393,5396,6188,6191,6197,6200,6220,6228,6232,6239,6334,6337,6533,6565,6570,6800,6859,6864,7186,7256,7261,7767,7864,7874,8250,8487,8491,8556,8868,8871,9839,10214,10217,10483,10578,10582,10585,10603,10609,10612,10615,10618],[26,27,21],"h1",{"id":28},"what-do-pascals-triangle-and-a-coin-have-in-common",[30,31,32],"p",{},"There is a triangle of numbers that has fascinated mathematicians across the globe for centuries. Its construction is so simple a child could draw it in a notebook. And yet, hidden within its rows lies a deep connection between three branches of mathematics that, at first glance, seem to have nothing in common: combinatorics, algebra, and probability. The most surprising part? That connection reveals itself through something as ordinary as flipping a coin.",[34,35,37],"h2",{"id":36},"a-bit-of-history","A bit of history",[30,39,40,41,45,46,50,51,54],{},"In the English-speaking world this pattern is known as ",[42,43,44],"strong",{},"Pascal's triangle",", after the French mathematician Blaise Pascal, who published a treatise in 1654 — the ",[47,48,49],"em",{},"Traité du triangle arithmétique"," — where he formalized its properties and, crucially, connected it for the first time to the problems of chance posed to him by the Chevalier de Méré. In many Spanish-speaking countries it is called the ",[42,52,53],{},"triangle of Tartaglia",", after the Italian mathematician Niccolò Fontana Tartaglia (1499–1557), who studied it in the context of solving cubic equations.",[30,56,57],{},"But the real history of this triangle goes much further back. In 11th-century China, the mathematician Jia Xian was already describing the same pattern for computing coefficients of powers. Later, in the 13th century, Yang Hui reproduced it in a diagram that survives to this day. In Persia, Omar Khayyam worked with the same ideas in the 12th century. And in India, Pingala explored similar combinatorial patterns centuries before the common era.",[30,59,60,61],{},"The fact that civilizations so distant in time and space independently discovered the same triangle tells us something important: this is not an arbitrary human invention, but a structure that emerges naturally whenever anyone asks ",[47,62,63],{},"how many ways can I arrange this?",[34,65,67],{"id":66},"how-to-build-the-triangle","How to build the triangle",[30,69,70],{},"The construction rule is elegantly simple:",[72,73,74,82,88],"ol",{},[75,76,77,78,81],"li",{},"Start by placing a ",[42,79,80],{},"1"," at the top.",[75,83,84,85,87],{},"On each new row, place a ",[42,86,80],{}," at each end.",[75,89,90,91,94],{},"Each inner number is the ",[42,92,93],{},"sum of the two numbers"," directly above it.",[30,96,97],{},"The first few rows look like this:",[99,100,106],"pre",{"className":101,"code":103,"language":104,"meta":105},[102],"language-text","            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","text","",[107,108,103],"code",{"__ignoreMap":105},[30,110,111,112,114,115,118],{},"The first row (the lone ",[42,113,80],{}," at the top) is called ",[42,116,117],{},"row 0",". The next one is row 1, and so on. This is not a quirk: numbering from zero will make everything fit neatly once we connect the triangle to the formulas.",[30,120,121,122,226,227,256,257,226,313,341,342,393],{},"Look at row 4: ",[123,124,127,170],"span",{"className":125},[126],"katex",[123,128,131],{"className":129},[130],"katex-mathml",[132,133,135],"math",{"xmlns":134},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[136,137,138,165],"semantics",{},[139,140,141,144,149,152,154,157,159,161,163],"mrow",{},[142,143,80],"mn",{},[145,146,148],"mo",{"separator":147},"true",",",[142,150,151],{},"4",[145,153,148],{"separator":147},[142,155,156],{},"6",[145,158,148],{"separator":147},[142,160,151],{},[145,162,148],{"separator":147},[142,164,80],{},[166,167,169],"annotation",{"encoding":168},"application\u002Fx-tex","1, 4, 6, 4, 1",[123,171,174],{"className":172,"ariaHidden":147},[173],"katex-html",[123,175,178,183,187,191,196,199,202,205,208,211,214,217,220,223],{"className":176},[177],"base",[123,179],{"className":180,"style":182},[181],"strut","height:0.8389em;vertical-align:-0.1944em;",[123,184,80],{"className":185},[186],"mord",[123,188,148],{"className":189},[190],"mpunct",[123,192],{"className":193,"style":195},[194],"mspace","margin-right:0.1667em;",[123,197,151],{"className":198},[186],[123,200,148],{"className":201},[190],[123,203],{"className":204,"style":195},[194],[123,206,156],{"className":207},[186],[123,209,148],{"className":210},[190],[123,212],{"className":213,"style":195},[194],[123,215,151],{"className":216},[186],[123,218,148],{"className":219},[190],[123,221],{"className":222,"style":195},[194],[123,224,80],{"className":225},[186],". The ",[123,228,230,243],{"className":229},[126],[123,231,233],{"className":232},[130],[132,234,235],{"xmlns":134},[136,236,237,241],{},[139,238,239],{},[142,240,156],{},[166,242,156],{"encoding":168},[123,244,246],{"className":245,"ariaHidden":147},[173],[123,247,249,253],{"className":248},[177],[123,250],{"className":251,"style":252},[181],"height:0.6444em;",[123,254,156],{"className":255},[186]," in the center comes from adding the two numbers above it: ",[123,258,260,280],{"className":259},[126],[123,261,263],{"className":262},[130],[132,264,265],{"xmlns":134},[136,266,267,277],{},[139,268,269,272,275],{},[142,270,271],{},"3",[145,273,274],{},"+",[142,276,271],{},[166,278,279],{"encoding":168},"3 + 3",[123,281,283,304],{"className":282,"ariaHidden":147},[173],[123,284,286,290,293,297,301],{"className":285},[177],[123,287],{"className":288,"style":289},[181],"height:0.7278em;vertical-align:-0.0833em;",[123,291,271],{"className":292},[186],[123,294],{"className":295,"style":296},[194],"margin-right:0.2222em;",[123,298,274],{"className":299},[300],"mbin",[123,302],{"className":303,"style":296},[194],[123,305,307,310],{"className":306},[177],[123,308],{"className":309,"style":252},[181],[123,311,271],{"className":312},[186],[123,314,316,329],{"className":315},[126],[123,317,319],{"className":318},[130],[132,320,321],{"xmlns":134},[136,322,323,327],{},[139,324,325],{},[142,326,151],{},[166,328,151],{"encoding":168},[123,330,332],{"className":331,"ariaHidden":147},[173],[123,333,335,338],{"className":334},[177],[123,336],{"className":337,"style":252},[181],[123,339,151],{"className":340},[186]," on the left comes from ",[123,343,345,363],{"className":344},[126],[123,346,348],{"className":347},[130],[132,349,350],{"xmlns":134},[136,351,352,360],{},[139,353,354,356,358],{},[142,355,80],{},[145,357,274],{},[142,359,271],{},[166,361,362],{"encoding":168},"1 + 3",[123,364,366,384],{"className":365,"ariaHidden":147},[173],[123,367,369,372,375,378,381],{"className":368},[177],[123,370],{"className":371,"style":289},[181],[123,373,80],{"className":374},[186],[123,376],{"className":377,"style":296},[194],[123,379,274],{"className":380},[300],[123,382],{"className":383,"style":296},[194],[123,385,387,390],{"className":386},[177],[123,388],{"className":389,"style":252},[181],[123,391,271],{"className":392},[186],". Always the same rule.",[395,396,398],"h3",{"id":397},"binomial-coefficients","Binomial coefficients",[30,400,401,402,405],{},"Every number in the triangle has a formal name: it is a ",[42,403,404],{},"binomial coefficient",". It is written as:",[30,407,410],{"className":408},[409],"math-block",[123,411,413,442],{"className":412},[126],[123,414,416],{"className":415},[130],[132,417,418],{"xmlns":134},[136,419,420,439],{},[139,421,422,425,436],{},[145,423,424],{"fence":147},"(",[426,427,429,433],"mfrac",{"linethickness":428},"0px",[430,431,432],"mi",{},"n",[430,434,435],{},"k",[145,437,438],{"fence":147},")",[166,440,441],{"encoding":168},"\\binom{n}{k}",[123,443,445],{"className":444,"ariaHidden":147},[173],[123,446,448,452],{"className":447},[177],[123,449],{"className":450,"style":451},[181],"height:1.2em;vertical-align:-0.35em;",[123,453,455,466,535],{"className":454},[186],[123,456,461],{"className":457,"style":460},[458,459],"mopen","delimcenter","top:0em;",[123,462,424],{"className":463},[464,465],"delimsizing","size1",[123,467,469],{"className":468},[426],[123,470,474,526],{"className":471},[472,473],"vlist-t","vlist-t2",[123,475,478,521],{"className":476},[477],"vlist-r",[123,479,483,506],{"className":480,"style":482},[481],"vlist","height:0.7454em;",[123,484,486,491],{"style":485},"top:-2.355em;",[123,487],{"className":488,"style":490},[489],"pstrut","height:2.7em;",[123,492,498],{"className":493},[494,495,496,497],"sizing","reset-size6","size3","mtight",[123,499,501],{"className":500},[186,497],[123,502,435],{"className":503,"style":505},[186,504,497],"mathnormal","margin-right:0.0315em;",[123,507,509,512],{"style":508},"top:-3.144em;",[123,510],{"className":511,"style":490},[489],[123,513,515],{"className":514},[494,495,496,497],[123,516,518],{"className":517},[186,497],[123,519,432],{"className":520},[186,504,497],[123,522,525],{"className":523},[524],"vlist-s","​",[123,527,529],{"className":528},[477],[123,530,533],{"className":531,"style":532},[481],"height:0.345em;",[123,534],{},[123,536,539],{"className":537,"style":460},[538,459],"mclose",[123,540,438],{"className":541},[464,465],[30,543,544,545,606,607,635,636,664],{},"and read \"",[42,546,547,576,577],{},[123,548,550,563],{"className":549},[126],[123,551,553],{"className":552},[130],[132,554,555],{"xmlns":134},[136,556,557,561],{},[139,558,559],{},[430,560,432],{},[166,562,432],{"encoding":168},[123,564,566],{"className":565,"ariaHidden":147},[173],[123,567,569,573],{"className":568},[177],[123,570],{"className":571,"style":572},[181],"height:0.4306em;",[123,574,432],{"className":575},[186,504]," choose ",[123,578,580,593],{"className":579},[126],[123,581,583],{"className":582},[130],[132,584,585],{"xmlns":134},[136,586,587,591],{},[139,588,589],{},[430,590,435],{},[166,592,435],{"encoding":168},[123,594,596],{"className":595,"ariaHidden":147},[173],[123,597,599,603],{"className":598},[177],[123,600],{"className":601,"style":602},[181],"height:0.6944em;",[123,604,435],{"className":605,"style":505},[186,504],"\". Its meaning is straightforward: it represents the number of ways to choose ",[123,608,610,623],{"className":609},[126],[123,611,613],{"className":612},[130],[132,614,615],{"xmlns":134},[136,616,617,621],{},[139,618,619],{},[430,620,435],{},[166,622,435],{"encoding":168},[123,624,626],{"className":625,"ariaHidden":147},[173],[123,627,629,632],{"className":628},[177],[123,630],{"className":631,"style":602},[181],[123,633,435],{"className":634,"style":505},[186,504]," elements from a set of ",[123,637,639,652],{"className":638},[126],[123,640,642],{"className":641},[130],[132,643,644],{"xmlns":134},[136,645,646,650],{},[139,647,648],{},[430,649,432],{},[166,651,432],{"encoding":168},[123,653,655],{"className":654,"ariaHidden":147},[173],[123,656,658,661],{"className":657},[177],[123,659],{"className":660,"style":572},[181],[123,662,432],{"className":663},[186,504],", regardless of order.",[30,666,667,668,798,799,827],{},"For example, ",[123,669,671,701],{"className":670},[126],[123,672,674],{"className":673},[130],[132,675,676],{"xmlns":134},[136,677,678,698],{},[139,679,680,693,696],{},[139,681,682,684,691],{},[145,683,424],{"fence":147},[426,685,686,688],{"linethickness":428},[142,687,151],{},[142,689,690],{},"2",[145,692,438],{"fence":147},[145,694,695],{},"=",[142,697,156],{},[166,699,700],{"encoding":168},"\\binom{4}{2} = 6",[123,702,704,789],{"className":703,"ariaHidden":147},[173],[123,705,707,711,778,782,786],{"className":706},[177],[123,708],{"className":709,"style":710},[181],"height:1.2451em;vertical-align:-0.35em;",[123,712,714,720,772],{"className":713},[186],[123,715,717],{"className":716,"style":460},[458,459],[123,718,424],{"className":719},[464,465],[123,721,723],{"className":722},[426],[123,724,726,764],{"className":725},[472,473],[123,727,729,761],{"className":728},[477],[123,730,733,747],{"className":731,"style":732},[481],"height:0.8951em;",[123,734,735,738],{"style":485},[123,736],{"className":737,"style":490},[489],[123,739,741],{"className":740},[494,495,496,497],[123,742,744],{"className":743},[186,497],[123,745,690],{"className":746},[186,497],[123,748,749,752],{"style":508},[123,750],{"className":751,"style":490},[489],[123,753,755],{"className":754},[494,495,496,497],[123,756,758],{"className":757},[186,497],[123,759,151],{"className":760},[186,497],[123,762,525],{"className":763},[524],[123,765,767],{"className":766},[477],[123,768,770],{"className":769,"style":532},[481],[123,771],{},[123,773,775],{"className":774,"style":460},[538,459],[123,776,438],{"className":777},[464,465],[123,779],{"className":780,"style":781},[194],"margin-right:0.2778em;",[123,783,695],{"className":784},[785],"mrel",[123,787],{"className":788,"style":781},[194],[123,790,792,795],{"className":791},[177],[123,793],{"className":794,"style":252},[181],[123,796,156],{"className":797},[186]," means there are 6 ways to choose 2 items from 4. And indeed, if you look at row 4 of the triangle, position 2 (counting from 0) is precisely ",[123,800,802,815],{"className":801},[126],[123,803,805],{"className":804},[130],[132,806,807],{"xmlns":134},[136,808,809,813],{},[139,810,811],{},[142,812,156],{},[166,814,156],{"encoding":168},[123,816,818],{"className":817,"ariaHidden":147},[173],[123,819,821,824],{"className":820},[177],[123,822],{"className":823,"style":252},[181],[123,825,156],{"className":826},[186],".",[30,829,830],{},"The formula to compute it is:",[30,832,834],{"className":833},[409],[123,835,837,895],{"className":836},[126],[123,838,840],{"className":839},[130],[132,841,842],{"xmlns":134},[136,843,844,892],{},[139,845,846,858,860],{},[139,847,848,850,856],{},[145,849,424],{"fence":147},[426,851,852,854],{"linethickness":428},[430,853,432],{},[430,855,435],{},[145,857,438],{"fence":147},[145,859,695],{},[426,861,862,870],{},[139,863,864,866],{},[430,865,432],{},[145,867,869],{"stretchy":868},"false","!",[139,871,872,874,876,879,881,883,886,888,890],{},[430,873,435],{},[145,875,869],{"stretchy":868},[145,877,878],{},"⋅",[145,880,424],{"stretchy":868},[430,882,432],{},[145,884,885],{},"−",[430,887,435],{},[145,889,438],{"stretchy":868},[145,891,869],{"stretchy":868},[166,893,894],{"encoding":168},"\\binom{n}{k} = \\frac{n!}{k! \\cdot (n-k)!}",[123,896,898,979],{"className":897,"ariaHidden":147},[173],[123,899,901,904,970,973,976],{"className":900},[177],[123,902],{"className":903,"style":451},[181],[123,905,907,913,964],{"className":906},[186],[123,908,910],{"className":909,"style":460},[458,459],[123,911,424],{"className":912},[464,465],[123,914,916],{"className":915},[426],[123,917,919,956],{"className":918},[472,473],[123,920,922,953],{"className":921},[477],[123,923,925,939],{"className":924,"style":482},[481],[123,926,927,930],{"style":485},[123,928],{"className":929,"style":490},[489],[123,931,933],{"className":932},[494,495,496,497],[123,934,936],{"className":935},[186,497],[123,937,435],{"className":938,"style":505},[186,504,497],[123,940,941,944],{"style":508},[123,942],{"className":943,"style":490},[489],[123,945,947],{"className":946},[494,495,496,497],[123,948,950],{"className":949},[186,497],[123,951,432],{"className":952},[186,504,497],[123,954,525],{"className":955},[524],[123,957,959],{"className":958},[477],[123,960,962],{"className":961,"style":532},[481],[123,963],{},[123,965,967],{"className":966,"style":460},[538,459],[123,968,438],{"className":969},[464,465],[123,971],{"className":972,"style":781},[194],[123,974,695],{"className":975},[785],[123,977],{"className":978,"style":781},[194],[123,980,982,986],{"className":981},[177],[123,983],{"className":984,"style":985},[181],"height:1.4001em;vertical-align:-0.52em;",[123,987,989,993,1085],{"className":988},[186],[123,990],{"className":991},[458,992],"nulldelimiter",[123,994,996],{"className":995},[426],[123,997,999,1076],{"className":998},[472,473],[123,1000,1002,1073],{"className":1001},[477],[123,1003,1006,1044,1055],{"className":1004,"style":1005},[481],"height:0.8801em;",[123,1007,1009,1013],{"style":1008},"top:-2.655em;",[123,1010],{"className":1011,"style":1012},[489],"height:3em;",[123,1014,1016],{"className":1015},[494,495,496,497],[123,1017,1019,1022,1025,1028,1031,1034,1037,1040],{"className":1018},[186,497],[123,1020,435],{"className":1021,"style":505},[186,504,497],[123,1023,869],{"className":1024},[538,497],[123,1026,878],{"className":1027},[300,497],[123,1029,424],{"className":1030},[458,497],[123,1032,432],{"className":1033},[186,504,497],[123,1035,885],{"className":1036},[300,497],[123,1038,435],{"className":1039,"style":505},[186,504,497],[123,1041,1043],{"className":1042},[538,497],")!",[123,1045,1047,1050],{"style":1046},"top:-3.23em;",[123,1048],{"className":1049,"style":1012},[489],[123,1051],{"className":1052,"style":1054},[1053],"frac-line","border-bottom-width:0.04em;",[123,1056,1058,1061],{"style":1057},"top:-3.394em;",[123,1059],{"className":1060,"style":1012},[489],[123,1062,1064],{"className":1063},[494,495,496,497],[123,1065,1067,1070],{"className":1066},[186,497],[123,1068,432],{"className":1069},[186,504,497],[123,1071,869],{"className":1072},[538,497],[123,1074,525],{"className":1075},[524],[123,1077,1079],{"className":1078},[477],[123,1080,1083],{"className":1081,"style":1082},[481],"height:0.52em;",[123,1084],{},[123,1086],{"className":1087},[538,992],[30,1089,1090,1091,1125,1126,1129,1130,1158,1159,1187,1188,1334,1335,827],{},"Here, the symbol ",[123,1092,1094,1110],{"className":1093},[126],[123,1095,1097],{"className":1096},[130],[132,1098,1099],{"xmlns":134},[136,1100,1101,1107],{},[139,1102,1103,1105],{},[430,1104,432],{},[145,1106,869],{"stretchy":868},[166,1108,1109],{"encoding":168},"n!",[123,1111,1113],{"className":1112,"ariaHidden":147},[173],[123,1114,1116,1119,1122],{"className":1115},[177],[123,1117],{"className":1118,"style":602},[181],[123,1120,432],{"className":1121},[186,504],[123,1123,869],{"className":1124},[538]," is called ",[42,1127,1128],{},"factorial"," and means multiplying all whole numbers from ",[123,1131,1133,1146],{"className":1132},[126],[123,1134,1136],{"className":1135},[130],[132,1137,1138],{"xmlns":134},[136,1139,1140,1144],{},[139,1141,1142],{},[142,1143,80],{},[166,1145,80],{"encoding":168},[123,1147,1149],{"className":1148,"ariaHidden":147},[173],[123,1150,1152,1155],{"className":1151},[177],[123,1153],{"className":1154,"style":252},[181],[123,1156,80],{"className":1157},[186]," up to ",[123,1160,1162,1175],{"className":1161},[126],[123,1163,1165],{"className":1164},[130],[132,1166,1167],{"xmlns":134},[136,1168,1169,1173],{},[139,1170,1171],{},[430,1172,432],{},[166,1174,432],{"encoding":168},[123,1176,1178],{"className":1177,"ariaHidden":147},[173],[123,1179,1181,1184],{"className":1180},[177],[123,1182],{"className":1183,"style":572},[181],[123,1185,432],{"className":1186},[186,504],". For instance, ",[123,1189,1191,1229],{"className":1190},[126],[123,1192,1194],{"className":1193},[130],[132,1195,1196],{"xmlns":134},[136,1197,1198,1226],{},[139,1199,1200,1202,1204,1206,1208,1211,1213,1215,1217,1219,1221,1223],{},[142,1201,151],{},[145,1203,869],{"stretchy":868},[145,1205,695],{},[142,1207,151],{},[145,1209,1210],{},"×",[142,1212,271],{},[145,1214,1210],{},[142,1216,690],{},[145,1218,1210],{},[142,1220,80],{},[145,1222,695],{},[142,1224,1225],{},"24",[166,1227,1228],{"encoding":168},"4! = 4 \\times 3 \\times 2 \\times 1 = 24",[123,1230,1232,1253,1271,1289,1307,1325],{"className":1231,"ariaHidden":147},[173],[123,1233,1235,1238,1241,1244,1247,1250],{"className":1234},[177],[123,1236],{"className":1237,"style":602},[181],[123,1239,151],{"className":1240},[186],[123,1242,869],{"className":1243},[538],[123,1245],{"className":1246,"style":781},[194],[123,1248,695],{"className":1249},[785],[123,1251],{"className":1252,"style":781},[194],[123,1254,1256,1259,1262,1265,1268],{"className":1255},[177],[123,1257],{"className":1258,"style":289},[181],[123,1260,151],{"className":1261},[186],[123,1263],{"className":1264,"style":296},[194],[123,1266,1210],{"className":1267},[300],[123,1269],{"className":1270,"style":296},[194],[123,1272,1274,1277,1280,1283,1286],{"className":1273},[177],[123,1275],{"className":1276,"style":289},[181],[123,1278,271],{"className":1279},[186],[123,1281],{"className":1282,"style":296},[194],[123,1284,1210],{"className":1285},[300],[123,1287],{"className":1288,"style":296},[194],[123,1290,1292,1295,1298,1301,1304],{"className":1291},[177],[123,1293],{"className":1294,"style":289},[181],[123,1296,690],{"className":1297},[186],[123,1299],{"className":1300,"style":296},[194],[123,1302,1210],{"className":1303},[300],[123,1305],{"className":1306,"style":296},[194],[123,1308,1310,1313,1316,1319,1322],{"className":1309},[177],[123,1311],{"className":1312,"style":252},[181],[123,1314,80],{"className":1315},[186],[123,1317],{"className":1318,"style":781},[194],[123,1320,695],{"className":1321},[785],[123,1323],{"className":1324,"style":781},[194],[123,1326,1328,1331],{"className":1327},[177],[123,1329],{"className":1330,"style":252},[181],[123,1332,1225],{"className":1333},[186],". By convention, ",[123,1336,1338,1359],{"className":1337},[126],[123,1339,1341],{"className":1340},[130],[132,1342,1343],{"xmlns":134},[136,1344,1345,1356],{},[139,1346,1347,1350,1352,1354],{},[142,1348,1349],{},"0",[145,1351,869],{"stretchy":868},[145,1353,695],{},[142,1355,80],{},[166,1357,1358],{"encoding":168},"0! = 1",[123,1360,1362,1383],{"className":1361,"ariaHidden":147},[173],[123,1363,1365,1368,1371,1374,1377,1380],{"className":1364},[177],[123,1366],{"className":1367,"style":602},[181],[123,1369,1349],{"className":1370},[186],[123,1372,869],{"className":1373},[538],[123,1375],{"className":1376,"style":781},[194],[123,1378,695],{"className":1379},[785],[123,1381],{"className":1382,"style":781},[194],[123,1384,1386,1389],{"className":1385},[177],[123,1387],{"className":1388,"style":252},[181],[123,1390,80],{"className":1391},[186],[30,1393,1394,1395,1838],{},"Let's verify: ",[123,1396,1398,1470],{"className":1397},[126],[123,1399,1401],{"className":1400},[130],[132,1402,1403],{"xmlns":134},[136,1404,1405,1467],{},[139,1406,1407,1419,1421,1441,1443,1455,1457,1463,1465],{},[139,1408,1409,1411,1417],{},[145,1410,424],{"fence":147},[426,1412,1413,1415],{"linethickness":428},[142,1414,151],{},[142,1416,690],{},[145,1418,438],{"fence":147},[145,1420,695],{},[426,1422,1423,1429],{},[139,1424,1425,1427],{},[142,1426,151],{},[145,1428,869],{"stretchy":868},[139,1430,1431,1433,1435,1437,1439],{},[142,1432,690],{},[145,1434,869],{"stretchy":868},[145,1436,878],{},[142,1438,690],{},[145,1440,869],{"stretchy":868},[145,1442,695],{},[426,1444,1445,1447],{},[142,1446,1225],{},[139,1448,1449,1451,1453],{},[142,1450,690],{},[145,1452,1210],{},[142,1454,690],{},[145,1456,695],{},[426,1458,1459,1461],{},[142,1460,1225],{},[142,1462,151],{},[145,1464,695],{},[142,1466,156],{},[166,1468,1469],{"encoding":168},"\\binom{4}{2} = \\frac{4!}{2! \\cdot 2!} = \\frac{24}{2 \\times 2} = \\frac{24}{4} = 6",[123,1471,1473,1554,1653,1745,1829],{"className":1472,"ariaHidden":147},[173],[123,1474,1476,1479,1545,1548,1551],{"className":1475},[177],[123,1477],{"className":1478,"style":710},[181],[123,1480,1482,1488,1539],{"className":1481},[186],[123,1483,1485],{"className":1484,"style":460},[458,459],[123,1486,424],{"className":1487},[464,465],[123,1489,1491],{"className":1490},[426],[123,1492,1494,1531],{"className":1493},[472,473],[123,1495,1497,1528],{"className":1496},[477],[123,1498,1500,1514],{"className":1499,"style":732},[481],[123,1501,1502,1505],{"style":485},[123,1503],{"className":1504,"style":490},[489],[123,1506,1508],{"className":1507},[494,495,496,497],[123,1509,1511],{"className":1510},[186,497],[123,1512,690],{"className":1513},[186,497],[123,1515,1516,1519],{"style":508},[123,1517],{"className":1518,"style":490},[489],[123,1520,1522],{"className":1521},[494,495,496,497],[123,1523,1525],{"className":1524},[186,497],[123,1526,151],{"className":1527},[186,497],[123,1529,525],{"className":1530},[524],[123,1532,1534],{"className":1533},[477],[123,1535,1537],{"className":1536,"style":532},[481],[123,1538],{},[123,1540,1542],{"className":1541,"style":460},[538,459],[123,1543,438],{"className":1544},[464,465],[123,1546],{"className":1547,"style":781},[194],[123,1549,695],{"className":1550},[785],[123,1552],{"className":1553,"style":781},[194],[123,1555,1557,1561,1644,1647,1650],{"className":1556},[177],[123,1558],{"className":1559,"style":1560},[181],"height:1.2251em;vertical-align:-0.345em;",[123,1562,1564,1567,1641],{"className":1563},[186],[123,1565],{"className":1566},[458,992],[123,1568,1570],{"className":1569},[426],[123,1571,1573,1633],{"className":1572},[472,473],[123,1574,1576,1630],{"className":1575},[477],[123,1577,1579,1605,1613],{"className":1578,"style":1005},[481],[123,1580,1581,1584],{"style":1008},[123,1582],{"className":1583,"style":1012},[489],[123,1585,1587],{"className":1586},[494,495,496,497],[123,1588,1590,1593,1596,1599,1602],{"className":1589},[186,497],[123,1591,690],{"className":1592},[186,497],[123,1594,869],{"className":1595},[538,497],[123,1597,878],{"className":1598},[300,497],[123,1600,690],{"className":1601},[186,497],[123,1603,869],{"className":1604},[538,497],[123,1606,1607,1610],{"style":1046},[123,1608],{"className":1609,"style":1012},[489],[123,1611],{"className":1612,"style":1054},[1053],[123,1614,1615,1618],{"style":1057},[123,1616],{"className":1617,"style":1012},[489],[123,1619,1621],{"className":1620},[494,495,496,497],[123,1622,1624,1627],{"className":1623},[186,497],[123,1625,151],{"className":1626},[186,497],[123,1628,869],{"className":1629},[538,497],[123,1631,525],{"className":1632},[524],[123,1634,1636],{"className":1635},[477],[123,1637,1639],{"className":1638,"style":532},[481],[123,1640],{},[123,1642],{"className":1643},[538,992],[123,1645],{"className":1646,"style":781},[194],[123,1648,695],{"className":1649},[785],[123,1651],{"className":1652,"style":781},[194],[123,1654,1656,1660,1736,1739,1742],{"className":1655},[177],[123,1657],{"className":1658,"style":1659},[181],"height:1.2484em;vertical-align:-0.4033em;",[123,1661,1663,1666,1733],{"className":1662},[186],[123,1664],{"className":1665},[458,992],[123,1667,1669],{"className":1668},[426],[123,1670,1672,1724],{"className":1671},[472,473],[123,1673,1675,1721],{"className":1674},[477],[123,1676,1679,1699,1707],{"className":1677,"style":1678},[481],"height:0.8451em;",[123,1680,1681,1684],{"style":1008},[123,1682],{"className":1683,"style":1012},[489],[123,1685,1687],{"className":1686},[494,495,496,497],[123,1688,1690,1693,1696],{"className":1689},[186,497],[123,1691,690],{"className":1692},[186,497],[123,1694,1210],{"className":1695},[300,497],[123,1697,690],{"className":1698},[186,497],[123,1700,1701,1704],{"style":1046},[123,1702],{"className":1703,"style":1012},[489],[123,1705],{"className":1706,"style":1054},[1053],[123,1708,1709,1712],{"style":1057},[123,1710],{"className":1711,"style":1012},[489],[123,1713,1715],{"className":1714},[494,495,496,497],[123,1716,1718],{"className":1717},[186,497],[123,1719,1225],{"className":1720},[186,497],[123,1722,525],{"className":1723},[524],[123,1725,1727],{"className":1726},[477],[123,1728,1731],{"className":1729,"style":1730},[481],"height:0.4033em;",[123,1732],{},[123,1734],{"className":1735},[538,992],[123,1737],{"className":1738,"style":781},[194],[123,1740,695],{"className":1741},[785],[123,1743],{"className":1744,"style":781},[194],[123,1746,1748,1752,1820,1823,1826],{"className":1747},[177],[123,1749],{"className":1750,"style":1751},[181],"height:1.1901em;vertical-align:-0.345em;",[123,1753,1755,1758,1817],{"className":1754},[186],[123,1756],{"className":1757},[458,992],[123,1759,1761],{"className":1760},[426],[123,1762,1764,1809],{"className":1763},[472,473],[123,1765,1767,1806],{"className":1766},[477],[123,1768,1770,1784,1792],{"className":1769,"style":1678},[481],[123,1771,1772,1775],{"style":1008},[123,1773],{"className":1774,"style":1012},[489],[123,1776,1778],{"className":1777},[494,495,496,497],[123,1779,1781],{"className":1780},[186,497],[123,1782,151],{"className":1783},[186,497],[123,1785,1786,1789],{"style":1046},[123,1787],{"className":1788,"style":1012},[489],[123,1790],{"className":1791,"style":1054},[1053],[123,1793,1794,1797],{"style":1057},[123,1795],{"className":1796,"style":1012},[489],[123,1798,1800],{"className":1799},[494,495,496,497],[123,1801,1803],{"className":1802},[186,497],[123,1804,1225],{"className":1805},[186,497],[123,1807,525],{"className":1808},[524],[123,1810,1812],{"className":1811},[477],[123,1813,1815],{"className":1814,"style":532},[481],[123,1816],{},[123,1818],{"className":1819},[538,992],[123,1821],{"className":1822,"style":781},[194],[123,1824,695],{"className":1825},[785],[123,1827],{"className":1828,"style":781},[194],[123,1830,1832,1835],{"className":1831},[177],[123,1833],{"className":1834,"style":252},[181],[123,1836,156],{"className":1837},[186],". Correct.",[30,1840,1841,1842,1870],{},"So row ",[123,1843,1845,1858],{"className":1844},[126],[123,1846,1848],{"className":1847},[130],[132,1849,1850],{"xmlns":134},[136,1851,1852,1856],{},[139,1853,1854],{},[430,1855,432],{},[166,1857,432],{"encoding":168},[123,1859,1861],{"className":1860,"ariaHidden":147},[173],[123,1862,1864,1867],{"className":1863},[177],[123,1865],{"className":1866,"style":572},[181],[123,1868,432],{"className":1869},[186,504]," of the triangle contains the values:",[30,1872,1874],{"className":1873},[409],[123,1875,1877,1958],{"className":1876},[126],[123,1878,1880],{"className":1879},[130],[132,1881,1882],{"xmlns":134},[136,1883,1884,1955],{},[139,1885,1886,1898,1900,1904,1916,1918,1920,1932,1934,1936,1939,1941,1943],{},[139,1887,1888,1890,1896],{},[145,1889,424],{"fence":147},[426,1891,1892,1894],{"linethickness":428},[430,1893,432],{},[142,1895,1349],{},[145,1897,438],{"fence":147},[145,1899,148],{"separator":147},[1901,1902,1903],"mtext",{},"  ",[139,1905,1906,1908,1914],{},[145,1907,424],{"fence":147},[426,1909,1910,1912],{"linethickness":428},[430,1911,432],{},[142,1913,80],{},[145,1915,438],{"fence":147},[145,1917,148],{"separator":147},[1901,1919,1903],{},[139,1921,1922,1924,1930],{},[145,1923,424],{"fence":147},[426,1925,1926,1928],{"linethickness":428},[430,1927,432],{},[142,1929,690],{},[145,1931,438],{"fence":147},[145,1933,148],{"separator":147},[1901,1935,1903],{},[145,1937,1938],{},"…",[145,1940,148],{"separator":147},[1901,1942,1903],{},[139,1944,1945,1947,1953],{},[145,1946,424],{"fence":147},[426,1948,1949,1951],{"linethickness":428},[430,1950,432],{},[430,1952,432],{},[145,1954,438],{"fence":147},[166,1956,1957],{"encoding":168},"\\binom{n}{0},\\; \\binom{n}{1},\\; \\binom{n}{2},\\; \\ldots,\\; \\binom{n}{n}",[123,1959,1961],{"className":1960,"ariaHidden":147},[173],[123,1962,1964,1967,2033,2036,2039,2042,2108,2111,2114,2117,2183,2186,2189,2192,2196,2199,2202,2205,2208],{"className":1963},[177],[123,1965],{"className":1966,"style":451},[181],[123,1968,1970,1976,2027],{"className":1969},[186],[123,1971,1973],{"className":1972,"style":460},[458,459],[123,1974,424],{"className":1975},[464,465],[123,1977,1979],{"className":1978},[426],[123,1980,1982,2019],{"className":1981},[472,473],[123,1983,1985,2016],{"className":1984},[477],[123,1986,1988,2002],{"className":1987,"style":482},[481],[123,1989,1990,1993],{"style":485},[123,1991],{"className":1992,"style":490},[489],[123,1994,1996],{"className":1995},[494,495,496,497],[123,1997,1999],{"className":1998},[186,497],[123,2000,1349],{"className":2001},[186,497],[123,2003,2004,2007],{"style":508},[123,2005],{"className":2006,"style":490},[489],[123,2008,2010],{"className":2009},[494,495,496,497],[123,2011,2013],{"className":2012},[186,497],[123,2014,432],{"className":2015},[186,504,497],[123,2017,525],{"className":2018},[524],[123,2020,2022],{"className":2021},[477],[123,2023,2025],{"className":2024,"style":532},[481],[123,2026],{},[123,2028,2030],{"className":2029,"style":460},[538,459],[123,2031,438],{"className":2032},[464,465],[123,2034,148],{"className":2035},[190],[123,2037],{"className":2038,"style":781},[194],[123,2040],{"className":2041,"style":195},[194],[123,2043,2045,2051,2102],{"className":2044},[186],[123,2046,2048],{"className":2047,"style":460},[458,459],[123,2049,424],{"className":2050},[464,465],[123,2052,2054],{"className":2053},[426],[123,2055,2057,2094],{"className":2056},[472,473],[123,2058,2060,2091],{"className":2059},[477],[123,2061,2063,2077],{"className":2062,"style":482},[481],[123,2064,2065,2068],{"style":485},[123,2066],{"className":2067,"style":490},[489],[123,2069,2071],{"className":2070},[494,495,496,497],[123,2072,2074],{"className":2073},[186,497],[123,2075,80],{"className":2076},[186,497],[123,2078,2079,2082],{"style":508},[123,2080],{"className":2081,"style":490},[489],[123,2083,2085],{"className":2084},[494,495,496,497],[123,2086,2088],{"className":2087},[186,497],[123,2089,432],{"className":2090},[186,504,497],[123,2092,525],{"className":2093},[524],[123,2095,2097],{"className":2096},[477],[123,2098,2100],{"className":2099,"style":532},[481],[123,2101],{},[123,2103,2105],{"className":2104,"style":460},[538,459],[123,2106,438],{"className":2107},[464,465],[123,2109,148],{"className":2110},[190],[123,2112],{"className":2113,"style":781},[194],[123,2115],{"className":2116,"style":195},[194],[123,2118,2120,2126,2177],{"className":2119},[186],[123,2121,2123],{"className":2122,"style":460},[458,459],[123,2124,424],{"className":2125},[464,465],[123,2127,2129],{"className":2128},[426],[123,2130,2132,2169],{"className":2131},[472,473],[123,2133,2135,2166],{"className":2134},[477],[123,2136,2138,2152],{"className":2137,"style":482},[481],[123,2139,2140,2143],{"style":485},[123,2141],{"className":2142,"style":490},[489],[123,2144,2146],{"className":2145},[494,495,496,497],[123,2147,2149],{"className":2148},[186,497],[123,2150,690],{"className":2151},[186,497],[123,2153,2154,2157],{"style":508},[123,2155],{"className":2156,"style":490},[489],[123,2158,2160],{"className":2159},[494,495,496,497],[123,2161,2163],{"className":2162},[186,497],[123,2164,432],{"className":2165},[186,504,497],[123,2167,525],{"className":2168},[524],[123,2170,2172],{"className":2171},[477],[123,2173,2175],{"className":2174,"style":532},[481],[123,2176],{},[123,2178,2180],{"className":2179,"style":460},[538,459],[123,2181,438],{"className":2182},[464,465],[123,2184,148],{"className":2185},[190],[123,2187],{"className":2188,"style":781},[194],[123,2190],{"className":2191,"style":195},[194],[123,2193,1938],{"className":2194},[2195],"minner",[123,2197],{"className":2198,"style":195},[194],[123,2200,148],{"className":2201},[190],[123,2203],{"className":2204,"style":781},[194],[123,2206],{"className":2207,"style":195},[194],[123,2209,2211,2217,2268],{"className":2210},[186],[123,2212,2214],{"className":2213,"style":460},[458,459],[123,2215,424],{"className":2216},[464,465],[123,2218,2220],{"className":2219},[426],[123,2221,2223,2260],{"className":2222},[472,473],[123,2224,2226,2257],{"className":2225},[477],[123,2227,2229,2243],{"className":2228,"style":482},[481],[123,2230,2231,2234],{"style":485},[123,2232],{"className":2233,"style":490},[489],[123,2235,2237],{"className":2236},[494,495,496,497],[123,2238,2240],{"className":2239},[186,497],[123,2241,432],{"className":2242},[186,504,497],[123,2244,2245,2248],{"style":508},[123,2246],{"className":2247,"style":490},[489],[123,2249,2251],{"className":2250},[494,495,496,497],[123,2252,2254],{"className":2253},[186,497],[123,2255,432],{"className":2256},[186,504,497],[123,2258,525],{"className":2259},[524],[123,2261,2263],{"className":2262},[477],[123,2264,2266],{"className":2265,"style":532},[481],[123,2267],{},[123,2269,2271],{"className":2270,"style":460},[538,459],[123,2272,438],{"className":2273},[464,465],[34,2275,2277],{"id":2276},"combinatorics-the-art-of-counting-without-listing","Combinatorics: the art of counting without listing",[30,2279,2280,2283,2284,827],{},[42,2281,2282],{},"Combinatorics"," is the branch of mathematics that studies how many distinct ways there are to arrange, group, or choose elements. Its power lies in letting us know the number of possibilities ",[47,2285,2286],{},"without having to list them one by one",[30,2288,2289,2290,2293,2294,2297,2298,2326,2327,2400,2401,827],{},"To see why this is useful, consider flipping a coin. Each flip has two possible outcomes: ",[42,2291,2292],{},"heads"," (O) or ",[42,2295,2296],{},"tails"," (X). If we flip the coin once, there are ",[123,2299,2301,2314],{"className":2300},[126],[123,2302,2304],{"className":2303},[130],[132,2305,2306],{"xmlns":134},[136,2307,2308,2312],{},[139,2309,2310],{},[142,2311,690],{},[166,2313,690],{"encoding":168},[123,2315,2317],{"className":2316,"ariaHidden":147},[173],[123,2318,2320,2323],{"className":2319},[177],[123,2321],{"className":2322,"style":252},[181],[123,2324,690],{"className":2325},[186]," outcomes. If we flip it twice, each of the 2 outcomes of the first flip combines with the 2 outcomes of the second, giving ",[123,2328,2330,2352],{"className":2329},[126],[123,2331,2333],{"className":2332},[130],[132,2334,2335],{"xmlns":134},[136,2336,2337,2349],{},[139,2338,2339,2341,2343,2345,2347],{},[142,2340,690],{},[145,2342,1210],{},[142,2344,690],{},[145,2346,695],{},[142,2348,151],{},[166,2350,2351],{"encoding":168},"2 \\times 2 = 4",[123,2353,2355,2373,2391],{"className":2354,"ariaHidden":147},[173],[123,2356,2358,2361,2364,2367,2370],{"className":2357},[177],[123,2359],{"className":2360,"style":289},[181],[123,2362,690],{"className":2363},[186],[123,2365],{"className":2366,"style":296},[194],[123,2368,1210],{"className":2369},[300],[123,2371],{"className":2372,"style":296},[194],[123,2374,2376,2379,2382,2385,2388],{"className":2375},[177],[123,2377],{"className":2378,"style":252},[181],[123,2380,690],{"className":2381},[186],[123,2383],{"className":2384,"style":781},[194],[123,2386,695],{"className":2387},[785],[123,2389],{"className":2390,"style":781},[194],[123,2392,2394,2397],{"className":2393},[177],[123,2395],{"className":2396,"style":252},[181],[123,2398,151],{"className":2399},[186]," outcomes. With three flips, ",[123,2402,2404,2431],{"className":2403},[126],[123,2405,2407],{"className":2406},[130],[132,2408,2409],{"xmlns":134},[136,2410,2411,2428],{},[139,2412,2413,2415,2417,2419,2421,2423,2425],{},[142,2414,690],{},[145,2416,1210],{},[142,2418,690],{},[145,2420,1210],{},[142,2422,690],{},[145,2424,695],{},[142,2426,2427],{},"8",[166,2429,2430],{"encoding":168},"2 \\times 2 \\times 2 = 8",[123,2432,2434,2452,2470,2488],{"className":2433,"ariaHidden":147},[173],[123,2435,2437,2440,2443,2446,2449],{"className":2436},[177],[123,2438],{"className":2439,"style":289},[181],[123,2441,690],{"className":2442},[186],[123,2444],{"className":2445,"style":296},[194],[123,2447,1210],{"className":2448},[300],[123,2450],{"className":2451,"style":296},[194],[123,2453,2455,2458,2461,2464,2467],{"className":2454},[177],[123,2456],{"className":2457,"style":289},[181],[123,2459,690],{"className":2460},[186],[123,2462],{"className":2463,"style":296},[194],[123,2465,1210],{"className":2466},[300],[123,2468],{"className":2469,"style":296},[194],[123,2471,2473,2476,2479,2482,2485],{"className":2472},[177],[123,2474],{"className":2475,"style":252},[181],[123,2477,690],{"className":2478},[186],[123,2480],{"className":2481,"style":781},[194],[123,2483,695],{"className":2484},[785],[123,2486],{"className":2487,"style":781},[194],[123,2489,2491,2494],{"className":2490},[177],[123,2492],{"className":2493,"style":252},[181],[123,2495,2427],{"className":2496},[186],[30,2498,2499,2500,2503],{},"That pattern is expressed with an ",[42,2501,2502],{},"exponent",":",[30,2505,2507],{"className":2506},[409],[123,2508,2510,2529],{"className":2509},[126],[123,2511,2513],{"className":2512},[130],[132,2514,2515],{"xmlns":134},[136,2516,2517,2526],{},[139,2518,2519],{},[2520,2521,2522,2524],"msup",{},[142,2523,690],{},[430,2525,432],{},[166,2527,2528],{"encoding":168},"2^n",[123,2530,2532],{"className":2531,"ariaHidden":147},[173],[123,2533,2535,2539],{"className":2534},[177],[123,2536],{"className":2537,"style":2538},[181],"height:0.6644em;",[123,2540,2542,2545],{"className":2541},[186],[123,2543,690],{"className":2544},[186],[123,2546,2549],{"className":2547},[2548],"msupsub",[123,2550,2552],{"className":2551},[472],[123,2553,2555],{"className":2554},[477],[123,2556,2558],{"className":2557,"style":2538},[481],[123,2559,2561,2564],{"style":2560},"top:-3.063em;margin-right:0.05em;",[123,2562],{"className":2563,"style":490},[489],[123,2565,2567],{"className":2566},[494,495,496,497],[123,2568,432],{"className":2569},[186,504,497],[30,2571,2572,2573,2601,2602,2630],{},"where ",[123,2574,2576,2589],{"className":2575},[126],[123,2577,2579],{"className":2578},[130],[132,2580,2581],{"xmlns":134},[136,2582,2583,2587],{},[139,2584,2585],{},[430,2586,432],{},[166,2588,432],{"encoding":168},[123,2590,2592],{"className":2591,"ariaHidden":147},[173],[123,2593,2595,2598],{"className":2594},[177],[123,2596],{"className":2597,"style":572},[181],[123,2599,432],{"className":2600},[186,504]," is the number of flips. The exponent tells us how many times we multiply the base (",[123,2603,2605,2618],{"className":2604},[126],[123,2606,2608],{"className":2607},[130],[132,2609,2610],{"xmlns":134},[136,2611,2612,2616],{},[139,2613,2614],{},[142,2615,690],{},[166,2617,690],{"encoding":168},[123,2619,2621],{"className":2620,"ariaHidden":147},[173],[123,2622,2624,2627],{"className":2623},[177],[123,2625],{"className":2626,"style":252},[181],[123,2628,690],{"className":2629},[186],") by itself. So:",[2632,2633,2634,2734],"table",{},[2635,2636,2637],"thead",{},[2638,2639,2640,2673],"tr",{},[2641,2642,2644,2645,438],"th",{"align":2643},"center","Flips (",[123,2646,2648,2661],{"className":2647},[126],[123,2649,2651],{"className":2650},[130],[132,2652,2653],{"xmlns":134},[136,2654,2655,2659],{},[139,2656,2657],{},[430,2658,432],{},[166,2660,432],{"encoding":168},[123,2662,2664],{"className":2663,"ariaHidden":147},[173],[123,2665,2667,2670],{"className":2666},[177],[123,2668],{"className":2669,"style":572},[181],[123,2671,432],{"className":2672},[186,504],[2641,2674,2675,2676,438],{"align":2643},"Possible outcomes (",[123,2677,2679,2696],{"className":2678},[126],[123,2680,2682],{"className":2681},[130],[132,2683,2684],{"xmlns":134},[136,2685,2686,2694],{},[139,2687,2688],{},[2520,2689,2690,2692],{},[142,2691,690],{},[430,2693,432],{},[166,2695,2528],{"encoding":168},[123,2697,2699],{"className":2698,"ariaHidden":147},[173],[123,2700,2702,2705],{"className":2701},[177],[123,2703],{"className":2704,"style":2538},[181],[123,2706,2708,2711],{"className":2707},[186],[123,2709,690],{"className":2710},[186],[123,2712,2714],{"className":2713},[2548],[123,2715,2717],{"className":2716},[472],[123,2718,2720],{"className":2719},[477],[123,2721,2723],{"className":2722,"style":2538},[481],[123,2724,2725,2728],{"style":2560},[123,2726],{"className":2727,"style":490},[489],[123,2729,2731],{"className":2730},[494,495,496,497],[123,2732,432],{"className":2733},[186,504,497],[2735,2736,2737,2744,2750,2756,2764],"tbody",{},[2638,2738,2739,2742],{},[2740,2741,80],"td",{"align":2643},[2740,2743,690],{"align":2643},[2638,2745,2746,2748],{},[2740,2747,690],{"align":2643},[2740,2749,151],{"align":2643},[2638,2751,2752,2754],{},[2740,2753,271],{"align":2643},[2740,2755,2427],{"align":2643},[2638,2757,2758,2761],{},[2740,2759,2760],{"align":2643},"5",[2740,2762,2763],{"align":2643},"32",[2638,2765,2766,2769],{},[2740,2767,2768],{"align":2643},"10",[2740,2770,2771],{"align":2643},"1,024",[30,2773,2774],{},"With just 10 flips there are already over a thousand possible combinations. Combinatorics helps us handle those numbers without losing our minds.",[30,2776,2777,2778,827],{},"If you need a refresher on what probability means and how it is measured, check out ",[2779,2780,2784],"a",{"href":2781,"rel":2782},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fprobability-basics\u002F",[2783],"nofollow","Probability Basics",[34,2786,2788],{"id":2787},"case-study-3-flips","Case study: 3 flips",[30,2790,2791,2792,2874],{},"Let's connect everything with a concrete example. We flip a coin 3 times. We know there are ",[123,2793,2795,2817],{"className":2794},[126],[123,2796,2798],{"className":2797},[130],[132,2799,2800],{"xmlns":134},[136,2801,2802,2814],{},[139,2803,2804,2810,2812],{},[2520,2805,2806,2808],{},[142,2807,690],{},[142,2809,271],{},[145,2811,695],{},[142,2813,2427],{},[166,2815,2816],{"encoding":168},"2^3 = 8",[123,2818,2820,2865],{"className":2819,"ariaHidden":147},[173],[123,2821,2823,2827,2856,2859,2862],{"className":2822},[177],[123,2824],{"className":2825,"style":2826},[181],"height:0.8141em;",[123,2828,2830,2833],{"className":2829},[186],[123,2831,690],{"className":2832},[186],[123,2834,2836],{"className":2835},[2548],[123,2837,2839],{"className":2838},[472],[123,2840,2842],{"className":2841},[477],[123,2843,2845],{"className":2844,"style":2826},[481],[123,2846,2847,2850],{"style":2560},[123,2848],{"className":2849,"style":490},[489],[123,2851,2853],{"className":2852},[494,495,496,497],[123,2854,271],{"className":2855},[186,497],[123,2857],{"className":2858,"style":781},[194],[123,2860,695],{"className":2861},[785],[123,2863],{"className":2864,"style":781},[194],[123,2866,2868,2871],{"className":2867},[177],[123,2869],{"className":2870,"style":252},[181],[123,2872,2427],{"className":2873},[186]," possible outcomes. Here they all are:",[99,2876,2879],{"className":2877,"code":2878,"language":104,"meta":105},[102],"OOO    OOX    OXO    XOO    OXX    XOX    XXO    XXX\n",[107,2880,2878],{"__ignoreMap":105},[30,2882,2883,2884,2503],{},"Now let's group these results by the ",[42,2885,2886],{},"number of heads",[2632,2888,2889,2903],{},[2635,2890,2891],{},[2638,2892,2893,2896,2900],{},[2641,2894,2895],{"align":2643},"Heads",[2641,2897,2899],{"align":2898},"left","Combinations",[2641,2901,2902],{"align":2643},"Count",[2735,2904,2905,2914,2923,2932],{},[2638,2906,2907,2909,2912],{},[2740,2908,1349],{"align":2643},[2740,2910,2911],{"align":2898},"XXX",[2740,2913,80],{"align":2643},[2638,2915,2916,2918,2921],{},[2740,2917,80],{"align":2643},[2740,2919,2920],{"align":2898},"OXX, XOX, XXO",[2740,2922,271],{"align":2643},[2638,2924,2925,2927,2930],{},[2740,2926,690],{"align":2643},[2740,2928,2929],{"align":2898},"OOX, OXO, XOO",[2740,2931,271],{"align":2643},[2638,2933,2934,2936,2939],{},[2740,2935,271],{"align":2643},[2740,2937,2938],{"align":2898},"OOO",[2740,2940,80],{"align":2643},[30,2942,2943,2944,3012,3013,3016],{},"Look at the count column: ",[123,2945,2947,2973],{"className":2946},[126],[123,2948,2950],{"className":2949},[130],[132,2951,2952],{"xmlns":134},[136,2953,2954,2970],{},[139,2955,2956,2958,2960,2962,2964,2966,2968],{},[142,2957,80],{},[145,2959,148],{"separator":147},[142,2961,271],{},[145,2963,148],{"separator":147},[142,2965,271],{},[145,2967,148],{"separator":147},[142,2969,80],{},[166,2971,2972],{"encoding":168},"1, 3, 3, 1",[123,2974,2976],{"className":2975,"ariaHidden":147},[173],[123,2977,2979,2982,2985,2988,2991,2994,2997,3000,3003,3006,3009],{"className":2978},[177],[123,2980],{"className":2981,"style":182},[181],[123,2983,80],{"className":2984},[186],[123,2986,148],{"className":2987},[190],[123,2989],{"className":2990,"style":195},[194],[123,2992,271],{"className":2993},[186],[123,2995,148],{"className":2996},[190],[123,2998],{"className":2999,"style":195},[194],[123,3001,271],{"className":3002},[186],[123,3004,148],{"className":3005},[190],[123,3007],{"className":3008,"style":195},[194],[123,3010,80],{"className":3011},[186],". Sound familiar? It is exactly ",[42,3014,3015],{},"row 3"," of Pascal's triangle.",[30,3018,3019,3020,3048,3049,3077,3078,3177,3178,3206,3207,3235],{},"And it is no coincidence. The number of ways to get exactly ",[123,3021,3023,3036],{"className":3022},[126],[123,3024,3026],{"className":3025},[130],[132,3027,3028],{"xmlns":134},[136,3029,3030,3034],{},[139,3031,3032],{},[430,3033,435],{},[166,3035,435],{"encoding":168},[123,3037,3039],{"className":3038,"ariaHidden":147},[173],[123,3040,3042,3045],{"className":3041},[177],[123,3043],{"className":3044,"style":602},[181],[123,3046,435],{"className":3047,"style":505},[186,504]," heads in ",[123,3050,3052,3065],{"className":3051},[126],[123,3053,3055],{"className":3054},[130],[132,3056,3057],{"xmlns":134},[136,3058,3059,3063],{},[139,3060,3061],{},[430,3062,432],{},[166,3064,432],{"encoding":168},[123,3066,3068],{"className":3067,"ariaHidden":147},[173],[123,3069,3071,3074],{"className":3070},[177],[123,3072],{"className":3073,"style":572},[181],[123,3075,432],{"className":3076},[186,504]," flips is precisely ",[123,3079,3081,3102],{"className":3080},[126],[123,3082,3084],{"className":3083},[130],[132,3085,3086],{"xmlns":134},[136,3087,3088,3100],{},[139,3089,3090,3092,3098],{},[145,3091,424],{"fence":147},[426,3093,3094,3096],{"linethickness":428},[430,3095,432],{},[430,3097,435],{},[145,3099,438],{"fence":147},[166,3101,441],{"encoding":168},[123,3103,3105],{"className":3104,"ariaHidden":147},[173],[123,3106,3108,3111],{"className":3107},[177],[123,3109],{"className":3110,"style":451},[181],[123,3112,3114,3120,3171],{"className":3113},[186],[123,3115,3117],{"className":3116,"style":460},[458,459],[123,3118,424],{"className":3119},[464,465],[123,3121,3123],{"className":3122},[426],[123,3124,3126,3163],{"className":3125},[472,473],[123,3127,3129,3160],{"className":3128},[477],[123,3130,3132,3146],{"className":3131,"style":482},[481],[123,3133,3134,3137],{"style":485},[123,3135],{"className":3136,"style":490},[489],[123,3138,3140],{"className":3139},[494,495,496,497],[123,3141,3143],{"className":3142},[186,497],[123,3144,435],{"className":3145,"style":505},[186,504,497],[123,3147,3148,3151],{"style":508},[123,3149],{"className":3150,"style":490},[489],[123,3152,3154],{"className":3153},[494,495,496,497],[123,3155,3157],{"className":3156},[186,497],[123,3158,432],{"className":3159},[186,504,497],[123,3161,525],{"className":3162},[524],[123,3164,3166],{"className":3165},[477],[123,3167,3169],{"className":3168,"style":532},[481],[123,3170],{},[123,3172,3174],{"className":3173,"style":460},[538,459],[123,3175,438],{"className":3176},[464,465]," — the binomial coefficient at position ",[123,3179,3181,3194],{"className":3180},[126],[123,3182,3184],{"className":3183},[130],[132,3185,3186],{"xmlns":134},[136,3187,3188,3192],{},[139,3189,3190],{},[430,3191,435],{},[166,3193,435],{"encoding":168},[123,3195,3197],{"className":3196,"ariaHidden":147},[173],[123,3198,3200,3203],{"className":3199},[177],[123,3201],{"className":3202,"style":602},[181],[123,3204,435],{"className":3205,"style":505},[186,504]," in row ",[123,3208,3210,3223],{"className":3209},[126],[123,3211,3213],{"className":3212},[130],[132,3214,3215],{"xmlns":134},[136,3216,3217,3221],{},[139,3218,3219],{},[430,3220,432],{},[166,3222,432],{"encoding":168},[123,3224,3226],{"className":3225,"ariaHidden":147},[173],[123,3227,3229,3232],{"className":3228},[177],[123,3230],{"className":3231,"style":572},[181],[123,3233,432],{"className":3234},[186,504]," of the triangle.",[395,3237,3239],{"id":3238},"probability-enters-the-scene","Probability enters the scene",[30,3241,3242,3245],{},[42,3243,3244],{},"Probability"," measures how likely an event is to occur. It is calculated as:",[30,3247,3249],{"className":3248},[409],[123,3250,3252,3277],{"className":3251},[126],[123,3253,3255],{"className":3254},[130],[132,3256,3257],{"xmlns":134},[136,3258,3259,3274],{},[139,3260,3261,3264,3266],{},[430,3262,3263],{},"P",[145,3265,695],{},[426,3267,3268,3271],{},[1901,3269,3270],{},"favorable outcomes",[1901,3272,3273],{},"total outcomes",[166,3275,3276],{"encoding":168},"P = \\frac{\\text{favorable outcomes}}{\\text{total outcomes}}",[123,3278,3280,3300],{"className":3279,"ariaHidden":147},[173],[123,3281,3283,3287,3291,3294,3297],{"className":3282},[177],[123,3284],{"className":3285,"style":3286},[181],"height:0.6833em;",[123,3288,3263],{"className":3289,"style":3290},[186,504],"margin-right:0.1389em;",[123,3292],{"className":3293,"style":781},[194],[123,3295,695],{"className":3296},[785],[123,3298],{"className":3299,"style":781},[194],[123,3301,3303,3306],{"className":3302},[177],[123,3304],{"className":3305,"style":1560},[181],[123,3307,3309,3312,3377],{"className":3308},[186],[123,3310],{"className":3311},[458,992],[123,3313,3315],{"className":3314},[426],[123,3316,3318,3369],{"className":3317},[472,473],[123,3319,3321,3366],{"className":3320},[477],[123,3322,3324,3341,3349],{"className":3323,"style":1005},[481],[123,3325,3326,3329],{"style":1008},[123,3327],{"className":3328,"style":1012},[489],[123,3330,3332],{"className":3331},[494,495,496,497],[123,3333,3335],{"className":3334},[186,497],[123,3336,3338],{"className":3337},[186,104,497],[123,3339,3273],{"className":3340},[186,497],[123,3342,3343,3346],{"style":1046},[123,3344],{"className":3345,"style":1012},[489],[123,3347],{"className":3348,"style":1054},[1053],[123,3350,3351,3354],{"style":1057},[123,3352],{"className":3353,"style":1012},[489],[123,3355,3357],{"className":3356},[494,495,496,497],[123,3358,3360],{"className":3359},[186,497],[123,3361,3363],{"className":3362},[186,104,497],[123,3364,3270],{"className":3365},[186,497],[123,3367,525],{"className":3368},[524],[123,3370,3372],{"className":3371},[477],[123,3373,3375],{"className":3374,"style":532},[481],[123,3376],{},[123,3378],{"className":3379},[538,992],[30,3381,3382,3383,3411],{},"If the coin is fair, each of the 8 outcomes is equally likely. So the probability of getting exactly ",[123,3384,3386,3399],{"className":3385},[126],[123,3387,3389],{"className":3388},[130],[132,3390,3391],{"xmlns":134},[136,3392,3393,3397],{},[139,3394,3395],{},[430,3396,435],{},[166,3398,435],{"encoding":168},[123,3400,3402],{"className":3401,"ariaHidden":147},[173],[123,3403,3405,3408],{"className":3404},[177],[123,3406],{"className":3407,"style":602},[181],[123,3409,435],{"className":3410,"style":505},[186,504]," heads in 3 flips is:",[30,3413,3415],{"className":3414},[409],[123,3416,3418,3481],{"className":3417},[126],[123,3419,3421],{"className":3420},[130],[132,3422,3423],{"xmlns":134},[136,3424,3425,3478],{},[139,3426,3427,3429,3431,3433,3436,3438,3440,3460,3462],{},[430,3428,3263],{},[145,3430,424],{"stretchy":868},[430,3432,435],{},[1901,3434,3435],{}," heads",[145,3437,438],{"stretchy":868},[145,3439,695],{},[426,3441,3442,3454],{},[139,3443,3444,3446,3452],{},[145,3445,424],{"fence":147},[426,3447,3448,3450],{"linethickness":428},[142,3449,271],{},[430,3451,435],{},[145,3453,438],{"fence":147},[2520,3455,3456,3458],{},[142,3457,690],{},[142,3459,271],{},[145,3461,695],{},[426,3463,3464,3476],{},[139,3465,3466,3468,3474],{},[145,3467,424],{"fence":147},[426,3469,3470,3472],{"linethickness":428},[142,3471,271],{},[430,3473,435],{},[145,3475,438],{"fence":147},[142,3477,2427],{},[166,3479,3480],{"encoding":168},"P(k \\text{ heads}) = \\frac{\\binom{3}{k}}{2^3} = \\frac{\\binom{3}{k}}{8}",[123,3482,3484,3518,3709],{"className":3483,"ariaHidden":147},[173],[123,3485,3487,3491,3494,3497,3500,3506,3509,3512,3515],{"className":3486},[177],[123,3488],{"className":3489,"style":3490},[181],"height:1em;vertical-align:-0.25em;",[123,3492,3263],{"className":3493,"style":3290},[186,504],[123,3495,424],{"className":3496},[458],[123,3498,435],{"className":3499,"style":505},[186,504],[123,3501,3503],{"className":3502},[186,104],[123,3504,3435],{"className":3505},[186],[123,3507,438],{"className":3508},[538],[123,3510],{"className":3511,"style":781},[194],[123,3513,695],{"className":3514},[785],[123,3516],{"className":3517,"style":781},[194],[123,3519,3521,3525,3700,3703,3706],{"className":3520},[177],[123,3522],{"className":3523,"style":3524},[181],"height:1.855em;vertical-align:-0.345em;",[123,3526,3528,3531,3697],{"className":3527},[186],[123,3529],{"className":3530},[458,992],[123,3532,3534],{"className":3533},[426],[123,3535,3537,3689],{"className":3536},[472,473],[123,3538,3540,3686],{"className":3539},[477],[123,3541,3544,3588,3596],{"className":3542,"style":3543},[481],"height:1.51em;",[123,3545,3546,3549],{"style":1008},[123,3547],{"className":3548,"style":1012},[489],[123,3550,3552],{"className":3551},[494,495,496,497],[123,3553,3555],{"className":3554},[186,497],[123,3556,3558,3561],{"className":3557},[186,497],[123,3559,690],{"className":3560},[186,497],[123,3562,3564],{"className":3563},[2548],[123,3565,3567],{"className":3566},[472],[123,3568,3570],{"className":3569},[477],[123,3571,3574],{"className":3572,"style":3573},[481],"height:0.7463em;",[123,3575,3577,3581],{"style":3576},"top:-2.786em;margin-right:0.0714em;",[123,3578],{"className":3579,"style":3580},[489],"height:2.5em;",[123,3582,3585],{"className":3583},[494,3584,465,497],"reset-size3",[123,3586,271],{"className":3587},[186,497],[123,3589,3590,3593],{"style":1046},[123,3591],{"className":3592,"style":1012},[489],[123,3594],{"className":3595,"style":1054},[1053],[123,3597,3599,3602],{"style":3598},"top:-3.7125em;",[123,3600],{"className":3601,"style":1012},[489],[123,3603,3605],{"className":3604},[494,495,496,497],[123,3606,3608],{"className":3607},[186,497],[123,3609,3611,3622,3677],{"className":3610},[186,497],[123,3612,3616],{"className":3613,"style":3615},[458,494,3584,3614,497,459],"size6","top:0.075em;",[123,3617,3619],{"className":3618},[464,465,497],[123,3620,424],{"className":3621},[497],[123,3623,3625],{"className":3624},[426],[123,3626,3628,3668],{"className":3627},[472,473],[123,3629,3631,3665],{"className":3630},[477],[123,3632,3635,3650],{"className":3633,"style":3634},[481],"height:0.9313em;",[123,3636,3638,3641],{"style":3637},"top:-2.156em;",[123,3639],{"className":3640,"style":3580},[489],[123,3642,3644],{"className":3643},[494,3584,465,497],[123,3645,3647],{"className":3646},[186,497],[123,3648,435],{"className":3649,"style":505},[186,504,497],[123,3651,3653,3656],{"style":3652},"top:-2.971em;",[123,3654],{"className":3655,"style":3580},[489],[123,3657,3659],{"className":3658},[494,3584,465,497],[123,3660,3662],{"className":3661},[186,497],[123,3663,271],{"className":3664},[186,497],[123,3666,525],{"className":3667},[524],[123,3669,3671],{"className":3670},[477],[123,3672,3675],{"className":3673,"style":3674},[481],"height:0.344em;",[123,3676],{},[123,3678,3680],{"className":3679,"style":3615},[538,494,3584,3614,497,459],[123,3681,3683],{"className":3682},[464,465,497],[123,3684,438],{"className":3685},[497],[123,3687,525],{"className":3688},[524],[123,3690,3692],{"className":3691},[477],[123,3693,3695],{"className":3694,"style":532},[481],[123,3696],{},[123,3698],{"className":3699},[538,992],[123,3701],{"className":3702,"style":781},[194],[123,3704,695],{"className":3705},[785],[123,3707],{"className":3708,"style":781},[194],[123,3710,3712,3715],{"className":3711},[177],[123,3713],{"className":3714,"style":3524},[181],[123,3716,3718,3721,3849],{"className":3717},[186],[123,3719],{"className":3720},[458,992],[123,3722,3724],{"className":3723},[426],[123,3725,3727,3841],{"className":3726},[472,473],[123,3728,3730,3838],{"className":3729},[477],[123,3731,3733,3747,3755],{"className":3732,"style":3543},[481],[123,3734,3735,3738],{"style":1008},[123,3736],{"className":3737,"style":1012},[489],[123,3739,3741],{"className":3740},[494,495,496,497],[123,3742,3744],{"className":3743},[186,497],[123,3745,2427],{"className":3746},[186,497],[123,3748,3749,3752],{"style":1046},[123,3750],{"className":3751,"style":1012},[489],[123,3753],{"className":3754,"style":1054},[1053],[123,3756,3757,3760],{"style":3598},[123,3758],{"className":3759,"style":1012},[489],[123,3761,3763],{"className":3762},[494,495,496,497],[123,3764,3766],{"className":3765},[186,497],[123,3767,3769,3778,3829],{"className":3768},[186,497],[123,3770,3772],{"className":3771,"style":3615},[458,494,3584,3614,497,459],[123,3773,3775],{"className":3774},[464,465,497],[123,3776,424],{"className":3777},[497],[123,3779,3781],{"className":3780},[426],[123,3782,3784,3821],{"className":3783},[472,473],[123,3785,3787,3818],{"className":3786},[477],[123,3788,3790,3804],{"className":3789,"style":3634},[481],[123,3791,3792,3795],{"style":3637},[123,3793],{"className":3794,"style":3580},[489],[123,3796,3798],{"className":3797},[494,3584,465,497],[123,3799,3801],{"className":3800},[186,497],[123,3802,435],{"className":3803,"style":505},[186,504,497],[123,3805,3806,3809],{"style":3652},[123,3807],{"className":3808,"style":3580},[489],[123,3810,3812],{"className":3811},[494,3584,465,497],[123,3813,3815],{"className":3814},[186,497],[123,3816,271],{"className":3817},[186,497],[123,3819,525],{"className":3820},[524],[123,3822,3824],{"className":3823},[477],[123,3825,3827],{"className":3826,"style":3674},[481],[123,3828],{},[123,3830,3832],{"className":3831,"style":3615},[538,494,3584,3614,497,459],[123,3833,3835],{"className":3834},[464,465,497],[123,3836,438],{"className":3837},[497],[123,3839,525],{"className":3840},[524],[123,3842,3844],{"className":3843},[477],[123,3845,3847],{"className":3846,"style":532},[481],[123,3848],{},[123,3850],{"className":3851},[538,992],[30,3853,3854],{},"Let's compute each case:",[2632,3856,3857,3999],{},[2635,3858,3859],{},[2638,3860,3861,3892,3994,3996],{},[2641,3862,3863,3891],{"align":2643},[123,3864,3866,3879],{"className":3865},[126],[123,3867,3869],{"className":3868},[130],[132,3870,3871],{"xmlns":134},[136,3872,3873,3877],{},[139,3874,3875],{},[430,3876,435],{},[166,3878,435],{"encoding":168},[123,3880,3882],{"className":3881,"ariaHidden":147},[173],[123,3883,3885,3888],{"className":3884},[177],[123,3886],{"className":3887,"style":602},[181],[123,3889,435],{"className":3890,"style":505},[186,504]," heads",[2641,3893,3894],{"align":2643},[123,3895,3897,3919],{"className":3896},[126],[123,3898,3900],{"className":3899},[130],[132,3901,3902],{"xmlns":134},[136,3903,3904,3916],{},[139,3905,3906,3908,3914],{},[145,3907,424],{"fence":147},[426,3909,3910,3912],{"linethickness":428},[142,3911,271],{},[430,3913,435],{},[145,3915,438],{"fence":147},[166,3917,3918],{"encoding":168},"\\binom{3}{k}",[123,3920,3922],{"className":3921,"ariaHidden":147},[173],[123,3923,3925,3928],{"className":3924},[177],[123,3926],{"className":3927,"style":710},[181],[123,3929,3931,3937,3988],{"className":3930},[186],[123,3932,3934],{"className":3933,"style":460},[458,459],[123,3935,424],{"className":3936},[464,465],[123,3938,3940],{"className":3939},[426],[123,3941,3943,3980],{"className":3942},[472,473],[123,3944,3946,3977],{"className":3945},[477],[123,3947,3949,3963],{"className":3948,"style":732},[481],[123,3950,3951,3954],{"style":485},[123,3952],{"className":3953,"style":490},[489],[123,3955,3957],{"className":3956},[494,495,496,497],[123,3958,3960],{"className":3959},[186,497],[123,3961,435],{"className":3962,"style":505},[186,504,497],[123,3964,3965,3968],{"style":508},[123,3966],{"className":3967,"style":490},[489],[123,3969,3971],{"className":3970},[494,495,496,497],[123,3972,3974],{"className":3973},[186,497],[123,3975,271],{"className":3976},[186,497],[123,3978,525],{"className":3979},[524],[123,3981,3983],{"className":3982},[477],[123,3984,3986],{"className":3985,"style":532},[481],[123,3987],{},[123,3989,3991],{"className":3990,"style":460},[538,459],[123,3992,438],{"className":3993},[464,465],[2641,3995,3244],{"align":2643},[2641,3997,3998],{"align":2643},"Percentage",[2735,4000,4001,4133,4265,4394],{},[2638,4002,4003,4005,4007,4130],{},[2740,4004,1349],{"align":2643},[2740,4006,80],{"align":2643},[2740,4008,4009],{"align":2643},[123,4010,4012,4035],{"className":4011},[126],[123,4013,4015],{"className":4014},[130],[132,4016,4017],{"xmlns":134},[136,4018,4019,4032],{},[139,4020,4021,4027,4029],{},[426,4022,4023,4025],{},[142,4024,80],{},[142,4026,2427],{},[145,4028,695],{},[142,4030,4031],{},"0.125",[166,4033,4034],{"encoding":168},"\\frac{1}{8} = 0.125",[123,4036,4038,4121],{"className":4037,"ariaHidden":147},[173],[123,4039,4041,4044,4112,4115,4118],{"className":4040},[177],[123,4042],{"className":4043,"style":1751},[181],[123,4045,4047,4050,4109],{"className":4046},[186],[123,4048],{"className":4049},[458,992],[123,4051,4053],{"className":4052},[426],[123,4054,4056,4101],{"className":4055},[472,473],[123,4057,4059,4098],{"className":4058},[477],[123,4060,4062,4076,4084],{"className":4061,"style":1678},[481],[123,4063,4064,4067],{"style":1008},[123,4065],{"className":4066,"style":1012},[489],[123,4068,4070],{"className":4069},[494,495,496,497],[123,4071,4073],{"className":4072},[186,497],[123,4074,2427],{"className":4075},[186,497],[123,4077,4078,4081],{"style":1046},[123,4079],{"className":4080,"style":1012},[489],[123,4082],{"className":4083,"style":1054},[1053],[123,4085,4086,4089],{"style":1057},[123,4087],{"className":4088,"style":1012},[489],[123,4090,4092],{"className":4091},[494,495,496,497],[123,4093,4095],{"className":4094},[186,497],[123,4096,80],{"className":4097},[186,497],[123,4099,525],{"className":4100},[524],[123,4102,4104],{"className":4103},[477],[123,4105,4107],{"className":4106,"style":532},[481],[123,4108],{},[123,4110],{"className":4111},[538,992],[123,4113],{"className":4114,"style":781},[194],[123,4116,695],{"className":4117},[785],[123,4119],{"className":4120,"style":781},[194],[123,4122,4124,4127],{"className":4123},[177],[123,4125],{"className":4126,"style":252},[181],[123,4128,4031],{"className":4129},[186],[2740,4131,4132],{"align":2643},"12.5%",[2638,4134,4135,4137,4139,4262],{},[2740,4136,80],{"align":2643},[2740,4138,271],{"align":2643},[2740,4140,4141],{"align":2643},[123,4142,4144,4167],{"className":4143},[126],[123,4145,4147],{"className":4146},[130],[132,4148,4149],{"xmlns":134},[136,4150,4151,4164],{},[139,4152,4153,4159,4161],{},[426,4154,4155,4157],{},[142,4156,271],{},[142,4158,2427],{},[145,4160,695],{},[142,4162,4163],{},"0.375",[166,4165,4166],{"encoding":168},"\\frac{3}{8} = 0.375",[123,4168,4170,4253],{"className":4169,"ariaHidden":147},[173],[123,4171,4173,4176,4244,4247,4250],{"className":4172},[177],[123,4174],{"className":4175,"style":1751},[181],[123,4177,4179,4182,4241],{"className":4178},[186],[123,4180],{"className":4181},[458,992],[123,4183,4185],{"className":4184},[426],[123,4186,4188,4233],{"className":4187},[472,473],[123,4189,4191,4230],{"className":4190},[477],[123,4192,4194,4208,4216],{"className":4193,"style":1678},[481],[123,4195,4196,4199],{"style":1008},[123,4197],{"className":4198,"style":1012},[489],[123,4200,4202],{"className":4201},[494,495,496,497],[123,4203,4205],{"className":4204},[186,497],[123,4206,2427],{"className":4207},[186,497],[123,4209,4210,4213],{"style":1046},[123,4211],{"className":4212,"style":1012},[489],[123,4214],{"className":4215,"style":1054},[1053],[123,4217,4218,4221],{"style":1057},[123,4219],{"className":4220,"style":1012},[489],[123,4222,4224],{"className":4223},[494,495,496,497],[123,4225,4227],{"className":4226},[186,497],[123,4228,271],{"className":4229},[186,497],[123,4231,525],{"className":4232},[524],[123,4234,4236],{"className":4235},[477],[123,4237,4239],{"className":4238,"style":532},[481],[123,4240],{},[123,4242],{"className":4243},[538,992],[123,4245],{"className":4246,"style":781},[194],[123,4248,695],{"className":4249},[785],[123,4251],{"className":4252,"style":781},[194],[123,4254,4256,4259],{"className":4255},[177],[123,4257],{"className":4258,"style":252},[181],[123,4260,4163],{"className":4261},[186],[2740,4263,4264],{"align":2643},"37.5%",[2638,4266,4267,4269,4271,4392],{},[2740,4268,690],{"align":2643},[2740,4270,271],{"align":2643},[2740,4272,4273],{"align":2643},[123,4274,4276,4297],{"className":4275},[126],[123,4277,4279],{"className":4278},[130],[132,4280,4281],{"xmlns":134},[136,4282,4283,4295],{},[139,4284,4285,4291,4293],{},[426,4286,4287,4289],{},[142,4288,271],{},[142,4290,2427],{},[145,4292,695],{},[142,4294,4163],{},[166,4296,4166],{"encoding":168},[123,4298,4300,4383],{"className":4299,"ariaHidden":147},[173],[123,4301,4303,4306,4374,4377,4380],{"className":4302},[177],[123,4304],{"className":4305,"style":1751},[181],[123,4307,4309,4312,4371],{"className":4308},[186],[123,4310],{"className":4311},[458,992],[123,4313,4315],{"className":4314},[426],[123,4316,4318,4363],{"className":4317},[472,473],[123,4319,4321,4360],{"className":4320},[477],[123,4322,4324,4338,4346],{"className":4323,"style":1678},[481],[123,4325,4326,4329],{"style":1008},[123,4327],{"className":4328,"style":1012},[489],[123,4330,4332],{"className":4331},[494,495,496,497],[123,4333,4335],{"className":4334},[186,497],[123,4336,2427],{"className":4337},[186,497],[123,4339,4340,4343],{"style":1046},[123,4341],{"className":4342,"style":1012},[489],[123,4344],{"className":4345,"style":1054},[1053],[123,4347,4348,4351],{"style":1057},[123,4349],{"className":4350,"style":1012},[489],[123,4352,4354],{"className":4353},[494,495,496,497],[123,4355,4357],{"className":4356},[186,497],[123,4358,271],{"className":4359},[186,497],[123,4361,525],{"className":4362},[524],[123,4364,4366],{"className":4365},[477],[123,4367,4369],{"className":4368,"style":532},[481],[123,4370],{},[123,4372],{"className":4373},[538,992],[123,4375],{"className":4376,"style":781},[194],[123,4378,695],{"className":4379},[785],[123,4381],{"className":4382,"style":781},[194],[123,4384,4386,4389],{"className":4385},[177],[123,4387],{"className":4388,"style":252},[181],[123,4390,4163],{"className":4391},[186],[2740,4393,4264],{"align":2643},[2638,4395,4396,4398,4400,4521],{},[2740,4397,271],{"align":2643},[2740,4399,80],{"align":2643},[2740,4401,4402],{"align":2643},[123,4403,4405,4426],{"className":4404},[126],[123,4406,4408],{"className":4407},[130],[132,4409,4410],{"xmlns":134},[136,4411,4412,4424],{},[139,4413,4414,4420,4422],{},[426,4415,4416,4418],{},[142,4417,80],{},[142,4419,2427],{},[145,4421,695],{},[142,4423,4031],{},[166,4425,4034],{"encoding":168},[123,4427,4429,4512],{"className":4428,"ariaHidden":147},[173],[123,4430,4432,4435,4503,4506,4509],{"className":4431},[177],[123,4433],{"className":4434,"style":1751},[181],[123,4436,4438,4441,4500],{"className":4437},[186],[123,4439],{"className":4440},[458,992],[123,4442,4444],{"className":4443},[426],[123,4445,4447,4492],{"className":4446},[472,473],[123,4448,4450,4489],{"className":4449},[477],[123,4451,4453,4467,4475],{"className":4452,"style":1678},[481],[123,4454,4455,4458],{"style":1008},[123,4456],{"className":4457,"style":1012},[489],[123,4459,4461],{"className":4460},[494,495,496,497],[123,4462,4464],{"className":4463},[186,497],[123,4465,2427],{"className":4466},[186,497],[123,4468,4469,4472],{"style":1046},[123,4470],{"className":4471,"style":1012},[489],[123,4473],{"className":4474,"style":1054},[1053],[123,4476,4477,4480],{"style":1057},[123,4478],{"className":4479,"style":1012},[489],[123,4481,4483],{"className":4482},[494,495,496,497],[123,4484,4486],{"className":4485},[186,497],[123,4487,80],{"className":4488},[186,497],[123,4490,525],{"className":4491},[524],[123,4493,4495],{"className":4494},[477],[123,4496,4498],{"className":4497,"style":532},[481],[123,4499],{},[123,4501],{"className":4502},[538,992],[123,4504],{"className":4505,"style":781},[194],[123,4507,695],{"className":4508},[785],[123,4510],{"className":4511,"style":781},[194],[123,4513,4515,4518],{"className":4514},[177],[123,4516],{"className":4517,"style":252},[181],[123,4519,4031],{"className":4520},[186],[2740,4522,4132],{"align":2643},[30,4524,4525,4526,4643,4644,4672],{},"If we add all probabilities: ",[123,4527,4529,4559],{"className":4528},[126],[123,4530,4532],{"className":4531},[130],[132,4533,4534],{"xmlns":134},[136,4535,4536,4556],{},[139,4537,4538,4540,4542,4544,4546,4548,4550,4552,4554],{},[142,4539,4031],{},[145,4541,274],{},[142,4543,4163],{},[145,4545,274],{},[142,4547,4163],{},[145,4549,274],{},[142,4551,4031],{},[145,4553,695],{},[142,4555,80],{},[166,4557,4558],{"encoding":168},"0.125 + 0.375 + 0.375 + 0.125 = 1",[123,4560,4562,4580,4598,4616,4634],{"className":4561,"ariaHidden":147},[173],[123,4563,4565,4568,4571,4574,4577],{"className":4564},[177],[123,4566],{"className":4567,"style":289},[181],[123,4569,4031],{"className":4570},[186],[123,4572],{"className":4573,"style":296},[194],[123,4575,274],{"className":4576},[300],[123,4578],{"className":4579,"style":296},[194],[123,4581,4583,4586,4589,4592,4595],{"className":4582},[177],[123,4584],{"className":4585,"style":289},[181],[123,4587,4163],{"className":4588},[186],[123,4590],{"className":4591,"style":296},[194],[123,4593,274],{"className":4594},[300],[123,4596],{"className":4597,"style":296},[194],[123,4599,4601,4604,4607,4610,4613],{"className":4600},[177],[123,4602],{"className":4603,"style":289},[181],[123,4605,4163],{"className":4606},[186],[123,4608],{"className":4609,"style":296},[194],[123,4611,274],{"className":4612},[300],[123,4614],{"className":4615,"style":296},[194],[123,4617,4619,4622,4625,4628,4631],{"className":4618},[177],[123,4620],{"className":4621,"style":252},[181],[123,4623,4031],{"className":4624},[186],[123,4626],{"className":4627,"style":781},[194],[123,4629,695],{"className":4630},[785],[123,4632],{"className":4633,"style":781},[194],[123,4635,4637,4640],{"className":4636},[177],[123,4638],{"className":4639,"style":252},[181],[123,4641,80],{"className":4642},[186],". The sum always equals ",[123,4645,4647,4660],{"className":4646},[126],[123,4648,4650],{"className":4649},[130],[132,4651,4652],{"xmlns":134},[136,4653,4654,4658],{},[139,4655,4656],{},[142,4657,80],{},[166,4659,80],{"encoding":168},[123,4661,4663],{"className":4662,"ariaHidden":147},[173],[123,4664,4666,4669],{"className":4665},[177],[123,4667],{"className":4668,"style":252},[181],[123,4670,80],{"className":4671},[186],", because together all cases cover 100% of the possibilities.",[30,4674,4675],{},"The general formula we have just discovered works for any number of flips:",[30,4677,4679],{"className":4678},[409],[123,4680,4682,4732],{"className":4681},[126],[123,4683,4685],{"className":4684},[130],[132,4686,4687],{"xmlns":134},[136,4688,4689,4729],{},[139,4690,4691,4693,4695,4697,4700,4702,4705,4707,4709],{},[430,4692,3263],{},[145,4694,424],{"stretchy":868},[430,4696,435],{},[1901,4698,4699],{}," heads in ",[430,4701,432],{},[1901,4703,4704],{}," flips",[145,4706,438],{"stretchy":868},[145,4708,695],{},[426,4710,4711,4723],{},[139,4712,4713,4715,4721],{},[145,4714,424],{"fence":147},[426,4716,4717,4719],{"linethickness":428},[430,4718,432],{},[430,4720,435],{},[145,4722,438],{"fence":147},[2520,4724,4725,4727],{},[142,4726,690],{},[430,4728,432],{},[166,4730,4731],{"encoding":168},"P(k \\text{ heads in } n \\text{ flips}) = \\frac{\\binom{n}{k}}{2^n}",[123,4733,4735,4777],{"className":4734,"ariaHidden":147},[173],[123,4736,4738,4741,4744,4747,4750,4756,4759,4765,4768,4771,4774],{"className":4737},[177],[123,4739],{"className":4740,"style":3490},[181],[123,4742,3263],{"className":4743,"style":3290},[186,504],[123,4745,424],{"className":4746},[458],[123,4748,435],{"className":4749,"style":505},[186,504],[123,4751,4753],{"className":4752},[186,104],[123,4754,4699],{"className":4755},[186],[123,4757,432],{"className":4758},[186,504],[123,4760,4762],{"className":4761},[186,104],[123,4763,4704],{"className":4764},[186],[123,4766,438],{"className":4767},[538],[123,4769],{"className":4770,"style":781},[194],[123,4772,695],{"className":4773},[785],[123,4775],{"className":4776,"style":781},[194],[123,4778,4780,4783],{"className":4779},[177],[123,4781],{"className":4782,"style":3524},[181],[123,4784,4786,4789,4945],{"className":4785},[186],[123,4787],{"className":4788},[458,992],[123,4790,4792],{"className":4791},[426],[123,4793,4795,4937],{"className":4794},[472,473],[123,4796,4798,4934],{"className":4797},[477],[123,4799,4801,4842,4850],{"className":4800,"style":3543},[481],[123,4802,4803,4806],{"style":1008},[123,4804],{"className":4805,"style":1012},[489],[123,4807,4809],{"className":4808},[494,495,496,497],[123,4810,4812],{"className":4811},[186,497],[123,4813,4815,4818],{"className":4814},[186,497],[123,4816,690],{"className":4817},[186,497],[123,4819,4821],{"className":4820},[2548],[123,4822,4824],{"className":4823},[472],[123,4825,4827],{"className":4826},[477],[123,4828,4831],{"className":4829,"style":4830},[481],"height:0.5935em;",[123,4832,4833,4836],{"style":3576},[123,4834],{"className":4835,"style":3580},[489],[123,4837,4839],{"className":4838},[494,3584,465,497],[123,4840,432],{"className":4841},[186,504,497],[123,4843,4844,4847],{"style":1046},[123,4845],{"className":4846,"style":1012},[489],[123,4848],{"className":4849,"style":1054},[1053],[123,4851,4852,4855],{"style":3598},[123,4853],{"className":4854,"style":1012},[489],[123,4856,4858],{"className":4857},[494,495,496,497],[123,4859,4861],{"className":4860},[186,497],[123,4862,4864,4873,4925],{"className":4863},[186,497],[123,4865,4867],{"className":4866,"style":3615},[458,494,3584,3614,497,459],[123,4868,4870],{"className":4869},[464,465,497],[123,4871,424],{"className":4872},[497],[123,4874,4876],{"className":4875},[426],[123,4877,4879,4917],{"className":4878},[472,473],[123,4880,4882,4914],{"className":4881},[477],[123,4883,4886,4900],{"className":4884,"style":4885},[481],"height:0.7785em;",[123,4887,4888,4891],{"style":3637},[123,4889],{"className":4890,"style":3580},[489],[123,4892,4894],{"className":4893},[494,3584,465,497],[123,4895,4897],{"className":4896},[186,497],[123,4898,435],{"className":4899,"style":505},[186,504,497],[123,4901,4902,4905],{"style":3652},[123,4903],{"className":4904,"style":3580},[489],[123,4906,4908],{"className":4907},[494,3584,465,497],[123,4909,4911],{"className":4910},[186,497],[123,4912,432],{"className":4913},[186,504,497],[123,4915,525],{"className":4916},[524],[123,4918,4920],{"className":4919},[477],[123,4921,4923],{"className":4922,"style":3674},[481],[123,4924],{},[123,4926,4928],{"className":4927,"style":3615},[538,494,3584,3614,497,459],[123,4929,4931],{"className":4930},[464,465,497],[123,4932,438],{"className":4933},[497],[123,4935,525],{"className":4936},[524],[123,4938,4940],{"className":4939},[477],[123,4941,4943],{"className":4942,"style":532},[481],[123,4944],{},[123,4946],{"className":4947},[538,992],[34,4949,4951],{"id":4950},"case-study-5-flips","Case study: 5 flips",[30,4953,4954],{},"Let's raise the stakes. With 5 flips, the corresponding row of the triangle is:",[30,4956,4958],{"className":4957},[409],[123,4959,4961,4996],{"className":4960},[126],[123,4962,4964],{"className":4963},[130],[132,4965,4966],{"xmlns":134},[136,4967,4968,4993],{},[139,4969,4970,4972,4975,4977,4979,4981,4983,4985,4987,4989,4991],{},[142,4971,80],{},[194,4973],{"width":4974},"1em",[142,4976,2760],{},[194,4978],{"width":4974},[142,4980,2768],{},[194,4982],{"width":4974},[142,4984,2768],{},[194,4986],{"width":4974},[142,4988,2760],{},[194,4990],{"width":4974},[142,4992,80],{},[166,4994,4995],{"encoding":168},"1 \\quad 5 \\quad 10 \\quad 10 \\quad 5 \\quad 1",[123,4997,4999],{"className":4998,"ariaHidden":147},[173],[123,5000,5002,5005,5008,5012,5015,5018,5021,5024,5027,5030,5033,5036],{"className":5001},[177],[123,5003],{"className":5004,"style":252},[181],[123,5006,80],{"className":5007},[186],[123,5009],{"className":5010,"style":5011},[194],"margin-right:1em;",[123,5013,2760],{"className":5014},[186],[123,5016],{"className":5017,"style":5011},[194],[123,5019,2768],{"className":5020},[186],[123,5022],{"className":5023,"style":5011},[194],[123,5025,2768],{"className":5026},[186],[123,5028],{"className":5029,"style":5011},[194],[123,5031,2760],{"className":5032},[186],[123,5034],{"className":5035,"style":5011},[194],[123,5037,80],{"className":5038},[186],[30,5040,5041,5042,827],{},"and the total number of possible outcomes is ",[123,5043,5045,5067],{"className":5044},[126],[123,5046,5048],{"className":5047},[130],[132,5049,5050],{"xmlns":134},[136,5051,5052,5064],{},[139,5053,5054,5060,5062],{},[2520,5055,5056,5058],{},[142,5057,690],{},[142,5059,2760],{},[145,5061,695],{},[142,5063,2763],{},[166,5065,5066],{"encoding":168},"2^5 = 32",[123,5068,5070,5114],{"className":5069,"ariaHidden":147},[173],[123,5071,5073,5076,5105,5108,5111],{"className":5072},[177],[123,5074],{"className":5075,"style":2826},[181],[123,5077,5079,5082],{"className":5078},[186],[123,5080,690],{"className":5081},[186],[123,5083,5085],{"className":5084},[2548],[123,5086,5088],{"className":5087},[472],[123,5089,5091],{"className":5090},[477],[123,5092,5094],{"className":5093,"style":2826},[481],[123,5095,5096,5099],{"style":2560},[123,5097],{"className":5098,"style":490},[489],[123,5100,5102],{"className":5101},[494,495,496,497],[123,5103,2760],{"className":5104},[186,497],[123,5106],{"className":5107,"style":781},[194],[123,5109,695],{"className":5110},[785],[123,5112],{"className":5113,"style":781},[194],[123,5115,5117,5120],{"className":5116},[177],[123,5118],{"className":5119,"style":252},[181],[123,5121,2763],{"className":5122},[186],[30,5124,5125,5126,5129,5130,5254],{},"Let's look at some concrete examples. How many ways can we get exactly ",[42,5127,5128],{},"2 heads","? The triangle tells us ",[123,5131,5133,5161],{"className":5132},[126],[123,5134,5136],{"className":5135},[130],[132,5137,5138],{"xmlns":134},[136,5139,5140,5158],{},[139,5141,5142,5154,5156],{},[139,5143,5144,5146,5152],{},[145,5145,424],{"fence":147},[426,5147,5148,5150],{"linethickness":428},[142,5149,2760],{},[142,5151,690],{},[145,5153,438],{"fence":147},[145,5155,695],{},[142,5157,2768],{},[166,5159,5160],{"encoding":168},"\\binom{5}{2} = 10",[123,5162,5164,5245],{"className":5163,"ariaHidden":147},[173],[123,5165,5167,5170,5236,5239,5242],{"className":5166},[177],[123,5168],{"className":5169,"style":710},[181],[123,5171,5173,5179,5230],{"className":5172},[186],[123,5174,5176],{"className":5175,"style":460},[458,459],[123,5177,424],{"className":5178},[464,465],[123,5180,5182],{"className":5181},[426],[123,5183,5185,5222],{"className":5184},[472,473],[123,5186,5188,5219],{"className":5187},[477],[123,5189,5191,5205],{"className":5190,"style":732},[481],[123,5192,5193,5196],{"style":485},[123,5194],{"className":5195,"style":490},[489],[123,5197,5199],{"className":5198},[494,495,496,497],[123,5200,5202],{"className":5201},[186,497],[123,5203,690],{"className":5204},[186,497],[123,5206,5207,5210],{"style":508},[123,5208],{"className":5209,"style":490},[489],[123,5211,5213],{"className":5212},[494,495,496,497],[123,5214,5216],{"className":5215},[186,497],[123,5217,2760],{"className":5218},[186,497],[123,5220,525],{"className":5221},[524],[123,5223,5225],{"className":5224},[477],[123,5226,5228],{"className":5227,"style":532},[481],[123,5229],{},[123,5231,5233],{"className":5232,"style":460},[538,459],[123,5234,438],{"className":5235},[464,465],[123,5237],{"className":5238,"style":781},[194],[123,5240,695],{"className":5241},[785],[123,5243],{"className":5244,"style":781},[194],[123,5246,5248,5251],{"className":5247},[177],[123,5249],{"className":5250,"style":252},[181],[123,5252,2768],{"className":5253},[186],". Here are all 10 combinations:",[99,5256,5259],{"className":5257,"code":5258,"language":104,"meta":105},[102],"OOXXX   OXOXX   OXXOX   OXXXO   XOOXX\nXOXOX   XOXXO   XXOOX   XXOXO   XXXOO\n",[107,5260,5258],{"__ignoreMap":105},[30,5262,5263,5264,5267,5268,5392],{},"And exactly ",[42,5265,5266],{},"3 heads","? Also ",[123,5269,5271,5299],{"className":5270},[126],[123,5272,5274],{"className":5273},[130],[132,5275,5276],{"xmlns":134},[136,5277,5278,5296],{},[139,5279,5280,5292,5294],{},[139,5281,5282,5284,5290],{},[145,5283,424],{"fence":147},[426,5285,5286,5288],{"linethickness":428},[142,5287,2760],{},[142,5289,271],{},[145,5291,438],{"fence":147},[145,5293,695],{},[142,5295,2768],{},[166,5297,5298],{"encoding":168},"\\binom{5}{3} = 10",[123,5300,5302,5383],{"className":5301,"ariaHidden":147},[173],[123,5303,5305,5308,5374,5377,5380],{"className":5304},[177],[123,5306],{"className":5307,"style":710},[181],[123,5309,5311,5317,5368],{"className":5310},[186],[123,5312,5314],{"className":5313,"style":460},[458,459],[123,5315,424],{"className":5316},[464,465],[123,5318,5320],{"className":5319},[426],[123,5321,5323,5360],{"className":5322},[472,473],[123,5324,5326,5357],{"className":5325},[477],[123,5327,5329,5343],{"className":5328,"style":732},[481],[123,5330,5331,5334],{"style":485},[123,5332],{"className":5333,"style":490},[489],[123,5335,5337],{"className":5336},[494,495,496,497],[123,5338,5340],{"className":5339},[186,497],[123,5341,271],{"className":5342},[186,497],[123,5344,5345,5348],{"style":508},[123,5346],{"className":5347,"style":490},[489],[123,5349,5351],{"className":5350},[494,495,496,497],[123,5352,5354],{"className":5353},[186,497],[123,5355,2760],{"className":5356},[186,497],[123,5358,525],{"className":5359},[524],[123,5361,5363],{"className":5362},[477],[123,5364,5366],{"className":5365,"style":532},[481],[123,5367],{},[123,5369,5371],{"className":5370,"style":460},[538,459],[123,5372,438],{"className":5373},[464,465],[123,5375],{"className":5376,"style":781},[194],[123,5378,695],{"className":5379},[785],[123,5381],{"className":5382,"style":781},[194],[123,5384,5386,5389],{"className":5385},[177],[123,5387],{"className":5388,"style":252},[181],[123,5390,2768],{"className":5391},[186],". The triangle is symmetric: there are just as many ways to get 2 heads (and 3 tails) as 3 heads (and 2 tails).",[30,5394,5395],{},"The complete probability table:",[2632,5397,5398,5538],{},[2635,5399,5400],{},[2638,5401,5402,5432,5534,5536],{},[2641,5403,5404,3891],{"align":2643},[123,5405,5407,5420],{"className":5406},[126],[123,5408,5410],{"className":5409},[130],[132,5411,5412],{"xmlns":134},[136,5413,5414,5418],{},[139,5415,5416],{},[430,5417,435],{},[166,5419,435],{"encoding":168},[123,5421,5423],{"className":5422,"ariaHidden":147},[173],[123,5424,5426,5429],{"className":5425},[177],[123,5427],{"className":5428,"style":602},[181],[123,5430,435],{"className":5431,"style":505},[186,504],[2641,5433,5434],{"align":2643},[123,5435,5437,5459],{"className":5436},[126],[123,5438,5440],{"className":5439},[130],[132,5441,5442],{"xmlns":134},[136,5443,5444,5456],{},[139,5445,5446,5448,5454],{},[145,5447,424],{"fence":147},[426,5449,5450,5452],{"linethickness":428},[142,5451,2760],{},[430,5453,435],{},[145,5455,438],{"fence":147},[166,5457,5458],{"encoding":168},"\\binom{5}{k}",[123,5460,5462],{"className":5461,"ariaHidden":147},[173],[123,5463,5465,5468],{"className":5464},[177],[123,5466],{"className":5467,"style":710},[181],[123,5469,5471,5477,5528],{"className":5470},[186],[123,5472,5474],{"className":5473,"style":460},[458,459],[123,5475,424],{"className":5476},[464,465],[123,5478,5480],{"className":5479},[426],[123,5481,5483,5520],{"className":5482},[472,473],[123,5484,5486,5517],{"className":5485},[477],[123,5487,5489,5503],{"className":5488,"style":732},[481],[123,5490,5491,5494],{"style":485},[123,5492],{"className":5493,"style":490},[489],[123,5495,5497],{"className":5496},[494,495,496,497],[123,5498,5500],{"className":5499},[186,497],[123,5501,435],{"className":5502,"style":505},[186,504,497],[123,5504,5505,5508],{"style":508},[123,5506],{"className":5507,"style":490},[489],[123,5509,5511],{"className":5510},[494,495,496,497],[123,5512,5514],{"className":5513},[186,497],[123,5515,2760],{"className":5516},[186,497],[123,5518,525],{"className":5519},[524],[123,5521,5523],{"className":5522},[477],[123,5524,5526],{"className":5525,"style":532},[481],[123,5527],{},[123,5529,5531],{"className":5530,"style":460},[538,459],[123,5532,438],{"className":5533},[464,465],[2641,5535,3244],{"align":2643},[2641,5537,3998],{"align":2643},[2735,5539,5540,5649,5758,5867,5974,6081],{},[2638,5541,5542,5544,5546,5646],{},[2740,5543,1349],{"align":2643},[2740,5545,80],{"align":2643},[2740,5547,5548],{"align":2643},[123,5549,5551,5569],{"className":5550},[126],[123,5552,5554],{"className":5553},[130],[132,5555,5556],{"xmlns":134},[136,5557,5558,5566],{},[139,5559,5560],{},[426,5561,5562,5564],{},[142,5563,80],{},[142,5565,2763],{},[166,5567,5568],{"encoding":168},"\\frac{1}{32}",[123,5570,5572],{"className":5571,"ariaHidden":147},[173],[123,5573,5575,5578],{"className":5574},[177],[123,5576],{"className":5577,"style":1751},[181],[123,5579,5581,5584,5643],{"className":5580},[186],[123,5582],{"className":5583},[458,992],[123,5585,5587],{"className":5586},[426],[123,5588,5590,5635],{"className":5589},[472,473],[123,5591,5593,5632],{"className":5592},[477],[123,5594,5596,5610,5618],{"className":5595,"style":1678},[481],[123,5597,5598,5601],{"style":1008},[123,5599],{"className":5600,"style":1012},[489],[123,5602,5604],{"className":5603},[494,495,496,497],[123,5605,5607],{"className":5606},[186,497],[123,5608,2763],{"className":5609},[186,497],[123,5611,5612,5615],{"style":1046},[123,5613],{"className":5614,"style":1012},[489],[123,5616],{"className":5617,"style":1054},[1053],[123,5619,5620,5623],{"style":1057},[123,5621],{"className":5622,"style":1012},[489],[123,5624,5626],{"className":5625},[494,495,496,497],[123,5627,5629],{"className":5628},[186,497],[123,5630,80],{"className":5631},[186,497],[123,5633,525],{"className":5634},[524],[123,5636,5638],{"className":5637},[477],[123,5639,5641],{"className":5640,"style":532},[481],[123,5642],{},[123,5644],{"className":5645},[538,992],[2740,5647,5648],{"align":2643},"3.1%",[2638,5650,5651,5653,5655,5755],{},[2740,5652,80],{"align":2643},[2740,5654,2760],{"align":2643},[2740,5656,5657],{"align":2643},[123,5658,5660,5678],{"className":5659},[126],[123,5661,5663],{"className":5662},[130],[132,5664,5665],{"xmlns":134},[136,5666,5667,5675],{},[139,5668,5669],{},[426,5670,5671,5673],{},[142,5672,2760],{},[142,5674,2763],{},[166,5676,5677],{"encoding":168},"\\frac{5}{32}",[123,5679,5681],{"className":5680,"ariaHidden":147},[173],[123,5682,5684,5687],{"className":5683},[177],[123,5685],{"className":5686,"style":1751},[181],[123,5688,5690,5693,5752],{"className":5689},[186],[123,5691],{"className":5692},[458,992],[123,5694,5696],{"className":5695},[426],[123,5697,5699,5744],{"className":5698},[472,473],[123,5700,5702,5741],{"className":5701},[477],[123,5703,5705,5719,5727],{"className":5704,"style":1678},[481],[123,5706,5707,5710],{"style":1008},[123,5708],{"className":5709,"style":1012},[489],[123,5711,5713],{"className":5712},[494,495,496,497],[123,5714,5716],{"className":5715},[186,497],[123,5717,2763],{"className":5718},[186,497],[123,5720,5721,5724],{"style":1046},[123,5722],{"className":5723,"style":1012},[489],[123,5725],{"className":5726,"style":1054},[1053],[123,5728,5729,5732],{"style":1057},[123,5730],{"className":5731,"style":1012},[489],[123,5733,5735],{"className":5734},[494,495,496,497],[123,5736,5738],{"className":5737},[186,497],[123,5739,2760],{"className":5740},[186,497],[123,5742,525],{"className":5743},[524],[123,5745,5747],{"className":5746},[477],[123,5748,5750],{"className":5749,"style":532},[481],[123,5751],{},[123,5753],{"className":5754},[538,992],[2740,5756,5757],{"align":2643},"15.6%",[2638,5759,5760,5762,5764,5864],{},[2740,5761,690],{"align":2643},[2740,5763,2768],{"align":2643},[2740,5765,5766],{"align":2643},[123,5767,5769,5787],{"className":5768},[126],[123,5770,5772],{"className":5771},[130],[132,5773,5774],{"xmlns":134},[136,5775,5776,5784],{},[139,5777,5778],{},[426,5779,5780,5782],{},[142,5781,2768],{},[142,5783,2763],{},[166,5785,5786],{"encoding":168},"\\frac{10}{32}",[123,5788,5790],{"className":5789,"ariaHidden":147},[173],[123,5791,5793,5796],{"className":5792},[177],[123,5794],{"className":5795,"style":1751},[181],[123,5797,5799,5802,5861],{"className":5798},[186],[123,5800],{"className":5801},[458,992],[123,5803,5805],{"className":5804},[426],[123,5806,5808,5853],{"className":5807},[472,473],[123,5809,5811,5850],{"className":5810},[477],[123,5812,5814,5828,5836],{"className":5813,"style":1678},[481],[123,5815,5816,5819],{"style":1008},[123,5817],{"className":5818,"style":1012},[489],[123,5820,5822],{"className":5821},[494,495,496,497],[123,5823,5825],{"className":5824},[186,497],[123,5826,2763],{"className":5827},[186,497],[123,5829,5830,5833],{"style":1046},[123,5831],{"className":5832,"style":1012},[489],[123,5834],{"className":5835,"style":1054},[1053],[123,5837,5838,5841],{"style":1057},[123,5839],{"className":5840,"style":1012},[489],[123,5842,5844],{"className":5843},[494,495,496,497],[123,5845,5847],{"className":5846},[186,497],[123,5848,2768],{"className":5849},[186,497],[123,5851,525],{"className":5852},[524],[123,5854,5856],{"className":5855},[477],[123,5857,5859],{"className":5858,"style":532},[481],[123,5860],{},[123,5862],{"className":5863},[538,992],[2740,5865,5866],{"align":2643},"31.3%",[2638,5868,5869,5871,5873,5972],{},[2740,5870,271],{"align":2643},[2740,5872,2768],{"align":2643},[2740,5874,5875],{"align":2643},[123,5876,5878,5895],{"className":5877},[126],[123,5879,5881],{"className":5880},[130],[132,5882,5883],{"xmlns":134},[136,5884,5885,5893],{},[139,5886,5887],{},[426,5888,5889,5891],{},[142,5890,2768],{},[142,5892,2763],{},[166,5894,5786],{"encoding":168},[123,5896,5898],{"className":5897,"ariaHidden":147},[173],[123,5899,5901,5904],{"className":5900},[177],[123,5902],{"className":5903,"style":1751},[181],[123,5905,5907,5910,5969],{"className":5906},[186],[123,5908],{"className":5909},[458,992],[123,5911,5913],{"className":5912},[426],[123,5914,5916,5961],{"className":5915},[472,473],[123,5917,5919,5958],{"className":5918},[477],[123,5920,5922,5936,5944],{"className":5921,"style":1678},[481],[123,5923,5924,5927],{"style":1008},[123,5925],{"className":5926,"style":1012},[489],[123,5928,5930],{"className":5929},[494,495,496,497],[123,5931,5933],{"className":5932},[186,497],[123,5934,2763],{"className":5935},[186,497],[123,5937,5938,5941],{"style":1046},[123,5939],{"className":5940,"style":1012},[489],[123,5942],{"className":5943,"style":1054},[1053],[123,5945,5946,5949],{"style":1057},[123,5947],{"className":5948,"style":1012},[489],[123,5950,5952],{"className":5951},[494,495,496,497],[123,5953,5955],{"className":5954},[186,497],[123,5956,2768],{"className":5957},[186,497],[123,5959,525],{"className":5960},[524],[123,5962,5964],{"className":5963},[477],[123,5965,5967],{"className":5966,"style":532},[481],[123,5968],{},[123,5970],{"className":5971},[538,992],[2740,5973,5866],{"align":2643},[2638,5975,5976,5978,5980,6079],{},[2740,5977,151],{"align":2643},[2740,5979,2760],{"align":2643},[2740,5981,5982],{"align":2643},[123,5983,5985,6002],{"className":5984},[126],[123,5986,5988],{"className":5987},[130],[132,5989,5990],{"xmlns":134},[136,5991,5992,6000],{},[139,5993,5994],{},[426,5995,5996,5998],{},[142,5997,2760],{},[142,5999,2763],{},[166,6001,5677],{"encoding":168},[123,6003,6005],{"className":6004,"ariaHidden":147},[173],[123,6006,6008,6011],{"className":6007},[177],[123,6009],{"className":6010,"style":1751},[181],[123,6012,6014,6017,6076],{"className":6013},[186],[123,6015],{"className":6016},[458,992],[123,6018,6020],{"className":6019},[426],[123,6021,6023,6068],{"className":6022},[472,473],[123,6024,6026,6065],{"className":6025},[477],[123,6027,6029,6043,6051],{"className":6028,"style":1678},[481],[123,6030,6031,6034],{"style":1008},[123,6032],{"className":6033,"style":1012},[489],[123,6035,6037],{"className":6036},[494,495,496,497],[123,6038,6040],{"className":6039},[186,497],[123,6041,2763],{"className":6042},[186,497],[123,6044,6045,6048],{"style":1046},[123,6046],{"className":6047,"style":1012},[489],[123,6049],{"className":6050,"style":1054},[1053],[123,6052,6053,6056],{"style":1057},[123,6054],{"className":6055,"style":1012},[489],[123,6057,6059],{"className":6058},[494,495,496,497],[123,6060,6062],{"className":6061},[186,497],[123,6063,2760],{"className":6064},[186,497],[123,6066,525],{"className":6067},[524],[123,6069,6071],{"className":6070},[477],[123,6072,6074],{"className":6073,"style":532},[481],[123,6075],{},[123,6077],{"className":6078},[538,992],[2740,6080,5757],{"align":2643},[2638,6082,6083,6085,6087,6186],{},[2740,6084,2760],{"align":2643},[2740,6086,80],{"align":2643},[2740,6088,6089],{"align":2643},[123,6090,6092,6109],{"className":6091},[126],[123,6093,6095],{"className":6094},[130],[132,6096,6097],{"xmlns":134},[136,6098,6099,6107],{},[139,6100,6101],{},[426,6102,6103,6105],{},[142,6104,80],{},[142,6106,2763],{},[166,6108,5568],{"encoding":168},[123,6110,6112],{"className":6111,"ariaHidden":147},[173],[123,6113,6115,6118],{"className":6114},[177],[123,6116],{"className":6117,"style":1751},[181],[123,6119,6121,6124,6183],{"className":6120},[186],[123,6122],{"className":6123},[458,992],[123,6125,6127],{"className":6126},[426],[123,6128,6130,6175],{"className":6129},[472,473],[123,6131,6133,6172],{"className":6132},[477],[123,6134,6136,6150,6158],{"className":6135,"style":1678},[481],[123,6137,6138,6141],{"style":1008},[123,6139],{"className":6140,"style":1012},[489],[123,6142,6144],{"className":6143},[494,495,496,497],[123,6145,6147],{"className":6146},[186,497],[123,6148,2763],{"className":6149},[186,497],[123,6151,6152,6155],{"style":1046},[123,6153],{"className":6154,"style":1012},[489],[123,6156],{"className":6157,"style":1054},[1053],[123,6159,6160,6163],{"style":1057},[123,6161],{"className":6162,"style":1012},[489],[123,6164,6166],{"className":6165},[494,495,496,497],[123,6167,6169],{"className":6168},[186,497],[123,6170,80],{"className":6171},[186,497],[123,6173,525],{"className":6174},[524],[123,6176,6178],{"className":6177},[477],[123,6179,6181],{"className":6180,"style":532},[481],[123,6182],{},[123,6184],{"className":6185},[538,992],[2740,6187,5648],{"align":2643},[30,6189,6190],{},"If we sketch this as a histogram, the distribution has a familiar shape:",[99,6192,6195],{"className":6193,"code":6194,"language":104,"meta":105},[102],"0 heads  ▌              3.1%\n1 head   ▌▌▌▌▌         15.6%\n2 heads  ▌▌▌▌▌▌▌▌▌▌    31.3%\n3 heads  ▌▌▌▌▌▌▌▌▌▌    31.3%\n4 heads  ▌▌▌▌▌         15.6%\n5 heads  ▌              3.1%\n",[107,6196,6194],{"__ignoreMap":105},[30,6198,6199],{},"Notice three things:",[72,6201,6202,6208,6214],{},[75,6203,6204,6207],{},[42,6205,6206],{},"Symmetry."," The distribution is a perfect mirror around the center. Getting 0 heads is just as unlikely as getting 5; getting 1 is just as likely as getting 4.",[75,6209,6210,6213],{},[42,6211,6212],{},"Central concentration."," The most probable results cluster around the center (2 or 3 heads), while the extremes (all heads or all tails) are very rare.",[75,6215,6216,6219],{},[42,6217,6218],{},"Bell shape."," The silhouette resembles the famous Gaussian curve. This is no accident: as we increase the number of flips, the distribution increasingly approximates that curve.",[30,6221,6222,6223,827],{},"If you are interested in exploring more of the statistics behind coin flips, read ",[2779,6224,6227],{"href":6225,"rel":6226},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fthe-unexpected-statistics-of-coin-flips\u002F",[2783],"The Unexpected Statistics of Coin Flips",[34,6229,6231],{"id":6230},"the-algebraic-connection-the-binomial-theorem","The algebraic connection: the binomial theorem",[30,6233,6234,6235,6238],{},"So far we have talked about triangles and coins. But there is a third protagonist that ties everything together: ",[42,6236,6237],{},"algebra",". Specifically, an expression you may remember from school:",[30,6240,6242],{"className":6241},[409],[123,6243,6245,6272],{"className":6244},[126],[123,6246,6248],{"className":6247},[130],[132,6249,6250],{"xmlns":134},[136,6251,6252,6269],{},[139,6253,6254,6256,6258,6260,6263],{},[145,6255,424],{"stretchy":868},[430,6257,2779],{},[145,6259,274],{},[430,6261,6262],{},"b",[2520,6264,6265,6267],{},[145,6266,438],{"stretchy":868},[430,6268,432],{},[166,6270,6271],{"encoding":168},"(a + b)^n",[123,6273,6275,6296],{"className":6274,"ariaHidden":147},[173],[123,6276,6278,6281,6284,6287,6290,6293],{"className":6277},[177],[123,6279],{"className":6280,"style":3490},[181],[123,6282,424],{"className":6283},[458],[123,6285,2779],{"className":6286},[186,504],[123,6288],{"className":6289,"style":296},[194],[123,6291,274],{"className":6292},[300],[123,6294],{"className":6295,"style":296},[194],[123,6297,6299,6302,6305],{"className":6298},[177],[123,6300],{"className":6301,"style":3490},[181],[123,6303,6262],{"className":6304},[186,504],[123,6306,6308,6311],{"className":6307},[538],[123,6309,438],{"className":6310},[538],[123,6312,6314],{"className":6313},[2548],[123,6315,6317],{"className":6316},[472],[123,6318,6320],{"className":6319},[477],[123,6321,6323],{"className":6322,"style":2538},[481],[123,6324,6325,6328],{"style":2560},[123,6326],{"className":6327,"style":490},[489],[123,6329,6331],{"className":6330},[494,495,496,497],[123,6332,432],{"className":6333},[186,504,497],[30,6335,6336],{},"Before expanding it, let's clarify a couple of concepts:",[6338,6339,6340,6462],"ul",{},[75,6341,6342,6343,6346,6347,827],{},"A ",[42,6344,6345],{},"polynomial"," is a mathematical expression made up of sums of terms with variables raised to various powers. For example: ",[123,6348,6350,6381],{"className":6349},[126],[123,6351,6353],{"className":6352},[130],[132,6354,6355],{"xmlns":134},[136,6356,6357,6378],{},[139,6358,6359,6361,6368,6370,6372,6374,6376],{},[142,6360,271],{},[2520,6362,6363,6366],{},[430,6364,6365],{},"x",[142,6367,690],{},[145,6369,274],{},[142,6371,690],{},[430,6373,6365],{},[145,6375,274],{},[142,6377,80],{},[166,6379,6380],{"encoding":168},"3x^2 + 2x + 1",[123,6382,6384,6432,6453],{"className":6383,"ariaHidden":147},[173],[123,6385,6387,6391,6394,6423,6426,6429],{"className":6386},[177],[123,6388],{"className":6389,"style":6390},[181],"height:0.8974em;vertical-align:-0.0833em;",[123,6392,271],{"className":6393},[186],[123,6395,6397,6400],{"className":6396},[186],[123,6398,6365],{"className":6399},[186,504],[123,6401,6403],{"className":6402},[2548],[123,6404,6406],{"className":6405},[472],[123,6407,6409],{"className":6408},[477],[123,6410,6412],{"className":6411,"style":2826},[481],[123,6413,6414,6417],{"style":2560},[123,6415],{"className":6416,"style":490},[489],[123,6418,6420],{"className":6419},[494,495,496,497],[123,6421,690],{"className":6422},[186,497],[123,6424],{"className":6425,"style":296},[194],[123,6427,274],{"className":6428},[300],[123,6430],{"className":6431,"style":296},[194],[123,6433,6435,6438,6441,6444,6447,6450],{"className":6434},[177],[123,6436],{"className":6437,"style":289},[181],[123,6439,690],{"className":6440},[186],[123,6442,6365],{"className":6443},[186,504],[123,6445],{"className":6446,"style":296},[194],[123,6448,274],{"className":6449},[300],[123,6451],{"className":6452,"style":296},[194],[123,6454,6456,6459],{"className":6455},[177],[123,6457],{"className":6458,"style":252},[181],[123,6460,80],{"className":6461},[186],[75,6463,6342,6464,6467,6468,6471,6472,827],{},[42,6465,6466],{},"binomial"," is a special case of a polynomial: it has exactly ",[42,6469,6470],{},"two"," terms. For example: ",[123,6473,6475,6497],{"className":6474},[126],[123,6476,6478],{"className":6477},[130],[132,6479,6480],{"xmlns":134},[136,6481,6482,6494],{},[139,6483,6484,6486,6488,6490,6492],{},[145,6485,424],{"stretchy":868},[430,6487,2779],{},[145,6489,274],{},[430,6491,6262],{},[145,6493,438],{"stretchy":868},[166,6495,6496],{"encoding":168},"(a + b)",[123,6498,6500,6521],{"className":6499,"ariaHidden":147},[173],[123,6501,6503,6506,6509,6512,6515,6518],{"className":6502},[177],[123,6504],{"className":6505,"style":3490},[181],[123,6507,424],{"className":6508},[458],[123,6510,2779],{"className":6511},[186,504],[123,6513],{"className":6514,"style":296},[194],[123,6516,274],{"className":6517},[300],[123,6519],{"className":6520,"style":296},[194],[123,6522,6524,6527,6530],{"className":6523},[177],[123,6525],{"className":6526,"style":3490},[181],[123,6528,6262],{"className":6529},[186,504],[123,6531,438],{"className":6532},[538],[30,6534,6535,6536,6564],{},"Raising a binomial to a power means multiplying it by itself ",[123,6537,6539,6552],{"className":6538},[126],[123,6540,6542],{"className":6541},[130],[132,6543,6544],{"xmlns":134},[136,6545,6546,6550],{},[139,6547,6548],{},[430,6549,432],{},[166,6551,432],{"encoding":168},[123,6553,6555],{"className":6554,"ariaHidden":147},[173],[123,6556,6558,6561],{"className":6557},[177],[123,6559],{"className":6560,"style":572},[181],[123,6562,432],{"className":6563},[186,504]," times. Let's start with the simplest cases:",[30,6566,6567],{},[42,6568,6569],{},"Power 2:",[30,6571,6573],{"className":6572},[409],[123,6574,6576,6626],{"className":6575},[126],[123,6577,6579],{"className":6578},[130],[132,6580,6581],{"xmlns":134},[136,6582,6583,6623],{},[139,6584,6585,6587,6589,6591,6593,6599,6601,6607,6609,6611,6613,6615,6617],{},[145,6586,424],{"stretchy":868},[430,6588,2779],{},[145,6590,274],{},[430,6592,6262],{},[2520,6594,6595,6597],{},[145,6596,438],{"stretchy":868},[142,6598,690],{},[145,6600,695],{},[2520,6602,6603,6605],{},[430,6604,2779],{},[142,6606,690],{},[145,6608,274],{},[142,6610,690],{},[430,6612,2779],{},[430,6614,6262],{},[145,6616,274],{},[2520,6618,6619,6621],{},[430,6620,6262],{},[142,6622,690],{},[166,6624,6625],{"encoding":168},"(a + b)^2 = a^2 + 2ab + b^2",[123,6627,6629,6650,6698,6742,6765],{"className":6628,"ariaHidden":147},[173],[123,6630,6632,6635,6638,6641,6644,6647],{"className":6631},[177],[123,6633],{"className":6634,"style":3490},[181],[123,6636,424],{"className":6637},[458],[123,6639,2779],{"className":6640},[186,504],[123,6642],{"className":6643,"style":296},[194],[123,6645,274],{"className":6646},[300],[123,6648],{"className":6649,"style":296},[194],[123,6651,6653,6657,6660,6689,6692,6695],{"className":6652},[177],[123,6654],{"className":6655,"style":6656},[181],"height:1.0641em;vertical-align:-0.25em;",[123,6658,6262],{"className":6659},[186,504],[123,6661,6663,6666],{"className":6662},[538],[123,6664,438],{"className":6665},[538],[123,6667,6669],{"className":6668},[2548],[123,6670,6672],{"className":6671},[472],[123,6673,6675],{"className":6674},[477],[123,6676,6678],{"className":6677,"style":2826},[481],[123,6679,6680,6683],{"style":2560},[123,6681],{"className":6682,"style":490},[489],[123,6684,6686],{"className":6685},[494,495,496,497],[123,6687,690],{"className":6688},[186,497],[123,6690],{"className":6691,"style":781},[194],[123,6693,695],{"className":6694},[785],[123,6696],{"className":6697,"style":781},[194],[123,6699,6701,6704,6733,6736,6739],{"className":6700},[177],[123,6702],{"className":6703,"style":6390},[181],[123,6705,6707,6710],{"className":6706},[186],[123,6708,2779],{"className":6709},[186,504],[123,6711,6713],{"className":6712},[2548],[123,6714,6716],{"className":6715},[472],[123,6717,6719],{"className":6718},[477],[123,6720,6722],{"className":6721,"style":2826},[481],[123,6723,6724,6727],{"style":2560},[123,6725],{"className":6726,"style":490},[489],[123,6728,6730],{"className":6729},[494,495,496,497],[123,6731,690],{"className":6732},[186,497],[123,6734],{"className":6735,"style":296},[194],[123,6737,274],{"className":6738},[300],[123,6740],{"className":6741,"style":296},[194],[123,6743,6745,6749,6752,6756,6759,6762],{"className":6744},[177],[123,6746],{"className":6747,"style":6748},[181],"height:0.7778em;vertical-align:-0.0833em;",[123,6750,690],{"className":6751},[186],[123,6753,6755],{"className":6754},[186,504],"ab",[123,6757],{"className":6758,"style":296},[194],[123,6760,274],{"className":6761},[300],[123,6763],{"className":6764,"style":296},[194],[123,6766,6768,6771],{"className":6767},[177],[123,6769],{"className":6770,"style":2826},[181],[123,6772,6774,6777],{"className":6773},[186],[123,6775,6262],{"className":6776},[186,504],[123,6778,6780],{"className":6779},[2548],[123,6781,6783],{"className":6782},[472],[123,6784,6786],{"className":6785},[477],[123,6787,6789],{"className":6788,"style":2826},[481],[123,6790,6791,6794],{"style":2560},[123,6792],{"className":6793,"style":490},[489],[123,6795,6797],{"className":6796},[494,495,496,497],[123,6798,690],{"className":6799},[186,497],[30,6801,6802,6803,6858],{},"The coefficients are ",[123,6804,6806,6828],{"className":6805},[126],[123,6807,6809],{"className":6808},[130],[132,6810,6811],{"xmlns":134},[136,6812,6813,6825],{},[139,6814,6815,6817,6819,6821,6823],{},[142,6816,80],{},[145,6818,148],{"separator":147},[142,6820,690],{},[145,6822,148],{"separator":147},[142,6824,80],{},[166,6826,6827],{"encoding":168},"1, 2, 1",[123,6829,6831],{"className":6830,"ariaHidden":147},[173],[123,6832,6834,6837,6840,6843,6846,6849,6852,6855],{"className":6833},[177],[123,6835],{"className":6836,"style":182},[181],[123,6838,80],{"className":6839},[186],[123,6841,148],{"className":6842},[190],[123,6844],{"className":6845,"style":195},[194],[123,6847,690],{"className":6848},[186],[123,6850,148],{"className":6851},[190],[123,6853],{"className":6854,"style":195},[194],[123,6856,80],{"className":6857},[186]," — row 2 of the triangle.",[30,6860,6861],{},[42,6862,6863],{},"Power 3:",[30,6865,6867],{"className":6866},[409],[123,6868,6870,6936],{"className":6869},[126],[123,6871,6873],{"className":6872},[130],[132,6874,6875],{"xmlns":134},[136,6876,6877,6933],{},[139,6878,6879,6881,6883,6885,6887,6893,6895,6901,6903,6905,6911,6913,6915,6917,6919,6925,6927],{},[145,6880,424],{"stretchy":868},[430,6882,2779],{},[145,6884,274],{},[430,6886,6262],{},[2520,6888,6889,6891],{},[145,6890,438],{"stretchy":868},[142,6892,271],{},[145,6894,695],{},[2520,6896,6897,6899],{},[430,6898,2779],{},[142,6900,271],{},[145,6902,274],{},[142,6904,271],{},[2520,6906,6907,6909],{},[430,6908,2779],{},[142,6910,690],{},[430,6912,6262],{},[145,6914,274],{},[142,6916,271],{},[430,6918,2779],{},[2520,6920,6921,6923],{},[430,6922,6262],{},[142,6924,690],{},[145,6926,274],{},[2520,6928,6929,6931],{},[430,6930,6262],{},[142,6932,271],{},[166,6934,6935],{"encoding":168},"(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3",[123,6937,6939,6960,7007,7051,7101,7151],{"className":6938,"ariaHidden":147},[173],[123,6940,6942,6945,6948,6951,6954,6957],{"className":6941},[177],[123,6943],{"className":6944,"style":3490},[181],[123,6946,424],{"className":6947},[458],[123,6949,2779],{"className":6950},[186,504],[123,6952],{"className":6953,"style":296},[194],[123,6955,274],{"className":6956},[300],[123,6958],{"className":6959,"style":296},[194],[123,6961,6963,6966,6969,6998,7001,7004],{"className":6962},[177],[123,6964],{"className":6965,"style":6656},[181],[123,6967,6262],{"className":6968},[186,504],[123,6970,6972,6975],{"className":6971},[538],[123,6973,438],{"className":6974},[538],[123,6976,6978],{"className":6977},[2548],[123,6979,6981],{"className":6980},[472],[123,6982,6984],{"className":6983},[477],[123,6985,6987],{"className":6986,"style":2826},[481],[123,6988,6989,6992],{"style":2560},[123,6990],{"className":6991,"style":490},[489],[123,6993,6995],{"className":6994},[494,495,496,497],[123,6996,271],{"className":6997},[186,497],[123,6999],{"className":7000,"style":781},[194],[123,7002,695],{"className":7003},[785],[123,7005],{"className":7006,"style":781},[194],[123,7008,7010,7013,7042,7045,7048],{"className":7009},[177],[123,7011],{"className":7012,"style":6390},[181],[123,7014,7016,7019],{"className":7015},[186],[123,7017,2779],{"className":7018},[186,504],[123,7020,7022],{"className":7021},[2548],[123,7023,7025],{"className":7024},[472],[123,7026,7028],{"className":7027},[477],[123,7029,7031],{"className":7030,"style":2826},[481],[123,7032,7033,7036],{"style":2560},[123,7034],{"className":7035,"style":490},[489],[123,7037,7039],{"className":7038},[494,495,496,497],[123,7040,271],{"className":7041},[186,497],[123,7043],{"className":7044,"style":296},[194],[123,7046,274],{"className":7047},[300],[123,7049],{"className":7050,"style":296},[194],[123,7052,7054,7057,7060,7089,7092,7095,7098],{"className":7053},[177],[123,7055],{"className":7056,"style":6390},[181],[123,7058,271],{"className":7059},[186],[123,7061,7063,7066],{"className":7062},[186],[123,7064,2779],{"className":7065},[186,504],[123,7067,7069],{"className":7068},[2548],[123,7070,7072],{"className":7071},[472],[123,7073,7075],{"className":7074},[477],[123,7076,7078],{"className":7077,"style":2826},[481],[123,7079,7080,7083],{"style":2560},[123,7081],{"className":7082,"style":490},[489],[123,7084,7086],{"className":7085},[494,495,496,497],[123,7087,690],{"className":7088},[186,497],[123,7090,6262],{"className":7091},[186,504],[123,7093],{"className":7094,"style":296},[194],[123,7096,274],{"className":7097},[300],[123,7099],{"className":7100,"style":296},[194],[123,7102,7104,7107,7110,7113,7142,7145,7148],{"className":7103},[177],[123,7105],{"className":7106,"style":6390},[181],[123,7108,271],{"className":7109},[186],[123,7111,2779],{"className":7112},[186,504],[123,7114,7116,7119],{"className":7115},[186],[123,7117,6262],{"className":7118},[186,504],[123,7120,7122],{"className":7121},[2548],[123,7123,7125],{"className":7124},[472],[123,7126,7128],{"className":7127},[477],[123,7129,7131],{"className":7130,"style":2826},[481],[123,7132,7133,7136],{"style":2560},[123,7134],{"className":7135,"style":490},[489],[123,7137,7139],{"className":7138},[494,495,496,497],[123,7140,690],{"className":7141},[186,497],[123,7143],{"className":7144,"style":296},[194],[123,7146,274],{"className":7147},[300],[123,7149],{"className":7150,"style":296},[194],[123,7152,7154,7157],{"className":7153},[177],[123,7155],{"className":7156,"style":2826},[181],[123,7158,7160,7163],{"className":7159},[186],[123,7161,6262],{"className":7162},[186,504],[123,7164,7166],{"className":7165},[2548],[123,7167,7169],{"className":7168},[472],[123,7170,7172],{"className":7171},[477],[123,7173,7175],{"className":7174,"style":2826},[481],[123,7176,7177,7180],{"style":2560},[123,7178],{"className":7179,"style":490},[489],[123,7181,7183],{"className":7182},[494,495,496,497],[123,7184,271],{"className":7185},[186,497],[30,7187,6802,7188,7255],{},[123,7189,7191,7216],{"className":7190},[126],[123,7192,7194],{"className":7193},[130],[132,7195,7196],{"xmlns":134},[136,7197,7198,7214],{},[139,7199,7200,7202,7204,7206,7208,7210,7212],{},[142,7201,80],{},[145,7203,148],{"separator":147},[142,7205,271],{},[145,7207,148],{"separator":147},[142,7209,271],{},[145,7211,148],{"separator":147},[142,7213,80],{},[166,7215,2972],{"encoding":168},[123,7217,7219],{"className":7218,"ariaHidden":147},[173],[123,7220,7222,7225,7228,7231,7234,7237,7240,7243,7246,7249,7252],{"className":7221},[177],[123,7223],{"className":7224,"style":182},[181],[123,7226,80],{"className":7227},[186],[123,7229,148],{"className":7230},[190],[123,7232],{"className":7233,"style":195},[194],[123,7235,271],{"className":7236},[186],[123,7238,148],{"className":7239},[190],[123,7241],{"className":7242,"style":195},[194],[123,7244,271],{"className":7245},[186],[123,7247,148],{"className":7248},[190],[123,7250],{"className":7251,"style":195},[194],[123,7253,80],{"className":7254},[186]," — row 3.",[30,7257,7258],{},[42,7259,7260],{},"Power 5:",[30,7262,7264],{"className":7263},[409],[123,7265,7267,7365],{"className":7266},[126],[123,7268,7270],{"className":7269},[130],[132,7271,7272],{"xmlns":134},[136,7273,7274,7362],{},[139,7275,7276,7278,7280,7282,7284,7290,7292,7298,7300,7302,7308,7310,7312,7314,7320,7326,7328,7330,7336,7342,7344,7346,7348,7354,7356],{},[145,7277,424],{"stretchy":868},[430,7279,2779],{},[145,7281,274],{},[430,7283,6262],{},[2520,7285,7286,7288],{},[145,7287,438],{"stretchy":868},[142,7289,2760],{},[145,7291,695],{},[2520,7293,7294,7296],{},[430,7295,2779],{},[142,7297,2760],{},[145,7299,274],{},[142,7301,2760],{},[2520,7303,7304,7306],{},[430,7305,2779],{},[142,7307,151],{},[430,7309,6262],{},[145,7311,274],{},[142,7313,2768],{},[2520,7315,7316,7318],{},[430,7317,2779],{},[142,7319,271],{},[2520,7321,7322,7324],{},[430,7323,6262],{},[142,7325,690],{},[145,7327,274],{},[142,7329,2768],{},[2520,7331,7332,7334],{},[430,7333,2779],{},[142,7335,690],{},[2520,7337,7338,7340],{},[430,7339,6262],{},[142,7341,271],{},[145,7343,274],{},[142,7345,2760],{},[430,7347,2779],{},[2520,7349,7350,7352],{},[430,7351,6262],{},[142,7353,151],{},[145,7355,274],{},[2520,7357,7358,7360],{},[430,7359,6262],{},[142,7361,2760],{},[166,7363,7364],{"encoding":168},"(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + 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Each term has a binomial coefficient ",[123,8388,8390,8411],{"className":8389},[126],[123,8391,8393],{"className":8392},[130],[132,8394,8395],{"xmlns":134},[136,8396,8397,8409],{},[139,8398,8399,8401,8407],{},[145,8400,424],{"fence":147},[426,8402,8403,8405],{"linethickness":428},[430,8404,432],{},[430,8406,435],{},[145,8408,438],{"fence":147},[166,8410,441],{"encoding":168},[123,8412,8414],{"className":8413,"ariaHidden":147},[173],[123,8415,8417,8420],{"className":8416},[177],[123,8418],{"className":8419,"style":451},[181],[123,8421,8423,8429,8480],{"className":8422},[186],[123,8424,8426],{"className":8425,"style":460},[458,459],[123,8427,424],{"className":8428},[464,465],[123,8430,8432],{"className":8431},[426],[123,8433,8435,8472],{"className":8434},[472,473],[123,8436,8438,8469],{"className":8437},[477],[123,8439,8441,8455],{"className":8440,"style":482},[481],[123,8442,8443,8446],{"style":485},[123,8444],{"className":8445,"style":490},[489],[123,8447,8449],{"className":8448},[494,495,496,497],[123,8450,8452],{"className":8451},[186,497],[123,8453,435],{"className":8454,"style":505},[186,504,497],[123,8456,8457,8460],{"style":508},[123,8458],{"className":8459,"style":490},[489],[123,8461,8463],{"className":8462},[494,495,496,497],[123,8464,8466],{"className":8465},[186,497],[123,8467,432],{"className":8468},[186,504,497],[123,8470,525],{"className":8471},[524],[123,8473,8475],{"className":8474},[477],[123,8476,8478],{"className":8477,"style":532},[481],[123,8479],{},[123,8481,8483],{"className":8482,"style":460},[538,459],[123,8484,438],{"className":8485},[464,465]," that comes directly from the triangle.",[395,8488,8490],{"id":8489},"the-bridge-between-algebra-and-probability","The bridge between algebra and probability",[30,8492,8493,8494,8522,8523,8525,8526,8522,8554,2503],{},"Now comes the fascinating part. Substitute ",[123,8495,8497,8510],{"className":8496},[126],[123,8498,8500],{"className":8499},[130],[132,8501,8502],{"xmlns":134},[136,8503,8504,8508],{},[139,8505,8506],{},[430,8507,2779],{},[166,8509,2779],{"encoding":168},[123,8511,8513],{"className":8512,"ariaHidden":147},[173],[123,8514,8516,8519],{"className":8515},[177],[123,8517],{"className":8518,"style":572},[181],[123,8520,2779],{"className":8521},[186,504]," with the probability of getting ",[42,8524,2292],{}," and ",[123,8527,8529,8542],{"className":8528},[126],[123,8530,8532],{"className":8531},[130],[132,8533,8534],{"xmlns":134},[136,8535,8536,8540],{},[139,8537,8538],{},[430,8539,6262],{},[166,8541,6262],{"encoding":168},[123,8543,8545],{"className":8544,"ariaHidden":147},[173],[123,8546,8548,8551],{"className":8547},[177],[123,8549],{"className":8550,"style":602},[181],[123,8552,6262],{"className":8553},[186,504],[42,8555,2296],{},[30,8557,8559],{"className":8558},[409],[123,8560,8562,8618],{"className":8561},[126],[123,8563,8565],{"className":8564},[130],[132,8566,8567],{"xmlns":134},[136,8568,8569,8615],{},[139,8570,8571,8573,8575,8577,8579,8581,8583,8585,8591,8593,8595,8597,8599,8601,8603,8605,8607,8609],{},[430,8572,2779],{},[145,8574,695],{},[430,8576,3263],{},[145,8578,424],{"stretchy":868},[1901,8580,2292],{},[145,8582,438],{"stretchy":868},[145,8584,695],{},[426,8586,8587,8589],{},[142,8588,80],{},[142,8590,690],{},[145,8592,148],{"separator":147},[194,8594],{"width":4974},[430,8596,6262],{},[145,8598,695],{},[430,8600,3263],{},[145,8602,424],{"stretchy":868},[1901,8604,2296],{},[145,8606,438],{"stretchy":868},[145,8608,695],{},[426,8610,8611,8613],{},[142,8612,80],{},[142,8614,690],{},[166,8616,8617],{"encoding":168},"a = P(\\text{heads}) = \\frac{1}{2}, \\quad b = P(\\text{tails}) = \\frac{1}{2}",[123,8619,8621,8639,8669,8764,8794],{"className":8620,"ariaHidden":147},[173],[123,8622,8624,8627,8630,8633,8636],{"className":8623},[177],[123,8625],{"className":8626,"style":572},[181],[123,8628,2779],{"className":8629},[186,504],[123,8631],{"className":8632,"style":781},[194],[123,8634,695],{"className":8635},[785],[123,8637],{"className":8638,"style":781},[194],[123,8640,8642,8645,8648,8651,8657,8660,8663,8666],{"className":8641},[177],[123,8643],{"className":8644,"style":3490},[181],[123,8646,3263],{"className":8647,"style":3290},[186,504],[123,8649,424],{"className":8650},[458],[123,8652,8654],{"className":8653},[186,104],[123,8655,2292],{"className":8656},[186],[123,8658,438],{"className":8659},[538],[123,8661],{"className":8662,"style":781},[194],[123,8664,695],{"className":8665},[785],[123,8667],{"className":8668,"style":781},[194],[123,8670,8672,8675,8743,8746,8749,8752,8755,8758,8761],{"className":8671},[177],[123,8673],{"className":8674,"style":1751},[181],[123,8676,8678,8681,8740],{"className":8677},[186],[123,8679],{"className":8680},[458,992],[123,8682,8684],{"className":8683},[426],[123,8685,8687,8732],{"className":8686},[472,473],[123,8688,8690,8729],{"className":8689},[477],[123,8691,8693,8707,8715],{"className":8692,"style":1678},[481],[123,8694,8695,8698],{"style":1008},[123,8696],{"className":8697,"style":1012},[489],[123,8699,8701],{"className":8700},[494,495,496,497],[123,8702,8704],{"className":8703},[186,497],[123,8705,690],{"className":8706},[186,497],[123,8708,8709,8712],{"style":1046},[123,8710],{"className":8711,"style":1012},[489],[123,8713],{"className":8714,"style":1054},[1053],[123,8716,8717,8720],{"style":1057},[123,8718],{"className":8719,"style":1012},[489],[123,8721,8723],{"className":8722},[494,495,496,497],[123,8724,8726],{"className":8725},[186,497],[123,8727,80],{"className":8728},[186,497],[123,8730,525],{"className":8731},[524],[123,8733,8735],{"className":8734},[477],[123,8736,8738],{"className":8737,"style":532},[481],[123,8739],{},[123,8741],{"className":8742},[538,992],[123,8744,148],{"className":8745},[190],[123,8747],{"className":8748,"style":5011},[194],[123,8750],{"className":8751,"style":195},[194],[123,8753,6262],{"className":8754},[186,504],[123,8756],{"className":8757,"style":781},[194],[123,8759,695],{"className":8760},[785],[123,8762],{"className":8763,"style":781},[194],[123,8765,8767,8770,8773,8776,8782,8785,8788,8791],{"className":8766},[177],[123,8768],{"className":8769,"style":3490},[181],[123,8771,3263],{"className":8772,"style":3290},[186,504],[123,8774,424],{"className":8775},[458],[123,8777,8779],{"className":8778},[186,104],[123,8780,2296],{"className":8781},[186],[123,8783,438],{"className":8784},[538],[123,8786],{"className":8787,"style":781},[194],[123,8789,695],{"className":8790},[785],[123,8792],{"className":8793,"style":781},[194],[123,8795,8797,8800],{"className":8796},[177],[123,8798],{"className":8799,"style":1751},[181],[123,8801,8803,8806,8865],{"className":8802},[186],[123,8804],{"className":8805},[458,992],[123,8807,8809],{"className":8808},[426],[123,8810,8812,8857],{"className":8811},[472,473],[123,8813,8815,8854],{"className":8814},[477],[123,8816,8818,8832,8840],{"className":8817,"style":1678},[481],[123,8819,8820,8823],{"style":1008},[123,8821],{"className":8822,"style":1012},[489],[123,8824,8826],{"className":8825},[494,495,496,497],[123,8827,8829],{"className":8828},[186,497],[123,8830,690],{"className":8831},[186,497],[123,8833,8834,8837],{"style":1046},[123,8835],{"className":8836,"style":1012},[489],[123,8838],{"className":8839,"style":1054},[1053],[123,8841,8842,8845],{"style":1057},[123,8843],{"className":8844,"style":1012},[489],[123,8846,8848],{"className":8847},[494,495,496,497],[123,8849,8851],{"className":8850},[186,497],[123,8852,80],{"className":8853},[186,497],[123,8855,525],{"className":8856},[524],[123,8858,8860],{"className":8859},[477],[123,8861,8863],{"className":8862,"style":532},[481],[123,8864],{},[123,8866],{"className":8867},[538,992],[30,8869,8870],{},"The binomial 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+ \\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}",[123,9016,9018,9222,9604],{"className":9017,"ariaHidden":147},[173],[123,9019,9021,9025,9213,9216,9219],{"className":9020},[177],[123,9022],{"className":9023,"style":9024},[181],"height:1.2543em;vertical-align:-0.35em;",[123,9026,9028,9188],{"className":9027},[2195],[123,9029,9031,9037,9105,9108,9111,9114,9182],{"className":9030},[2195],[123,9032,9034],{"className":9033,"style":460},[458,459],[123,9035,424],{"className":9036},[464,465],[123,9038,9040,9043,9102],{"className":9039},[186],[123,9041],{"className":9042},[458,992],[123,9044,9046],{"className":9045},[426],[123,9047,9049,9094],{"className":9048},[472,473],[123,9050,9052,9091],{"className":9051},[477],[123,9053,9055,9069,9077],{"className":9054,"style":1678},[481],[123,9056,9057,9060],{"style":1008},[123,9058],{"className":9059,"style":1012},[489],[123,9061,9063],{"className":9062},[494,495,496,497],[123,9064,9066],{"className":9065},[186,497],[123,9067,690],{"className":9068},[186,497],[123,9070,9071,9074],{"style":1046},[123,9072],{"className":9073,"style":1012},[489],[123,9075],{"className":9076,"style":1054},[1053],[123,9078,9079,9082],{"style":1057},[123,9080],{"className":9081,"style":1012},[489],[123,9083,9085],{"className":9084},[494,495,496,497],[123,9086,9088],{"className":9087},[186,497],[123,9089,80],{"className":9090},[186,497],[123,9092,525],{"className":9093},[524],[123,9095,9097],{"className":9096},[477],[123,9098,9100],{"className":9099,"style":532},[481],[123,9101],{},[123,9103],{"className":9104},[538,992],[123,9106],{"className":9107,"style":296},[194],[123,9109,274],{"className":9110},[300],[123,9112],{"className":9113,"style":296},[194],[123,9115,9117,9120,9179],{"className":9116},[186],[123,9118],{"className":9119},[458,992],[123,9121,9123],{"className":9122},[426],[123,9124,9126,9171],{"className":9125},[472,473],[123,9127,9129,9168],{"className":9128},[477],[123,9130,9132,9146,9154],{"className":9131,"style":1678},[481],[123,9133,9134,9137],{"style":1008},[123,9135],{"className":9136,"style":1012},[489],[123,9138,9140],{"className":9139},[494,495,496,497],[123,9141,9143],{"className":9142},[186,497],[123,9144,690],{"className":9145},[186,497],[123,9147,9148,9151],{"style":1046},[123,9149],{"className":9150,"style":1012},[489],[123,9152],{"className":9153,"style":1054},[1053],[123,9155,9156,9159],{"style":1057},[123,9157],{"className":9158,"style":1012},[489],[123,9160,9162],{"className":9161},[494,495,496,497],[123,9163,9165],{"className":9164},[186,497],[123,9166,80],{"className":9167},[186,497],[123,9169,525],{"className":9170},[524],[123,9172,9174],{"className":9173},[477],[123,9175,9177],{"className":9176,"style":532},[481],[123,9178],{},[123,9180],{"className":9181},[538,992],[123,9183,9185],{"className":9184,"style":460},[538,459],[123,9186,438],{"className":9187},[464,465],[123,9189,9191],{"className":9190},[2548],[123,9192,9194],{"className":9193},[472],[123,9195,9197],{"className":9196},[477],[123,9198,9201],{"className":9199,"style":9200},[481],"height:0.9043em;",[123,9202,9204,9207],{"style":9203},"top:-3.3029em;margin-right:0.05em;",[123,9205],{"className":9206,"style":490},[489],[123,9208,9210],{"className":9209},[494,495,496,497],[123,9211,432],{"className":9212},[186,504,497],[123,9214],{"className":9215,"style":781},[194],[123,9217,695],{"className":9218},[785],[123,9220],{"className":9221,"style":781},[194],[123,9223,9225,9229,9292,9295,9361,9364,9483,9486,9595,9598,9601],{"className":9224},[177],[123,9226],{"className":9227,"style":9228},[181],"height:1.439em;vertical-align:-0.35em;",[123,9230,9232,9235],{"className":9231},[8039],[123,9233,7910],{"className":9234,"style":8045},[8039,8043,8044],[123,9236,9238],{"className":9237},[2548],[123,9239,9241,9284],{"className":9240},[472,473],[123,9242,9244,9281],{"className":9243},[477],[123,9245,9247,9267],{"className":9246,"style":8058},[481],[123,9248,9249,9252],{"style":8061},[123,9250],{"className":9251,"style":490},[489],[123,9253,9255],{"className":9254},[494,495,496,497],[123,9256,9258,9261,9264],{"className":9257},[186,497],[123,9259,435],{"className":9260,"style":505},[186,504,497],[123,9262,695],{"className":9263},[785,497],[123,9265,1349],{"className":9266},[186,497],[123,9268,9269,9272],{"style":8082},[123,9270],{"className":9271,"style":490},[489],[123,9273,9275],{"className":9274},[494,495,496,497],[123,9276,9278],{"className":9277},[186,497],[123,9279,432],{"className":9280},[186,504,497],[123,9282,525],{"className":9283},[524],[123,9285,9287],{"className":9286},[477],[123,9288,9290],{"className":9289,"style":8104},[481],[123,9291],{},[123,9293],{"className":9294,"style":195},[194],[123,9296,9298,9304,9355],{"className":9297},[186],[123,9299,9301],{"className":9300,"style":460},[458,459],[123,9302,424],{"className":9303},[464,465],[123,9305,9307],{"className":9306},[426],[123,9308,9310,9347],{"className":9309},[472,473],[123,9311,9313,9344],{"className":9312},[477],[123,9314,9316,9330],{"className":9315,"style":482},[481]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left side is simply ",[123,9843,9845,9867],{"className":9844},[126],[123,9846,9848],{"className":9847},[130],[132,9849,9850],{"xmlns":134},[136,9851,9852,9864],{},[139,9853,9854,9860,9862],{},[2520,9855,9856,9858],{},[142,9857,80],{},[430,9859,432],{},[145,9861,695],{},[142,9863,80],{},[166,9865,9866],{"encoding":168},"1^n = 1",[123,9868,9870,9914],{"className":9869,"ariaHidden":147},[173],[123,9871,9873,9876,9905,9908,9911],{"className":9872},[177],[123,9874],{"className":9875,"style":2538},[181],[123,9877,9879,9882],{"className":9878},[186],[123,9880,80],{"className":9881},[186],[123,9883,9885],{"className":9884},[2548],[123,9886,9888],{"className":9887},[472],[123,9889,9891],{"className":9890},[477],[123,9892,9894],{"className":9893,"style":2538},[481],[123,9895,9896,9899],{"style":2560},[123,9897],{"className":9898,"style":490},[489],[123,9900,9902],{"className":9901},[494,495,496,497],[123,9903,432],{"className":9904},[186,504,497],[123,9906],{"className":9907,"style":781},[194],[123,9909,695],{"className":9910},[785],[123,9912],{"className":9913,"style":781},[194],[123,9915,9917,9920],{"className":9916},[177],[123,9918],{"className":9919,"style":252},[181],[123,9921,80],{"className":9922},[186],": total certainty. The right side is the sum of all individual probabilities. And each term ",[123,9925,9927,9959],{"className":9926},[126],[123,9928,9930],{"className":9929},[130],[132,9931,9932],{"xmlns":134},[136,9933,9934,9956],{},[139,9935,9936],{},[426,9937,9938,9950],{},[139,9939,9940,9942,9948],{},[145,9941,424],{"fence":147},[426,9943,9944,9946],{"linethickness":428},[430,9945,432],{},[430,9947,435],{},[145,9949,438],{"fence":147},[2520,9951,9952,9954],{},[142,9953,690],{},[430,9955,432],{},[166,9957,9958],{"encoding":168},"\\frac{\\binom{n}{k}}{2^n}",[123,9960,9962],{"className":9961,"ariaHidden":147},[173],[123,9963,9965,9968],{"className":9964},[177],[123,9966],{"className":9967,"style":3524},[181],[123,9969,9971,9974,10128],{"className":9970},[186],[123,9972],{"className":9973},[458,992],[123,9975,9977],{"className":9976},[426],[123,9978,9980,10120],{"className":9979},[472,473],[123,9981,9983,10117],{"className":9982},[477],[123,9984,9986,10026,10034],{"className":9985,"style":3543},[481],[123,9987,9988,9991],{"style":1008},[123,9989],{"className":9990,"style":1012},[489],[123,9992,9994],{"className":9993},[494,495,496,497],[123,9995,9997],{"className":9996},[186,497],[123,9998,10000,10003],{"className":9999},[186,497],[123,10001,690],{"className":10002},[186,497],[123,10004,10006],{"className":10005},[2548],[123,10007,10009],{"className":10008},[472],[123,10010,10012],{"className":10011},[477],[123,10013,10015],{"className":10014,"style":4830},[481],[123,10016,10017,10020],{"style":3576},[123,10018],{"className":10019,"style":3580},[489],[123,10021,10023],{"className":10022},[494,3584,465,497],[123,10024,432],{"className":10025},[186,504,497],[123,10027,10028,10031],{"style":1046},[123,10029],{"className":10030,"style":1012},[489],[123,10032],{"className":10033,"style":1054},[1053],[123,10035,10036,10039],{"style":3598},[123,10037],{"className":10038,"style":1012},[489],[123,10040,10042],{"className":10041},[494,495,496,497],[123,10043,10045],{"className":10044},[186,497],[123,10046,10048,10057,10108],{"className":10047},[186,497],[123,10049,10051],{"className":10050,"style":3615},[458,494,3584,3614,497,459],[123,10052,10054],{"className":10053},[464,465,497],[123,10055,424],{"className":10056},[497],[123,10058,10060],{"className":10059},[426],[123,10061,10063,10100],{"className":10062},[472,473],[123,10064,10066,10097],{"className":10065},[477],[123,10067,10069,10083],{"className":10068,"style":4885},[481],[123,10070,10071,10074],{"style":3637},[123,10072],{"className":10073,"style":3580},[489],[123,10075,10077],{"className":10076},[494,3584,465,497],[123,10078,10080],{"className":10079},[186,497],[123,10081,435],{"className":10082,"style":505},[186,504,497],[123,10084,10085,10088],{"style":3652},[123,10086],{"className":10087,"style":3580},[489],[123,10089,10091],{"className":10090},[494,3584,465,497],[123,10092,10094],{"className":10093},[186,497],[123,10095,432],{"className":10096},[186,504,497],[123,10098,525],{"className":10099},[524],[123,10101,10103],{"className":10102},[477],[123,10104,10106],{"className":10105,"style":3674},[481],[123,10107],{},[123,10109,10111],{"className":10110,"style":3615},[538,494,3584,3614,497,459],[123,10112,10114],{"className":10113},[464,465,497],[123,10115,438],{"className":10116},[497],[123,10118,525],{"className":10119},[524],[123,10121,10123],{"className":10122},[477],[123,10124,10126],{"className":10125,"style":532},[481],[123,10127],{},[123,10129],{"className":10130},[538,992]," is exactly the probability of getting ",[123,10133,10135,10148],{"className":10134},[126],[123,10136,10138],{"className":10137},[130],[132,10139,10140],{"xmlns":134},[136,10141,10142,10146],{},[139,10143,10144],{},[430,10145,435],{},[166,10147,435],{"encoding":168},[123,10149,10151],{"className":10150,"ariaHidden":147},[173],[123,10152,10154,10157],{"className":10153},[177],[123,10155],{"className":10156,"style":602},[181],[123,10158,435],{"className":10159,"style":505},[186,504]," tails (or equivalently, ",[123,10162,10164,10182],{"className":10163},[126],[123,10165,10167],{"className":10166},[130],[132,10168,10169],{"xmlns":134},[136,10170,10171,10179],{},[139,10172,10173,10175,10177],{},[430,10174,432],{},[145,10176,885],{},[430,10178,435],{},[166,10180,10181],{"encoding":168},"n - k",[123,10183,10185,10204],{"className":10184,"ariaHidden":147},[173],[123,10186,10188,10192,10195,10198,10201],{"className":10187},[177],[123,10189],{"className":10190,"style":10191},[181],"height:0.6667em;vertical-align:-0.0833em;",[123,10193,432],{"className":10194},[186,504],[123,10196],{"className":10197,"style":296},[194],[123,10199,885],{"className":10200},[300],[123,10202],{"className":10203,"style":296},[194],[123,10205,10207,10210],{"className":10206},[177],[123,10208],{"className":10209,"style":602},[181],[123,10211,435],{"className":10212,"style":505},[186,504]," heads).",[30,10215,10216],{},"Let's verify with 5 flips:",[30,10218,10220],{"className":10219},[409],[123,10221,10223,10275],{"className":10222},[126],[123,10224,10226],{"className":10225},[130],[132,10227,10228],{"xmlns":134},[136,10229,10230,10272],{},[139,10231,10232,10260,10262,10268,10270],{},[426,10233,10234,10258],{},[139,10235,10236,10238,10240,10242,10244,10246,10248,10250,10252,10254,10256],{},[142,10237,80],{},[145,10239,274],{},[142,10241,2760],{},[145,10243,274],{},[142,10245,2768],{},[145,10247,274],{},[142,10249,2768],{},[145,10251,274],{},[142,10253,2760],{},[145,10255,274],{},[142,10257,80],{},[142,10259,2763],{},[145,10261,695],{},[426,10263,10264,10266],{},[142,10265,2763],{},[142,10267,2763],{},[145,10269,695],{},[142,10271,80],{},[166,10273,10274],{"encoding":168},"\\frac{1 + 5 + 10 + 10 + 5 + 1}{32} = \\frac{32}{32} = 1",[123,10276,10278,10391,10474],{"className":10277,"ariaHidden":147},[173],[123,10279,10281,10284,10382,10385,10388],{"className":10280},[177],[123,10282],{"className":10283,"style":1751},[181],[123,10285,10287,10290,10379],{"className":10286},[186],[123,10288],{"className":10289},[458,992],[123,10291,10293],{"className":10292},[426],[123,10294,10296,10371],{"className":10295},[472,473],[123,10297,10299,10368],{"className":10298},[477],[123,10300,10302,10316,10324],{"className":10301,"style":1678},[481],[123,10303,10304,10307],{"style":1008},[123,10305],{"className":10306,"style":1012},[489],[123,10308,10310],{"className":10309},[494,495,496,497],[123,10311,10313],{"className":10312},[186,497],[123,10314,2763],{"className":10315},[186,497],[123,10317,10318,10321],{"style":1046},[123,10319],{"className":10320,"style":1012},[489],[123,10322],{"className":10323,"style":1054},[1053],[123,10325,10326,10329],{"style":1057},[123,10327],{"className":10328,"style":1012},[489],[123,10330,10332],{"className":10331},[494,495,496,497],[123,10333,10335,10338,10341,10344,10347,10350,10353,10356,10359,10362,10365],{"className":10334},[186,497],[123,10336,80],{"className":10337},[186,497],[123,10339,274],{"className":10340},[300,497],[123,10342,2760],{"className":10343},[186,497],[123,10345,274],{"className":10346},[300,497],[123,10348,2768],{"className":10349},[186,497],[123,10351,274],{"className":10352},[300,497],[123,10354,2768],{"className":10355},[186,497],[123,10357,274],{"className":10358},[300,497],[123,10360,2760],{"className":10361},[186,497],[123,10363,274],{"className":10364},[300,497],[123,10366,80],{"className":10367},[186,497],[123,10369,525],{"className":10370},[524],[123,10372,10374],{"className":10373},[477],[123,10375,10377],{"className":10376,"style":532},[481],[123,10378],{},[123,10380],{"className":10381},[538,992],[123,10383],{"className":10384,"style":781},[194],[123,10386,695],{"className":10387},[785],[123,10389],{"className":10390,"style":781},[194],[123,10392,10394,10397,10465,10468,10471],{"className":10393},[177],[123,10395],{"className":10396,"style":1751},[181],[123,10398,10400,10403,10462],{"className":10399},[186],[123,10401],{"className":10402},[458,992],[123,10404,10406],{"className":10405},[426],[123,10407,10409,10454],{"className":10408},[472,473],[123,10410,10412,10451],{"className":10411},[477],[123,10413,10415,10429,10437],{"className":10414,"style":1678},[481],[123,10416,10417,10420],{"style":1008},[123,10418],{"className":10419,"style":1012},[489],[123,10421,10423],{"className":10422},[494,495,496,497],[123,10424,10426],{"className":10425},[186,497],[123,10427,2763],{"className":10428},[186,497],[123,10430,10431,10434],{"style":1046},[123,10432],{"className":10433,"style":1012},[489],[123,10435],{"className":10436,"style":1054},[1053],[123,10438,10439,10442],{"style":1057},[123,10440],{"className":10441,"style":1012},[489],[123,10443,10445],{"className":10444},[494,495,496,497],[123,10446,10448],{"className":10447},[186,497],[123,10449,2763],{"className":10450},[186,497],[123,10452,525],{"className":10453},[524],[123,10455,10457],{"className":10456},[477],[123,10458,10460],{"className":10459,"style":532},[481],[123,10461],{},[123,10463],{"className":10464},[538,992],[123,10466],{"className":10467,"style":781},[194],[123,10469,695],{"className":10470},[785],[123,10472],{"className":10473,"style":781},[194],[123,10475,10477,10480],{"className":10476},[177],[123,10478],{"className":10479,"style":252},[181],[123,10481,80],{"className":10482},[186],[30,10484,10485,10486,10545,10546,10574,10575,827],{},"Each number in row 5 divided by ",[123,10487,10489,10507],{"className":10488},[126],[123,10490,10492],{"className":10491},[130],[132,10493,10494],{"xmlns":134},[136,10495,10496,10504],{},[139,10497,10498],{},[2520,10499,10500,10502],{},[142,10501,690],{},[142,10503,2760],{},[166,10505,10506],{"encoding":168},"2^5",[123,10508,10510],{"className":10509,"ariaHidden":147},[173],[123,10511,10513,10516],{"className":10512},[177],[123,10514],{"className":10515,"style":2826},[181],[123,10517,10519,10522],{"className":10518},[186],[123,10520,690],{"className":10521},[186],[123,10523,10525],{"className":10524},[2548],[123,10526,10528],{"className":10527},[472],[123,10529,10531],{"className":10530},[477],[123,10532,10534],{"className":10533,"style":2826},[481],[123,10535,10536,10539],{"style":2560},[123,10537],{"className":10538,"style":490},[489],[123,10540,10542],{"className":10541},[494,495,496,497],[123,10543,2760],{"className":10544},[186,497]," gives us a probability, and together they sum to ",[123,10547,10549,10562],{"className":10548},[126],[123,10550,10552],{"className":10551},[130],[132,10553,10554],{"xmlns":134},[136,10555,10556,10560],{},[139,10557,10558],{},[142,10559,80],{},[166,10561,80],{"encoding":168},[123,10563,10565],{"className":10564,"ariaHidden":147},[173],[123,10566,10568,10571],{"className":10567},[177],[123,10569],{"className":10570,"style":252},[181],[123,10572,80],{"className":10573},[186],". The triangle, combinatorics, and algebra are not three different stories: ",[42,10576,10577],{},"they are the same story told in three different languages",[34,10579,10581],{"id":10580},"final-reflection","Final reflection",[30,10583,10584],{},"Throughout this article we have followed a thread that connects three branches of mathematics through a single object:",[6338,10586,10587,10592,10598],{},[75,10588,10589,10591],{},[42,10590,2282],{}," tells us how many ways we can arrange a set of outcomes.",[75,10593,10594,10595,10597],{},"The ",[42,10596,6237],{}," of the binomial generates those same quantities as coefficients in a formula.",[75,10599,10600,10602],{},[42,10601,3244],{}," transforms those numbers into predictions about the real world.",[30,10604,10605,10606,10608],{},"And ",[42,10607,44],{}," is the object that unites them: a visual table where all three disciplines converge.",[30,10610,10611],{},"What started as a triangle for counting combinations — drawn in 11th-century China — ends up describing with precision what happens when you flip a coin five, ten, or a hundred times. An abstract structure created for counting objects turns out to describe random phenomena.",[30,10613,10614],{},"And there is one more detail worth mentioning. As the number of flips grows, the rows of the triangle increasingly resemble the Gaussian bell curve: that ubiquitous curve that appears in statistics, physics, and biology. The deep reason for that convergence is another story, but the fact that a triangle built from such simple sums contains the seed of one of the most important distributions in science is, at the very least, beautiful.",[30,10616,10617],{},"Mathematics has a unique quality: it reveals hidden patterns behind what appears to be pure randomness. A single coin flip is unpredictable. Ten flips already hint at an order. A hundred flips draw an almost perfect curve. And it was all written, from the very beginning, in a triangle.",[30,10619,10620,10621,10626],{},"Put theory into practice: ",[2779,10622,10625],{"href":10623,"rel":10624},"https:\u002F\u002Fflipthecoin.app\u002Fplay\u002F",[2783],"flip a coin online"," and watch combinatorics in action.",{"title":105,"searchDepth":10628,"depth":10628,"links":10629},2,[10630,10631,10635,10636,10639,10640,10643],{"id":36,"depth":10628,"text":37},{"id":66,"depth":10628,"text":67,"children":10632},[10633],{"id":397,"depth":10634,"text":398},3,{"id":2276,"depth":10628,"text":2277},{"id":2787,"depth":10628,"text":2788,"children":10637},[10638],{"id":3238,"depth":10634,"text":3239},{"id":4950,"depth":10628,"text":4951},{"id":6230,"depth":10628,"text":6231,"children":10641},[10642],{"id":8489,"depth":10634,"text":8490},{"id":10580,"depth":10628,"text":10581},"2026-05-24T10:00:00Z","Discover how Pascal's triangle connects combinatorics, algebra, and probability — explained step by step through coin flips.",{"type":23,"value":10647},[10648,10650],[26,10649,21],{"id":28},[30,10651,32],{},"md","intermediate",{},true,"\u002Fblog\u002Fpascals-triangle-and-probability.en",{"title":21,"description":10645},1,"pascals-triangle-combinatorics-and-coin-flips","blog\u002Fpascals-triangle-and-probability.en",[44,10662,10663,10664,10665],"Bell curve","Omar Khayyam","Jia Xian","Blaise Pascal","RHFZ2dq4_tJb1RmUf79baGCus2eTOvbDi6eQDFGNzAA",{"id":10668,"title":10669,"_locale":8,"author":3,"body":10670,"date":13989,"description":13990,"excerpt":13991,"extension":10652,"level":13997,"meta":13998,"navigation":10655,"path":13999,"published":10655,"seo":14000,"series":15,"seriesOrder":10628,"slug":14001,"stem":14002,"tags":14003,"__hash__":14012},"blog\u002Fblog\u002Ffair-coin-looks-unfair.en.md","How a Fair Coin Spends Most of Its Life Looking Unfair?",{"type":23,"value":10671,"toc":13980},[10672,10675,10678,10686,10693,10699,10703,10769,10775,10785,10791,10802,10805,10827,10830,10834,10837,10909,10915,10922,10928,10940,10943,10947,10950,10953,10959,10974,10979,10986,11000,11003,11014,11020,11030,11040,11044,11050,11059,11273,11276,11282,11293,11296,11344,11351,11355,11419,11663,11875,11948,12328,12750,13470,13477,13484,13490,13520,13523,13873,13877,13883,13886,13930,13937,13941,13947,13964,13971,13974],[26,10673,10669],{"id":10674},"how-a-fair-coin-spends-most-of-its-life-looking-unfair",[30,10676,10677],{},"Here is a question that sounds simple and isn't. Suppose two friends, Alice and Bob, play a game. A fair coin is tossed once per second. Heads, Alice scores a point; tails, Bob does. They keep playing for a whole year — more than thirty million tosses. At every moment, one of them is \"in the lead\" (has more points so far). What is more likely?",[6338,10679,10680,10683],{},[75,10681,10682],{},"A) The lead changes hands often, so each player leads for roughly half the year.",[75,10684,10685],{},"B) One player grabs the lead early and keeps it for almost the entire year, while the other spends month after month behind.",[30,10687,10688,10689,10692],{},"Almost everyone picks (A). It feels obvious: the coin is fair, so the lead should be shared fairly. Yet the surprising truth — proven in the middle of the twentieth century — is that ",[42,10690,10691],{},"(B) is far more likely than (A)",". A perfectly fair coin spends most of its life looking deeply unfair.",[30,10694,10695,10696,827],{},"This article is about that paradox, and about the beautiful chain of ideas needed to explain it: random walks, the bell curve, Brownian motion, and a strange U-shaped curve called the ",[42,10697,10698],{},"arcsine law",[34,10700,10702],{"id":10701},"a-walk-built-from-coin-tosses","A walk built from coin tosses",[30,10704,10705,10706,10740,10741,827],{},"Let's turn the game into a picture. Start a running total at zero. Every heads adds ",[123,10707,10709,10725],{"className":10708},[126],[123,10710,10712],{"className":10711},[130],[132,10713,10714],{"xmlns":134},[136,10715,10716,10722],{},[139,10717,10718,10720],{},[145,10719,274],{},[142,10721,80],{},[166,10723,10724],{"encoding":168},"+1",[123,10726,10728],{"className":10727,"ariaHidden":147},[173],[123,10729,10731,10734,10737],{"className":10730},[177],[123,10732],{"className":10733,"style":289},[181],[123,10735,274],{"className":10736},[186],[123,10738,80],{"className":10739},[186],"; every tails subtracts ",[123,10742,10744,10757],{"className":10743},[126],[123,10745,10747],{"className":10746},[130],[132,10748,10749],{"xmlns":134},[136,10750,10751,10755],{},[139,10752,10753],{},[142,10754,80],{},[166,10756,80],{"encoding":168},[123,10758,10760],{"className":10759,"ariaHidden":147},[173],[123,10761,10763,10766],{"className":10762},[177],[123,10764],{"className":10765,"style":252},[181],[123,10767,80],{"className":10768},[186],[99,10770,10773],{"className":10771,"code":10772,"language":104,"meta":105},[102],"Toss:     H   T   H   H   T   T   T   H\nStep:    +1  -1  +1  +1  -1  -1  -1  +1\nTotal:    1   0   1   2   1   0  -1   0\n",[107,10774,10772],{"__ignoreMap":105},[30,10776,10777,10778,827],{},"If the total is positive, Alice is ahead; if negative, Bob is. Plot the total against time and you get a jagged path that wanders up and down. Mathematicians call this a ",[2779,10779,10782],{"href":10780,"rel":10781},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FRandom_walk",[2783],[42,10783,10784],{},"random walk",[99,10786,10789],{"className":10787,"code":10788,"language":104,"meta":105},[102]," +3 |\n +2 |        ___\n +1 |    ___\u002F   \\___        ___\n  0 |___\u002F           \\___   \u002F\n -1 |                   \\_\u002F\n    +------------------------- time\n",[107,10790,10788],{"__ignoreMap":105},[30,10792,10793,10794,10797,10798,10801],{},"A random walk is the simplest example of a ",[42,10795,10796],{},"stochastic process"," — a fancy term for ",[47,10799,10800],{},"a system that changes step by step, with a dash of randomness at every step",". You already know many of them. The price of a stock is a stochastic process: every tick it nudges up or down, and nobody can say which. The position of a tiny pollen grain jiggling in water is one too. So is the path of a person who has had a little too much to drink, deciding at each step whether to lurch left or right — the classic \"drunkard's walk.\"",[30,10803,10804],{},"Now here is the single most important idea in the whole article, so hold onto it:",[10806,10807,10808],"blockquote",{},[30,10809,10810,10811,10814,10815,10818,10819,10822,10823,10826],{},"There is a difference between the ",[42,10812,10813],{},"destination"," and the ",[42,10816,10817],{},"journey",". Where the walk ",[47,10820,10821],{},"ends up"," is one question. What the walk ",[47,10824,10825],{},"does along the way"," is a completely different question.",[30,10828,10829],{},"Keep both of those in mind. The paradox lives in the gap between them.",[34,10831,10833],{"id":10832},"the-bell-curve-appears","The bell curve appears",[30,10835,10836],{},"Let's start with the destination. After a million tosses, roughly where is the total?",[30,10838,10839,10840,10873,10874,10908],{},"Each toss is equally likely to be ",[123,10841,10843,10858],{"className":10842},[126],[123,10844,10846],{"className":10845},[130],[132,10847,10848],{"xmlns":134},[136,10849,10850,10856],{},[139,10851,10852,10854],{},[145,10853,274],{},[142,10855,80],{},[166,10857,10724],{"encoding":168},[123,10859,10861],{"className":10860,"ariaHidden":147},[173],[123,10862,10864,10867,10870],{"className":10863},[177],[123,10865],{"className":10866,"style":289},[181],[123,10868,274],{"className":10869},[186],[123,10871,80],{"className":10872},[186]," or ",[123,10875,10877,10893],{"className":10876},[126],[123,10878,10880],{"className":10879},[130],[132,10881,10882],{"xmlns":134},[136,10883,10884,10890],{},[139,10885,10886,10888],{},[145,10887,885],{},[142,10889,80],{},[166,10891,10892],{"encoding":168},"-1",[123,10894,10896],{"className":10895,"ariaHidden":147},[173],[123,10897,10899,10902,10905],{"className":10898},[177],[123,10900],{"className":10901,"style":289},[181],[123,10903,885],{"className":10904},[186],[123,10906,80],{"className":10907},[186],", so on average the total stays near zero. But \"on average\" hides a lot of wandering. If you ran thousands of these million-toss games and made a histogram of the final totals, a familiar shape would emerge:",[99,10910,10913],{"className":10911,"code":10912,"language":104,"meta":105},[102],"            .:||:.\n          .:||||||:.\n        .:||||||||||:.\n      .:||||||||||||||:.\n   ..:||||||||||||||||||:..\n -----------------0----------------\n        final total after n tosses\n",[107,10914,10912],{"__ignoreMap":105},[30,10916,10917,10918,10921],{},"This bell shape is the famous ",[42,10919,10920],{},"Central Limit Theorem"," (CLT) in action: add up a large number of small independent random nudges, and the total follows a bell curve (a \"normal distribution\"), no matter what the individual nudges looked like.",[30,10923,10924,10925,2503],{},"But notice what the CLT actually told us — and what it did ",[42,10926,10927],{},"not",[10806,10929,10930],{},[30,10931,10932,10933,10936,10937,827],{},"The Central Limit Theorem tells us where most walkers ",[47,10934,10935],{},"end up",". It says nothing about ",[47,10938,10939],{},"how they got there",[30,10941,10942],{},"That is the loophole the paradox sneaks through.",[34,10944,10946],{"id":10945},"zooming-out-from-random-walks-to-brownian-motion","Zooming out: from random walks to Brownian motion",[30,10948,10949],{},"Before we return to the puzzle, let's zoom out — literally.",[30,10951,10952],{},"Take a random walk and shrink it. Make the steps smaller and more frequent: instead of one big step per second, take a hundred tiny steps, then ten thousand even tinier ones. As the steps shrink toward nothing, the jagged staircase smooths into a continuous, infinitely wiggly curve.",[99,10954,10957],{"className":10955,"code":10956,"language":104,"meta":105},[102],"   coarse walk        finer walk          finest → Brownian motion\n   _                  _                    .·'·.\n  | |_      _        | |_   _.            ·    '·._.·'·.\n  |   |_   | |       |   '·' |.            ·             '·.\n _|     |__| |     _·'         '·_       ·'                  '·.\n",[107,10958,10956],{"__ignoreMap":105},[30,10960,10961,10962,10965,10966,10969,10970,10973],{},"That limit has a name: ",[42,10963,10964],{},"Brownian motion",", after the botanist Robert Brown, who in 1827 watched pollen grains tremble endlessly in water without knowing why. Einstein explained the trembling in 1905 (the grains are kicked around by invisible water molecules), and the mathematician Norbert Wiener built the exact mathematical model in the 1920s. The deep fact that a shrinking random walk ",[47,10967,10968],{},"becomes"," Brownian motion is known, informally, as ",[42,10971,10972],{},"Donsker's theorem",". We will only mention it for now.",[10806,10975,10976],{},[30,10977,10978],{},"Brownian motion is what a random walk looks like when you zoom out forever.",[30,10980,10981,10982,10985],{},"But why should specks floating in water have anything to do with a coin-flip game? Because the speck is playing one. A grain of pollen is far too big to move on its own — until the invisible water molecules around it, jittering with heat, hammer into it from every direction at once. At each instant a few more happen to strike from the left than from the right, and the grain lurches slightly to the right; a moment later the imbalance reverses and it lurches back. Every collision is a tiny random nudge — a coin toss repeated millions of times a second. Follow the grain's position over time and it traces the very same jagged, wandering path we drew for Alice and Bob. The abstract coin-flip walk and the trembling fleck of pollen are not merely alike; they are the ",[47,10983,10984],{},"same"," mathematical object — and that object is Brownian motion.",[30,10987,10988,10989,10995,10996,10999],{},"Have you ever flipped a few coins on our ",[2779,10990,10992],{"href":10623,"rel":10991},[2783],[107,10993,10994],{},"\u002Fplay"," page, watched one side surge ahead and ",[47,10997,10998],{},"stay"," ahead, and wondered whether the randomness might be broken?",[30,11001,11002],{},"Let's go back to the water. A single drop of river water has the answer.",[30,11004,11005,11006,11009,11010,11013],{},"Picture a molecule of water drifting down a wide, slow river. Ignore the downstream current for a moment and watch only how it wanders ",[47,11007,11008],{},"across"," the river, from one bank toward the other. Each collision shoves it a little to the left or a little to the right — a coin-flip walk all over again — so if you plot that side-to-side position as the molecule floats downstream (with distance travelled downstream standing in for time), you are drawing our graph yet again. Now: would you ever expect the molecule to glide politely down the ",[42,11011,11012],{},"exact middle"," of the river for its whole journey? Of course not. It drifts to one bank and lingers there for a long stretch, wanders across, hugs the other bank for a while. Staying balanced in the center is the one thing it almost never does.",[99,11015,11018],{"className":11016,"code":11017,"language":104,"meta":105},[102]," +3 |               __\n +2 |        ___   \u002F  \\\n +1 |    ___\u002F   \\_\u002F\n  0 |___\u002F\n -1 |\n    +------------------- time\n",[107,11019,11017],{"__ignoreMap":105},[30,11021,11022,11023,11025,11026,11029],{},"A fair coin behaves the same way. When you flip coins on ",[107,11024,10994],{}," and see one side win eight times in a row, or watch \"heads\" stay in the lead for a hundred flips straight, it can feel like the coin is broken or the app is rigged. It isn't. ",[42,11027,11028],{},"Long stretches of one-sided dominance are not a glitch in fairness — they are exactly what fairness looks like up close."," The molecule does not stay mid-river, and the coin does not stay near a tie.",[10806,11031,11032],{},[30,11033,11034,11035,827],{},"If you want to know more about how we flip the coin in FlipTheCoin.app, you can read ",[2779,11036,11039],{"href":11037,"rel":11038},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fhow-we-toss-a-coin",[2783],"How we toss a coin in FlipTheCoin.app",[34,11041,11043],{"id":11042},"lévys-discovery-and-the-chungfeller-surprise","Lévy's discovery and the Chung–Feller surprise",[30,11045,11046,11047,11049],{},"Now back to Alice and Bob. We want to measure the ",[47,11048,10817],{},", not the destination. So we ask:",[10806,11051,11052],{},[30,11053,11054,11055,11058],{},"Out of the whole year, what ",[42,11056,11057],{},"fraction of the time"," was Alice in the lead?",[30,11060,11061,11062,11090,11091,11119,11120,11148,11149,11272],{},"Call that fraction ",[123,11063,11065,11078],{"className":11064},[126],[123,11066,11068],{"className":11067},[130],[132,11069,11070],{"xmlns":134},[136,11071,11072,11076],{},[139,11073,11074],{},[430,11075,6365],{},[166,11077,6365],{"encoding":168},[123,11079,11081],{"className":11080,"ariaHidden":147},[173],[123,11082,11084,11087],{"className":11083},[177],[123,11085],{"className":11086,"style":572},[181],[123,11088,6365],{"className":11089},[186,504],", a number between ",[123,11092,11094,11107],{"className":11093},[126],[123,11095,11097],{"className":11096},[130],[132,11098,11099],{"xmlns":134},[136,11100,11101,11105],{},[139,11102,11103],{},[142,11104,1349],{},[166,11106,1349],{"encoding":168},[123,11108,11110],{"className":11109,"ariaHidden":147},[173],[123,11111,11113,11116],{"className":11112},[177],[123,11114],{"className":11115,"style":252},[181],[123,11117,1349],{"className":11118},[186]," (Alice was never ahead) and ",[123,11121,11123,11136],{"className":11122},[126],[123,11124,11126],{"className":11125},[130],[132,11127,11128],{"xmlns":134},[136,11129,11130,11134],{},[139,11131,11132],{},[142,11133,80],{},[166,11135,80],{"encoding":168},[123,11137,11139],{"className":11138,"ariaHidden":147},[173],[123,11140,11142,11145],{"className":11141},[177],[123,11143],{"className":11144,"style":252},[181],[123,11146,80],{"className":11147},[186]," (Alice was ahead the entire time). Since we just learned about the bell curve, your instinct probably says: this fraction should pile up around ",[123,11150,11152,11177],{"className":11151},[126],[123,11153,11155],{"className":11154},[130],[132,11156,11157],{"xmlns":134},[136,11158,11159,11174],{},[139,11160,11161,11163,11165],{},[430,11162,6365],{},[145,11164,695],{},[11166,11167,11168],"mstyle",{"scriptlevel":1349,"displaystyle":868},[426,11169,11170,11172],{},[142,11171,80],{},[142,11173,690],{},[166,11175,11176],{"encoding":168},"x = \\tfrac{1}{2}",[123,11178,11180,11198],{"className":11179,"ariaHidden":147},[173],[123,11181,11183,11186,11189,11192,11195],{"className":11182},[177],[123,11184],{"className":11185,"style":572},[181],[123,11187,6365],{"className":11188},[186,504],[123,11190],{"className":11191,"style":781},[194],[123,11193,695],{"className":11194},[785],[123,11196],{"className":11197,"style":781},[194],[123,11199,11201,11204],{"className":11200},[177],[123,11202],{"className":11203,"style":1751},[181],[123,11205,11207,11210,11269],{"className":11206},[186],[123,11208],{"className":11209},[458,992],[123,11211,11213],{"className":11212},[426],[123,11214,11216,11261],{"className":11215},[472,473],[123,11217,11219,11258],{"className":11218},[477],[123,11220,11222,11236,11244],{"className":11221,"style":1678},[481],[123,11223,11224,11227],{"style":1008},[123,11225],{"className":11226,"style":1012},[489],[123,11228,11230],{"className":11229},[494,495,496,497],[123,11231,11233],{"className":11232},[186,497],[123,11234,690],{"className":11235},[186,497],[123,11237,11238,11241],{"style":1046},[123,11239],{"className":11240,"style":1012},[489],[123,11242],{"className":11243,"style":1054},[1053],[123,11245,11246,11249],{"style":1057},[123,11247],{"className":11248,"style":1012},[489],[123,11250,11252],{"className":11251},[494,495,496,497],[123,11253,11255],{"className":11254},[186,497],[123,11256,80],{"className":11257},[186,497],[123,11259,525],{"className":11260},[524],[123,11262,11264],{"className":11263},[477],[123,11265,11267],{"className":11266,"style":532},[481],[123,11268],{},[123,11270],{"className":11271},[538,992],", with a nice hump in the middle. Half the time for Alice, half for Bob — fair is fair.",[30,11274,11275],{},"That instinct is wrong. The real distribution looks like this:",[99,11277,11280],{"className":11278,"code":11279,"language":104,"meta":105},[102]," likely  |\\                            \u002F|\n         | \\                          \u002F |\n         |  \\                        \u002F  |\n         |   \\__                  __\u002F   |\n rare    |      \\________________\u002F      |\n         +------------------------------+\n         0     1\u002F4     1\u002F2    3\u002F4       1\n           Fraction of time Alice leads\n",[107,11281,11279],{"__ignoreMap":105},[30,11283,11284,11285,11288,11289,11292],{},"It is a ",[42,11286,11287],{},"U-shape",". The most likely outcomes are the extremes — Alice leading almost all the time, or almost none of the time. The ",[47,11290,11291],{},"least"," likely outcome is the \"fair\" one, where the lead is split evenly. The fraction of time in the lead behaves in the exact opposite way to the final score.",[30,11294,11295],{},"This was pieced together by several mathematicians:",[6338,11297,11298,11309,11327],{},[75,11299,11300,11301,11304,11305,11308],{},"In the late 1930s, ",[42,11302,11303],{},"Paul Lévy"," — one of the founders of modern probability — studied how long a Brownian motion spends on the positive side. He found this strange U-shaped law for the ",[47,11306,11307],{},"continuous"," case.",[75,11310,11311,11314,11315,11322,11323,11326],{},[42,11312,11313],{},"William Feller",", in his legendary textbook ",[2779,11316,11319],{"href":11317,"rel":11318},"https:\u002F\u002Fbitcoinwords.github.io\u002Fassets\u002Fpapers\u002Fan-introduction-to-probability-theory-and-its-applications.pdf",[2783],[47,11320,11321],{},"An Introduction to Probability Theory and its Applications",", worked out the ",[47,11324,11325],{},"discrete"," coin-tossing version in clean, countable detail.",[75,11328,11329,11330,11333,11334,11339,11340,11343],{},"And the ",[42,11331,11332],{},"Chung–Feller theorem"," (",[2779,11335,11338],{"href":11336,"rel":11337},"https:\u002F\u002Fwww.pnas.org\u002Fdoi\u002Fpdf\u002F10.1073\u002Fpnas.35.10.605",[2783],"Kai Lai Chung and William Feller, 1949",") nailed down a precise combinatorial fact underneath it all: among all the ways a walk of a given length can return to zero, the number of them spending a given amount of time positive is astonishingly ",[47,11341,11342],{},"constant"," — the lead duration is, in a deep sense, spread out rather than concentrated.",[30,11345,11346,11347,11350],{},"Here's a metaphor to make the U-shape feel real. Two marathon runners cross the finish line in a dead tie. You might assume they must have traded the lead back and forth the whole race. But it is far more common for one runner to surge ahead in the first mile and lead for ",[47,11348,11349],{},"almost the entire course",", only to be caught right at the tape. A tied finish does not mean a shared race.",[34,11352,11354],{"id":11353},"the-arcsine-laws","The arcsine laws",[30,11356,11357,11358,11361,11362,8334,11390,11418],{},"The U-shaped curve has a name and a formula. It is called the ",[42,11359,11360],{},"arcsine distribution",", and on the interval from ",[123,11363,11365,11378],{"className":11364},[126],[123,11366,11368],{"className":11367},[130],[132,11369,11370],{"xmlns":134},[136,11371,11372,11376],{},[139,11373,11374],{},[142,11375,1349],{},[166,11377,1349],{"encoding":168},[123,11379,11381],{"className":11380,"ariaHidden":147},[173],[123,11382,11384,11387],{"className":11383},[177],[123,11385],{"className":11386,"style":252},[181],[123,11388,1349],{"className":11389},[186],[123,11391,11393,11406],{"className":11392},[126],[123,11394,11396],{"className":11395},[130],[132,11397,11398],{"xmlns":134},[136,11399,11400,11404],{},[139,11401,11402],{},[142,11403,80],{},[166,11405,80],{"encoding":168},[123,11407,11409],{"className":11408,"ariaHidden":147},[173],[123,11410,11412,11415],{"className":11411},[177],[123,11413],{"className":11414,"style":252},[181],[123,11416,80],{"className":11417},[186]," its density is:",[30,11420,11422],{"className":11421},[409],[123,11423,11425,11474],{"className":11424},[126],[123,11426,11428],{"className":11427},[130],[132,11429,11430],{"xmlns":134},[136,11431,11432,11471],{},[139,11433,11434,11437,11439,11441,11443,11445],{},[430,11435,11436],{},"f",[145,11438,424],{"stretchy":868},[430,11440,6365],{},[145,11442,438],{"stretchy":868},[145,11444,695],{},[426,11446,11447,11449],{},[142,11448,80],{},[139,11450,11451,11454],{},[430,11452,11453],{},"π",[11455,11456,11457],"msqrt",{},[139,11458,11459,11461,11463,11465,11467,11469],{},[430,11460,6365],{},[145,11462,424],{"stretchy":868},[142,11464,80],{},[145,11466,885],{},[430,11468,6365],{},[145,11470,438],{"stretchy":868},[166,11472,11473],{"encoding":168},"f(x) = \\frac{1}{\\pi\\sqrt{x(1-x)}}",[123,11475,11477,11505],{"className":11476,"ariaHidden":147},[173],[123,11478,11480,11483,11487,11490,11493,11496,11499,11502],{"className":11479},[177],[123,11481],{"className":11482,"style":3490},[181],[123,11484,11436],{"className":11485,"style":11486},[186,504],"margin-right:0.1076em;",[123,11488,424],{"className":11489},[458],[123,11491,6365],{"className":11492},[186,504],[123,11494,438],{"className":11495},[538],[123,11497],{"className":11498,"style":781},[194],[123,11500,695],{"className":11501},[785],[123,11503],{"className":11504,"style":781},[194],[123,11506,11508,11512],{"className":11507},[177],[123,11509],{"className":11510,"style":11511},[181],"height:1.6747em;vertical-align:-0.8296em;",[123,11513,11515,11518,11660],{"className":11514},[186],[123,11516],{"className":11517},[458,992],[123,11519,11521],{"className":11520},[426],[123,11522,11524,11651],{"className":11523},[472,473],[123,11525,11527,11648],{"className":11526},[477],[123,11528,11530,11626,11634],{"className":11529,"style":1678},[481],[123,11531,11533,11536],{"style":11532},"top:-2.4642em;",[123,11534],{"className":11535,"style":1012},[489],[123,11537,11539],{"className":11538},[494,495,496,497],[123,11540,11542,11546],{"className":11541},[186,497],[123,11543,11453],{"className":11544,"style":11545},[186,504,497],"margin-right:0.0359em;",[123,11547,11550],{"className":11548},[186,11549,497],"sqrt",[123,11551,11553,11617],{"className":11552},[472,473],[123,11554,11556,11614],{"className":11555},[477],[123,11557,11560,11591],{"className":11558,"style":11559},[481],"height:1.0369em;",[123,11561,11565,11569],{"className":11562,"style":11564},[11563],"svg-align","top:-3.4286em;",[123,11566],{"className":11567,"style":11568},[489],"height:3.4286em;",[123,11570,11573,11576,11579,11582,11585,11588],{"className":11571,"style":11572},[186,497],"padding-left:1.19em;",[123,11574,6365],{"className":11575},[186,504,497],[123,11577,424],{"className":11578},[458,497],[123,11580,80],{"className":11581},[186,497],[123,11583,885],{"className":11584},[300,497],[123,11586,6365],{"className":11587},[186,504,497],[123,11589,438],{"className":11590},[538,497],[123,11592,11594,11597],{"style":11593},"top:-3.0089em;",[123,11595],{"className":11596,"style":11568},[489],[123,11598,11602],{"className":11599,"style":11601},[11600,497],"hide-tail","min-width:0.853em;height:1.5429em;",[11603,11604,11610],"svg",{"xmlns":11605,"width":11606,"height":11607,"viewBox":11608,"preserveAspectRatio":11609},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.5429em","0 0 400000 1080","xMinYMin slice",[11611,11612],"path",{"d":11613},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[123,11615,525],{"className":11616},[524],[123,11618,11620],{"className":11619},[477],[123,11621,11624],{"className":11622,"style":11623},[481],"height:0.4197em;",[123,11625],{},[123,11627,11628,11631],{"style":1046},[123,11629],{"className":11630,"style":1012},[489],[123,11632],{"className":11633,"style":1054},[1053],[123,11635,11636,11639],{"style":1057},[123,11637],{"className":11638,"style":1012},[489],[123,11640,11642],{"className":11641},[494,495,496,497],[123,11643,11645],{"className":11644},[186,497],[123,11646,80],{"className":11647},[186,497],[123,11649,525],{"className":11650},[524],[123,11652,11654],{"className":11653},[477],[123,11655,11658],{"className":11656,"style":11657},[481],"height:0.8296em;",[123,11659],{},[123,11661],{"className":11662},[538,992],[30,11664,11665,11666,11694,11695,10873,11723,11751,11752,11874],{},"You can read the U-shape straight off the formula: as ",[123,11667,11669,11682],{"className":11668},[126],[123,11670,11672],{"className":11671},[130],[132,11673,11674],{"xmlns":134},[136,11675,11676,11680],{},[139,11677,11678],{},[430,11679,6365],{},[166,11681,6365],{"encoding":168},[123,11683,11685],{"className":11684,"ariaHidden":147},[173],[123,11686,11688,11691],{"className":11687},[177],[123,11689],{"className":11690,"style":572},[181],[123,11692,6365],{"className":11693},[186,504]," approaches ",[123,11696,11698,11711],{"className":11697},[126],[123,11699,11701],{"className":11700},[130],[132,11702,11703],{"xmlns":134},[136,11704,11705,11709],{},[139,11706,11707],{},[142,11708,1349],{},[166,11710,1349],{"encoding":168},[123,11712,11714],{"className":11713,"ariaHidden":147},[173],[123,11715,11717,11720],{"className":11716},[177],[123,11718],{"className":11719,"style":252},[181],[123,11721,1349],{"className":11722},[186],[123,11724,11726,11739],{"className":11725},[126],[123,11727,11729],{"className":11728},[130],[132,11730,11731],{"xmlns":134},[136,11732,11733,11737],{},[139,11734,11735],{},[142,11736,80],{},[166,11738,80],{"encoding":168},[123,11740,11742],{"className":11741,"ariaHidden":147},[173],[123,11743,11745,11748],{"className":11744},[177],[123,11746],{"className":11747,"style":252},[181],[123,11749,80],{"className":11750},[186],", the bottom of the fraction shrinks toward zero, so the curve shoots up. The middle, around ",[123,11753,11755,11779],{"className":11754},[126],[123,11756,11758],{"className":11757},[130],[132,11759,11760],{"xmlns":134},[136,11761,11762,11776],{},[139,11763,11764,11766,11768],{},[430,11765,6365],{},[145,11767,695],{},[11166,11769,11770],{"scriptlevel":1349,"displaystyle":868},[426,11771,11772,11774],{},[142,11773,80],{},[142,11775,690],{},[166,11777,11778],{"encoding":168},"x=\\tfrac12",[123,11780,11782,11800],{"className":11781,"ariaHidden":147},[173],[123,11783,11785,11788,11791,11794,11797],{"className":11784},[177],[123,11786],{"className":11787,"style":572},[181],[123,11789,6365],{"className":11790},[186,504],[123,11792],{"className":11793,"style":781},[194],[123,11795,695],{"className":11796},[785],[123,11798],{"className":11799,"style":781},[194],[123,11801,11803,11806],{"className":11802},[177],[123,11804],{"className":11805,"style":1751},[181],[123,11807,11809,11812,11871],{"className":11808},[186],[123,11810],{"className":11811},[458,992],[123,11813,11815],{"className":11814},[426],[123,11816,11818,11863],{"className":11817},[472,473],[123,11819,11821,11860],{"className":11820},[477],[123,11822,11824,11838,11846],{"className":11823,"style":1678},[481],[123,11825,11826,11829],{"style":1008},[123,11827],{"className":11828,"style":1012},[489],[123,11830,11832],{"className":11831},[494,495,496,497],[123,11833,11835],{"className":11834},[186,497],[123,11836,690],{"className":11837},[186,497],[123,11839,11840,11843],{"style":1046},[123,11841],{"className":11842,"style":1012},[489],[123,11844],{"className":11845,"style":1054},[1053],[123,11847,11848,11851],{"style":1057},[123,11849],{"className":11850,"style":1012},[489],[123,11852,11854],{"className":11853},[494,495,496,497],[123,11855,11857],{"className":11856},[186,497],[123,11858,80],{"className":11859},[186,497],[123,11861,525],{"className":11862},[524],[123,11864,11866],{"className":11865},[477],[123,11867,11869],{"className":11868,"style":532},[481],[123,11870],{},[123,11872],{"className":11873},[538,992],", is where it sags lowest.",[30,11876,11877,11878,11912,11913,11947],{},"For the actual coins, before any zooming out, the discrete version is just as clean. If a fair coin is tossed ",[123,11879,11881,11897],{"className":11880},[126],[123,11882,11884],{"className":11883},[130],[132,11885,11886],{"xmlns":134},[136,11887,11888,11894],{},[139,11889,11890,11892],{},[142,11891,690],{},[430,11893,432],{},[166,11895,11896],{"encoding":168},"2n",[123,11898,11900],{"className":11899,"ariaHidden":147},[173],[123,11901,11903,11906,11909],{"className":11902},[177],[123,11904],{"className":11905,"style":252},[181],[123,11907,690],{"className":11908},[186],[123,11910,432],{"className":11911},[186,504]," times, the probability that the walk spends exactly ",[123,11914,11916,11932],{"className":11915},[126],[123,11917,11919],{"className":11918},[130],[132,11920,11921],{"xmlns":134},[136,11922,11923,11929],{},[139,11924,11925,11927],{},[142,11926,690],{},[430,11928,435],{},[166,11930,11931],{"encoding":168},"2k",[123,11933,11935],{"className":11934,"ariaHidden":147},[173],[123,11936,11938,11941,11944],{"className":11937},[177],[123,11939],{"className":11940,"style":602},[181],[123,11942,690],{"className":11943},[186],[123,11945,435],{"className":11946,"style":505},[186,504]," of those steps on the positive side is:",[30,11949,11951],{"className":11950},[409],[123,11952,11954,12045],{"className":11953},[126],[123,11955,11957],{"className":11956},[130],[132,11958,11959],{"xmlns":134},[136,11960,11961,12042],{},[139,11962,11963,11982,11984,12000,12028,12030],{},[11964,11965,11966,11968],"msub",{},[430,11967,3263],{},[139,11969,11970,11972,11974,11976,11978,11980],{},[142,11971,690],{},[430,11973,435],{},[145,11975,148],{"separator":147},[1901,11977,7935],{},[142,11979,690],{},[430,11981,432],{},[145,11983,695],{},[139,11985,11986,11988,11998],{},[145,11987,424],{"fence":147},[426,11989,11990,11996],{"linethickness":428},[139,11991,11992,11994],{},[142,11993,690],{},[430,11995,435],{},[430,11997,435],{},[145,11999,438],{"fence":147},[139,12001,12002,12004,12026],{},[145,12003,424],{"fence":147},[426,12005,12006,12018],{"linethickness":428},[139,12007,12008,12010,12012,12014,12016],{},[142,12009,690],{},[430,12011,432],{},[145,12013,885],{},[142,12015,690],{},[430,12017,435],{},[139,12019,12020,12022,12024],{},[430,12021,432],{},[145,12023,885],{},[430,12025,435],{},[145,12027,438],{"fence":147},[1901,12029,7935],{},[2520,12031,12032,12034],{},[142,12033,690],{},[139,12035,12036,12038,12040],{},[145,12037,885],{},[142,12039,690],{},[430,12041,432],{},[166,12043,12044],{"encoding":168},"P_{2k,\\,2n} = \\binom{2k}{k}\\binom{2n-2k}{n-k}\\,2^{-2n}",[123,12046,12048,12126],{"className":12047,"ariaHidden":147},[173],[123,12049,12051,12055,12117,12120,12123],{"className":12050},[177],[123,12052],{"className":12053,"style":12054},[181],"height:0.9694em;vertical-align:-0.2861em;",[123,12056,12058,12061],{"className":12057},[186],[123,12059,3263],{"className":12060,"style":3290},[186,504],[123,12062,12064],{"className":12063},[2548],[123,12065,12067,12108],{"className":12066},[472,473],[123,12068,12070,12105],{"className":12069},[477],[123,12071,12074],{"className":12072,"style":12073},[481],"height:0.3361em;",[123,12075,12077,12080],{"style":12076},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[123,12078],{"className":12079,"style":490},[489],[123,12081,12083],{"className":12082},[494,495,496,497],[123,12084,12086,12089,12092,12095,12099,12102],{"className":12085},[186,497],[123,12087,690],{"className":12088},[186,497],[123,12090,435],{"className":12091,"style":505},[186,504,497],[123,12093,148],{"className":12094},[190,497],[123,12096],{"className":12097,"style":12098},[194,497],"margin-right:0.1952em;",[123,12100,690],{"className":12101},[186,497],[123,12103,432],{"className":12104},[186,504,497],[123,12106,525],{"className":12107},[524],[123,12109,12111],{"className":12110},[477],[123,12112,12115],{"className":12113,"style":12114},[481],"height:0.2861em;",[123,12116],{},[123,12118],{"className":12119,"style":781},[194],[123,12121,695],{"className":12122},[785],[123,12124],{"className":12125,"style":781},[194],[123,12127,12129,12133,12203,12287,12290],{"className":12128},[177],[123,12130],{"className":12131,"style":12132},[181],"height:1.3334em;vertical-align:-0.4033em;",[123,12134,12136,12142,12197],{"className":12135},[186],[123,12137,12139],{"className":12138,"style":460},[458,459],[123,12140,424],{"className":12141},[464,465],[123,12143,12145],{"className":12144},[426],[123,12146,12148,12189],{"className":12147},[472,473],[123,12149,12151,12186],{"className":12150},[477],[123,12152,12155,12169],{"className":12153,"style":12154},[481],"height:0.9301em;",[123,12156,12157,12160],{"style":485},[123,12158],{"className":12159,"style":490},[489],[123,12161,12163],{"className":12162},[494,495,496,497],[123,12164,12166],{"className":12165},[186,497],[123,12167,435],{"className":12168,"style":505},[186,504,497],[123,12170,12171,12174],{"style":508},[123,12172],{"className":12173,"style":490},[489],[123,12175,12177],{"className":12176},[494,495,496,497],[123,12178,12180,12183],{"className":12179},[186,497],[123,12181,690],{"className":12182},[186,497],[123,12184,435],{"className":12185,"style":505},[186,504,497],[123,12187,525],{"className":12188},[524],[123,12190,12192],{"className":12191},[477],[123,12193,12195],{"className":12194,"style":532},[481],[123,12196],{},[123,12198,12200],{"className":12199,"style":460},[538,459],[123,12201,438],{"className":12202},[464,465],[123,12204,12206,12212,12281],{"className":12205},[186],[123,12207,12209],{"className":12208,"style":460},[458,459],[123,12210,424],{"className":12211},[464,465],[123,12213,12215],{"className":12214},[426],[123,12216,12218,12273],{"className":12217},[472,473],[123,12219,12221,12270],{"className":12220},[477],[123,12222,12224,12244],{"className":12223,"style":12154},[481],[123,12225,12226,12229],{"style":485},[123,12227],{"className":12228,"style":490},[489],[123,12230,12232],{"className":12231},[494,495,496,497],[123,12233,12235,12238,12241],{"className":12234},[186,497],[123,12236,432],{"className":12237},[186,504,497],[123,12239,885],{"className":12240},[300,497],[123,12242,435],{"className":12243,"style":505},[186,504,497],[123,12245,12246,12249],{"style":508},[123,12247],{"className":12248,"style":490},[489],[123,12250,12252],{"className":12251},[494,495,496,497],[123,12253,12255,12258,12261,12264,12267],{"className":12254},[186,497],[123,12256,690],{"className":12257},[186,497],[123,12259,432],{"className":12260},[186,504,497],[123,12262,885],{"className":12263},[300,497],[123,12265,690],{"className":12266},[186,497],[123,12268,435],{"className":12269,"style":505},[186,504,497],[123,12271,525],{"className":12272},[524],[123,12274,12276],{"className":12275},[477],[123,12277,12279],{"className":12278,"style":1730},[481],[123,12280],{},[123,12282,12284],{"className":12283,"style":460},[538,459],[123,12285,438],{"className":12286},[464,465],[123,12288],{"className":12289,"style":195},[194],[123,12291,12293,12296],{"className":12292},[186],[123,12294,690],{"className":12295},[186],[123,12297,12299],{"className":12298},[2548],[123,12300,12302],{"className":12301},[472],[123,12303,12305],{"className":12304},[477],[123,12306,12308],{"className":12307,"style":2826},[481],[123,12309,12310,12313],{"style":2560},[123,12311],{"className":12312,"style":490},[489],[123,12314,12316],{"className":12315},[494,495,496,497],[123,12317,12319,12322,12325],{"className":12318},[186,497],[123,12320,885],{"className":12321},[186,497],[123,12323,690],{"className":12324},[186,497],[123,12326,432],{"className":12327},[186,504,497],[30,12329,12330,12331,8525,12359,12387,12388,12421,12422,12455,12456,12484,12485,12513,12514,12542,12543,827],{},"Here ",[123,12332,12334,12347],{"className":12333},[126],[123,12335,12337],{"className":12336},[130],[132,12338,12339],{"xmlns":134},[136,12340,12341,12345],{},[139,12342,12343],{},[430,12344,432],{},[166,12346,432],{"encoding":168},[123,12348,12350],{"className":12349,"ariaHidden":147},[173],[123,12351,12353,12356],{"className":12352},[177],[123,12354],{"className":12355,"style":572},[181],[123,12357,432],{"className":12358},[186,504],[123,12360,12362,12375],{"className":12361},[126],[123,12363,12365],{"className":12364},[130],[132,12366,12367],{"xmlns":134},[136,12368,12369,12373],{},[139,12370,12371],{},[430,12372,435],{},[166,12374,435],{"encoding":168},[123,12376,12378],{"className":12377,"ariaHidden":147},[173],[123,12379,12381,12384],{"className":12380},[177],[123,12382],{"className":12383,"style":602},[181],[123,12385,435],{"className":12386,"style":505},[186,504]," are just counters — plain whole numbers. We deliberately write the number of tosses as an even number ",[123,12389,12391,12406],{"className":12390},[126],[123,12392,12394],{"className":12393},[130],[132,12395,12396],{"xmlns":134},[136,12397,12398,12404],{},[139,12399,12400,12402],{},[142,12401,690],{},[430,12403,432],{},[166,12405,11896],{"encoding":168},[123,12407,12409],{"className":12408,"ariaHidden":147},[173],[123,12410,12412,12415,12418],{"className":12411},[177],[123,12413],{"className":12414,"style":252},[181],[123,12416,690],{"className":12417},[186],[123,12419,432],{"className":12420},[186,504]," and the time spent in the lead as an even number ",[123,12423,12425,12440],{"className":12424},[126],[123,12426,12428],{"className":12427},[130],[132,12429,12430],{"xmlns":134},[136,12431,12432,12438],{},[139,12433,12434,12436],{},[142,12435,690],{},[430,12437,435],{},[166,12439,11931],{"encoding":168},[123,12441,12443],{"className":12442,"ariaHidden":147},[173],[123,12444,12446,12449,12452],{"className":12445},[177],[123,12447],{"className":12448,"style":602},[181],[123,12450,690],{"className":12451},[186],[123,12453,435],{"className":12454,"style":505},[186,504],", because in this kind of walk the score can only return to a tie after an even number of tosses, and the time one player spends ahead always works out to be even as well. The counter ",[123,12457,12459,12472],{"className":12458},[126],[123,12460,12462],{"className":12461},[130],[132,12463,12464],{"xmlns":134},[136,12465,12466,12470],{},[139,12467,12468],{},[430,12469,435],{},[166,12471,435],{"encoding":168},[123,12473,12475],{"className":12474,"ariaHidden":147},[173],[123,12476,12478,12481],{"className":12477},[177],[123,12479],{"className":12480,"style":602},[181],[123,12482,435],{"className":12483,"style":505},[186,504]," ranges from ",[123,12486,12488,12501],{"className":12487},[126],[123,12489,12491],{"className":12490},[130],[132,12492,12493],{"xmlns":134},[136,12494,12495,12499],{},[139,12496,12497],{},[142,12498,1349],{},[166,12500,1349],{"encoding":168},[123,12502,12504],{"className":12503,"ariaHidden":147},[173],[123,12505,12507,12510],{"className":12506},[177],[123,12508],{"className":12509,"style":252},[181],[123,12511,1349],{"className":12512},[186]," (never in the lead) up to ",[123,12515,12517,12530],{"className":12516},[126],[123,12518,12520],{"className":12519},[130],[132,12521,12522],{"xmlns":134},[136,12523,12524,12528],{},[139,12525,12526],{},[430,12527,432],{},[166,12529,432],{"encoding":168},[123,12531,12533],{"className":12532,"ariaHidden":147},[173],[123,12534,12536,12539],{"className":12535},[177],[123,12537],{"className":12538,"style":572},[181],[123,12540,432],{"className":12541},[186,504]," (in the lead the whole game), so the fraction of time ahead is simply ",[123,12544,12546,12584],{"className":12545},[126],[123,12547,12549],{"className":12548},[130],[132,12550,12551],{"xmlns":134},[136,12552,12553,12581],{},[139,12554,12555,12571,12573],{},[11166,12556,12557],{"scriptlevel":1349,"displaystyle":868},[426,12558,12559,12565],{},[139,12560,12561,12563],{},[142,12562,690],{},[430,12564,435],{},[139,12566,12567,12569],{},[142,12568,690],{},[430,12570,432],{},[145,12572,695],{},[11166,12574,12575],{"scriptlevel":1349,"displaystyle":868},[426,12576,12577,12579],{},[430,12578,435],{},[430,12580,432],{},[166,12582,12583],{"encoding":168},"\\tfrac{2k}{2n} = \\tfrac{k}{n}",[123,12585,12587,12676],{"className":12586,"ariaHidden":147},[173],[123,12588,12590,12593,12667,12670,12673],{"className":12589},[177],[123,12591],{"className":12592,"style":1560},[181],[123,12594,12596,12599,12664],{"className":12595},[186],[123,12597],{"className":12598},[458,992],[123,12600,12602],{"className":12601},[426],[123,12603,12605,12656],{"className":12604},[472,473],[123,12606,12608,12653],{"className":12607},[477],[123,12609,12611,12628,12636],{"className":12610,"style":1005},[481],[123,12612,12613,12616],{"style":1008},[123,12614],{"className":12615,"style":1012},[489],[123,12617,12619],{"className":12618},[494,495,496,497],[123,12620,12622,12625],{"className":12621},[186,497],[123,12623,690],{"className":12624},[186,497],[123,12626,432],{"className":12627},[186,504,497],[123,12629,12630,12633],{"style":1046},[123,12631],{"className":12632,"style":1012},[489],[123,12634],{"className":12635,"style":1054},[1053],[123,12637,12638,12641],{"style":1057},[123,12639],{"className":12640,"style":1012},[489],[123,12642,12644],{"className":12643},[494,495,496,497],[123,12645,12647,12650],{"className":12646},[186,497],[123,12648,690],{"className":12649},[186,497],[123,12651,435],{"className":12652,"style":505},[186,504,497],[123,12654,525],{"className":12655},[524],[123,12657,12659],{"className":12658},[477],[123,12660,12662],{"className":12661,"style":532},[481],[123,12663],{},[123,12665],{"className":12666},[538,992],[123,12668],{"className":12669,"style":781},[194],[123,12671,695],{"className":12672},[785],[123,12674],{"className":12675,"style":781},[194],[123,12677,12679,12682],{"className":12678},[177],[123,12680],{"className":12681,"style":1560},[181],[123,12683,12685,12688,12747],{"className":12684},[186],[123,12686],{"className":12687},[458,992],[123,12689,12691],{"className":12690},[426],[123,12692,12694,12739],{"className":12693},[472,473],[123,12695,12697,12736],{"className":12696},[477],[123,12698,12700,12714,12722],{"className":12699,"style":1005},[481],[123,12701,12702,12705],{"style":1008},[123,12703],{"className":12704,"style":1012},[489],[123,12706,12708],{"className":12707},[494,495,496,497],[123,12709,12711],{"className":12710},[186,497],[123,12712,432],{"className":12713},[186,504,497],[123,12715,12716,12719],{"style":1046},[123,12717],{"className":12718,"style":1012},[489],[123,12720],{"className":12721,"style":1054},[1053],[123,12723,12724,12727],{"style":1057},[123,12725],{"className":12726,"style":1012},[489],[123,12728,12730],{"className":12729},[494,495,496,497],[123,12731,12733],{"className":12732},[186,497],[123,12734,435],{"className":12735,"style":505},[186,504,497],[123,12737,525],{"className":12738},[524],[123,12740,12742],{"className":12741},[477],[123,12743,12745],{"className":12744,"style":532},[481],[123,12746],{},[123,12748],{"className":12749},[538,992],[30,12751,12752,12753,12781,12782,8525,12838,12889,12890,12918,12919,12947,12948,12976,12977,13005,13006,12947,13034,12976,13134,13162,13163,13264,13265,13365,13366,13465,13466,13469],{},"Let's check it for a tiny game of just ",[123,12754,12756,12769],{"className":12755},[126],[123,12757,12759],{"className":12758},[130],[132,12760,12761],{"xmlns":134},[136,12762,12763,12767],{},[139,12764,12765],{},[142,12766,151],{},[166,12768,151],{"encoding":168},[123,12770,12772],{"className":12771,"ariaHidden":147},[173],[123,12773,12775,12778],{"className":12774},[177],[123,12776],{"className":12777,"style":252},[181],[123,12779,151],{"className":12780},[186]," tosses, so ",[123,12783,12785,12805],{"className":12784},[126],[123,12786,12788],{"className":12787},[130],[132,12789,12790],{"xmlns":134},[136,12791,12792,12802],{},[139,12793,12794,12796,12798,12800],{},[142,12795,690],{},[430,12797,432],{},[145,12799,695],{},[142,12801,151],{},[166,12803,12804],{"encoding":168},"2n = 4",[123,12806,12808,12829],{"className":12807,"ariaHidden":147},[173],[123,12809,12811,12814,12817,12820,12823,12826],{"className":12810},[177],[123,12812],{"className":12813,"style":252},[181],[123,12815,690],{"className":12816},[186],[123,12818,432],{"className":12819},[186,504],[123,12821],{"className":12822,"style":781},[194],[123,12824,695],{"className":12825},[785],[123,12827],{"className":12828,"style":781},[194],[123,12830,12832,12835],{"className":12831},[177],[123,12833],{"className":12834,"style":252},[181],[123,12836,151],{"className":12837},[186],[123,12839,12841,12859],{"className":12840},[126],[123,12842,12844],{"className":12843},[130],[132,12845,12846],{"xmlns":134},[136,12847,12848,12856],{},[139,12849,12850,12852,12854],{},[430,12851,432],{},[145,12853,695],{},[142,12855,690],{},[166,12857,12858],{"encoding":168},"n = 2",[123,12860,12862,12880],{"className":12861,"ariaHidden":147},[173],[123,12863,12865,12868,12871,12874,12877],{"className":12864},[177],[123,12866],{"className":12867,"style":572},[181],[123,12869,432],{"className":12870},[186,504],[123,12872],{"className":12873,"style":781},[194],[123,12875,695],{"className":12876},[785],[123,12878],{"className":12879,"style":781},[194],[123,12881,12883,12886],{"className":12882},[177],[123,12884],{"className":12885,"style":252},[181],[123,12887,690],{"className":12888},[186],". Now ",[123,12891,12893,12906],{"className":12892},[126],[123,12894,12896],{"className":12895},[130],[132,12897,12898],{"xmlns":134},[136,12899,12900,12904],{},[139,12901,12902],{},[430,12903,435],{},[166,12905,435],{"encoding":168},[123,12907,12909],{"className":12908,"ariaHidden":147},[173],[123,12910,12912,12915],{"className":12911},[177],[123,12913],{"className":12914,"style":602},[181],[123,12916,435],{"className":12917,"style":505},[186,504]," can only be ",[123,12920,12922,12935],{"className":12921},[126],[123,12923,12925],{"className":12924},[130],[132,12926,12927],{"xmlns":134},[136,12928,12929,12933],{},[139,12930,12931],{},[142,12932,1349],{},[166,12934,1349],{"encoding":168},[123,12936,12938],{"className":12937,"ariaHidden":147},[173],[123,12939,12941,12944],{"className":12940},[177],[123,12942],{"className":12943,"style":252},[181],[123,12945,1349],{"className":12946},[186],", ",[123,12949,12951,12964],{"className":12950},[126],[123,12952,12954],{"className":12953},[130],[132,12955,12956],{"xmlns":134},[136,12957,12958,12962],{},[139,12959,12960],{},[142,12961,80],{},[166,12963,80],{"encoding":168},[123,12965,12967],{"className":12966,"ariaHidden":147},[173],[123,12968,12970,12973],{"className":12969},[177],[123,12971],{"className":12972,"style":252},[181],[123,12974,80],{"className":12975},[186],", or ",[123,12978,12980,12993],{"className":12979},[126],[123,12981,12983],{"className":12982},[130],[132,12984,12985],{"xmlns":134},[136,12986,12987,12991],{},[139,12988,12989],{},[142,12990,690],{},[166,12992,690],{"encoding":168},[123,12994,12996],{"className":12995,"ariaHidden":147},[173],[123,12997,12999,13002],{"className":12998},[177],[123,13000],{"className":13001,"style":252},[181],[123,13003,690],{"className":13004},[186],", so the fraction of time in the lead is ",[123,13007,13009,13022],{"className":13008},[126],[123,13010,13012],{"className":13011},[130],[132,13013,13014],{"xmlns":134},[136,13015,13016,13020],{},[139,13017,13018],{},[142,13019,1349],{},[166,13021,1349],{"encoding":168},[123,13023,13025],{"className":13024,"ariaHidden":147},[173],[123,13026,13028,13031],{"className":13027},[177],[123,13029],{"className":13030,"style":252},[181],[123,13032,1349],{"className":13033},[186],[123,13035,13037,13057],{"className":13036},[126],[123,13038,13040],{"className":13039},[130],[132,13041,13042],{"xmlns":134},[136,13043,13044,13054],{},[139,13045,13046],{},[11166,13047,13048],{"scriptlevel":1349,"displaystyle":868},[426,13049,13050,13052],{},[142,13051,80],{},[142,13053,690],{},[166,13055,13056],{"encoding":168},"\\tfrac12",[123,13058,13060],{"className":13059,"ariaHidden":147},[173],[123,13061,13063,13066],{"className":13062},[177],[123,13064],{"className":13065,"style":1751},[181],[123,13067,13069,13072,13131],{"className":13068},[186],[123,13070],{"className":13071},[458,992],[123,13073,13075],{"className":13074},[426],[123,13076,13078,13123],{"className":13077},[472,473],[123,13079,13081,13120],{"className":13080},[477],[123,13082,13084,13098,13106],{"className":13083,"style":1678},[481],[123,13085,13086,13089],{"style":1008},[123,13087],{"className":13088,"style":1012},[489],[123,13090,13092],{"className":13091},[494,495,496,497],[123,13093,13095],{"className":13094},[186,497],[123,13096,690],{"className":13097},[186,497],[123,13099,13100,13103],{"style":1046},[123,13101],{"className":13102,"style":1012},[489],[123,13104],{"className":13105,"style":1054},[1053],[123,13107,13108,13111],{"style":1057},[123,13109],{"className":13110,"style":1012},[489],[123,13112,13114],{"className":13113},[494,495,496,497],[123,13115,13117],{"className":13116},[186,497],[123,13118,80],{"className":13119},[186,497],[123,13121,525],{"className":13122},[524],[123,13124,13126],{"className":13125},[477],[123,13127,13129],{"className":13128,"style":532},[481],[123,13130],{},[123,13132],{"className":13133},[538,992],[123,13135,13137,13150],{"className":13136},[126],[123,13138,13140],{"className":13139},[130],[132,13141,13142],{"xmlns":134},[136,13143,13144,13148],{},[139,13145,13146],{},[142,13147,80],{},[166,13149,80],{"encoding":168},[123,13151,13153],{"className":13152,"ariaHidden":147},[173],[123,13154,13156,13159],{"className":13155},[177],[123,13157],{"className":13158,"style":252},[181],[123,13160,80],{"className":13161},[186],". The three probabilities come out as ",[123,13164,13166,13187],{"className":13165},[126],[123,13167,13169],{"className":13168},[130],[132,13170,13171],{"xmlns":134},[136,13172,13173,13184],{},[139,13174,13175],{},[11166,13176,13177],{"scriptlevel":1349,"displaystyle":868},[426,13178,13179,13181],{},[142,13180,156],{},[142,13182,13183],{},"16",[166,13185,13186],{"encoding":168},"\\tfrac{6}{16}",[123,13188,13190],{"className":13189,"ariaHidden":147},[173],[123,13191,13193,13196],{"className":13192},[177],[123,13194],{"className":13195,"style":1751},[181],[123,13197,13199,13202,13261],{"className":13198},[186],[123,13200],{"className":13201},[458,992],[123,13203,13205],{"className":13204},[426],[123,13206,13208,13253],{"className":13207},[472,473],[123,13209,13211,13250],{"className":13210},[477],[123,13212,13214,13228,13236],{"className":13213,"style":1678},[481],[123,13215,13216,13219],{"style":1008},[123,13217],{"className":13218,"style":1012},[489],[123,13220,13222],{"className":13221},[494,495,496,497],[123,13223,13225],{"className":13224},[186,497],[123,13226,13183],{"className":13227},[186,497],[123,13229,13230,13233],{"style":1046},[123,13231],{"className":13232,"style":1012},[489],[123,13234],{"className":13235,"style":1054},[1053],[123,13237,13238,13241],{"style":1057},[123,13239],{"className":13240,"style":1012},[489],[123,13242,13244],{"className":13243},[494,495,496,497],[123,13245,13247],{"className":13246},[186,497],[123,13248,156],{"className":13249},[186,497],[123,13251,525],{"className":13252},[524],[123,13254,13256],{"className":13255},[477],[123,13257,13259],{"className":13258,"style":532},[481],[123,13260],{},[123,13262],{"className":13263},[538,992]," for \"never ahead\", ",[123,13266,13268,13288],{"className":13267},[126],[123,13269,13271],{"className":13270},[130],[132,13272,13273],{"xmlns":134},[136,13274,13275,13285],{},[139,13276,13277],{},[11166,13278,13279],{"scriptlevel":1349,"displaystyle":868},[426,13280,13281,13283],{},[142,13282,151],{},[142,13284,13183],{},[166,13286,13287],{"encoding":168},"\\tfrac{4}{16}",[123,13289,13291],{"className":13290,"ariaHidden":147},[173],[123,13292,13294,13297],{"className":13293},[177],[123,13295],{"className":13296,"style":1751},[181],[123,13298,13300,13303,13362],{"className":13299},[186],[123,13301],{"className":13302},[458,992],[123,13304,13306],{"className":13305},[426],[123,13307,13309,13354],{"className":13308},[472,473],[123,13310,13312,13351],{"className":13311},[477],[123,13313,13315,13329,13337],{"className":13314,"style":1678},[481],[123,13316,13317,13320],{"style":1008},[123,13318],{"className":13319,"style":1012},[489],[123,13321,13323],{"className":13322},[494,495,496,497],[123,13324,13326],{"className":13325},[186,497],[123,13327,13183],{"className":13328},[186,497],[123,13330,13331,13334],{"style":1046},[123,13332],{"className":13333,"style":1012},[489],[123,13335],{"className":13336,"style":1054},[1053],[123,13338,13339,13342],{"style":1057},[123,13340],{"className":13341,"style":1012},[489],[123,13343,13345],{"className":13344},[494,495,496,497],[123,13346,13348],{"className":13347},[186,497],[123,13349,151],{"className":13350},[186,497],[123,13352,525],{"className":13353},[524],[123,13355,13357],{"className":13356},[477],[123,13358,13360],{"className":13359,"style":532},[481],[123,13361],{},[123,13363],{"className":13364},[538,992]," for \"ahead exactly half the time\", and ",[123,13367,13369,13388],{"className":13368},[126],[123,13370,13372],{"className":13371},[130],[132,13373,13374],{"xmlns":134},[136,13375,13376,13386],{},[139,13377,13378],{},[11166,13379,13380],{"scriptlevel":1349,"displaystyle":868},[426,13381,13382,13384],{},[142,13383,156],{},[142,13385,13183],{},[166,13387,13186],{"encoding":168},[123,13389,13391],{"className":13390,"ariaHidden":147},[173],[123,13392,13394,13397],{"className":13393},[177],[123,13395],{"className":13396,"style":1751},[181],[123,13398,13400,13403,13462],{"className":13399},[186],[123,13401],{"className":13402},[458,992],[123,13404,13406],{"className":13405},[426],[123,13407,13409,13454],{"className":13408},[472,473],[123,13410,13412,13451],{"className":13411},[477],[123,13413,13415,13429,13437],{"className":13414,"style":1678},[481],[123,13416,13417,13420],{"style":1008},[123,13418],{"className":13419,"style":1012},[489],[123,13421,13423],{"className":13422},[494,495,496,497],[123,13424,13426],{"className":13425},[186,497],[123,13427,13183],{"className":13428},[186,497],[123,13430,13431,13434],{"style":1046},[123,13432],{"className":13433,"style":1012},[489],[123,13435],{"className":13436,"style":1054},[1053],[123,13438,13439,13442],{"style":1057},[123,13440],{"className":13441,"style":1012},[489],[123,13443,13445],{"className":13444},[494,495,496,497],[123,13446,13448],{"className":13447},[186,497],[123,13449,156],{"className":13450},[186,497],[123,13452,525],{"className":13453},[524],[123,13455,13457],{"className":13456},[477],[123,13458,13460],{"className":13459,"style":532},[481],[123,13461],{},[123,13463],{"className":13464},[538,992]," for \"ahead the whole game\". Even in this baby example the U is already there: the two extremes are each ",[47,13467,13468],{},"more"," likely than the fair-looking middle. Make the game longer and the U only deepens.",[30,13471,13472,13473,13476],{},"Here is the part that always makes us smile — quite literally. Plot these probabilities for a small game and you get just a handful of points tracing the U. Because the values at the two ends tower over the sagging middle, those end points sit like a pair of wide eyes above a low, curved mouth: the arcsine distribution appears to be ",[47,13474,13475],{},"grinning"," back at you. It is a fitting little joke — the curve whose whole message is that a fair coin will almost always look unfair wears a face that seems rather pleased about it.",[30,13478,13479],{},[13480,13481],"img",{"alt":13482,"src":13483},"Probabilities of the arcsine distribution p(2k,2n) plotted against the fraction k\u002Fn for a small number of tosses; the few points form a U-shape that resembles a smiley face (the tall extremes are the eyes)","\u002Fimg\u002Fuploads\u002FPlot-of-the-probabilities-p-2k-2n-that-a-particle-spends-a-fraction-k-n-of-its-time-on.png",[30,13485,13486,13487,13489],{},"The truly magical part is that the ",[42,13488,10984],{}," arcsine law shows up for three different questions about the very same walk:",[72,13491,13492,13502,13511],{},[75,13493,13494,13497,13498,13501],{},[42,13495,13496],{},"First arcsine law"," — the ",[47,13499,13500],{},"fraction of time spent in the lead"," (above zero). Extremes win.",[75,13503,13504,13497,13507,13510],{},[42,13505,13506],{},"Second arcsine law",[47,13508,13509],{},"moment the walk hits its maximum",". The peak is most likely to happen near the very start or the very end, and least likely in the middle.",[75,13512,13513,13497,13516,13519],{},[42,13514,13515],{},"Third arcsine law",[47,13517,13518],{},"last time the walk touches zero"," (the final tie). Again: an early or a late final tie is far more likely than one halfway through.",[30,13521,13522],{},"Three different questions, one weird curve. That repetition is what makes the arcsine laws feel like a piece of hidden machinery in randomness itself.",[30,13524,13525,13526,13531,13532,13535,13536,13539,13540,13568,13569,13764,13765,13872],{},"One delightful footnote on the name. We have met the word \"arcsine\" on this blog before, in ",[2779,13527,13530],{"href":13528,"rel":13529},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fflipping-a-coin-until-you-get-pi\u002F",[2783],"the article about flipping a coin until you reach π",". That piece is about the arcsine ",[47,13533,13534],{},"function"," from trigonometry; this one is about the arcsine ",[47,13537,13538],{},"distribution",". They are different objects — but not strangers. The distribution earns its name because the chance of leading less than a fraction ",[123,13541,13543,13556],{"className":13542},[126],[123,13544,13546],{"className":13545},[130],[132,13547,13548],{"xmlns":134},[136,13549,13550,13554],{},[139,13551,13552],{},[430,13553,6365],{},[166,13555,6365],{"encoding":168},[123,13557,13559],{"className":13558,"ariaHidden":147},[173],[123,13560,13562,13565],{"className":13561},[177],[123,13563],{"className":13564,"style":572},[181],[123,13566,6365],{"className":13567},[186,504]," of the time is exactly ",[123,13570,13572,13611],{"className":13571},[126],[123,13573,13575],{"className":13574},[130],[132,13576,13577],{"xmlns":134},[136,13578,13579,13608],{},[139,13580,13581,13589,13592,13595,13598],{},[11166,13582,13583],{"scriptlevel":1349,"displaystyle":868},[426,13584,13585,13587],{},[142,13586,690],{},[430,13588,11453],{},[430,13590,13591],{},"arcsin",[145,13593,13594],{},"⁡",[1901,13596,13597],{}," ⁣",[139,13599,13600,13602,13606],{},[145,13601,424],{"fence":147},[11455,13603,13604],{},[430,13605,6365],{},[145,13607,438],{"fence":147},[166,13609,13610],{"encoding":168},"\\tfrac{2}{\\pi}\\arcsin\\!\\left(\\sqrt{x}\\right)",[123,13612,13614],{"className":13613,"ariaHidden":147},[173],[123,13615,13617,13620,13688,13691,13694,13698,13701],{"className":13616},[177],[123,13618],{"className":13619,"style":1751},[181],[123,13621,13623,13626,13685],{"className":13622},[186],[123,13624],{"className":13625},[458,992],[123,13627,13629],{"className":13628},[426],[123,13630,13632,13677],{"className":13631},[472,473],[123,13633,13635,13674],{"className":13634},[477],[123,13636,13638,13652,13660],{"className":13637,"style":1678},[481],[123,13639,13640,13643],{"style":1008},[123,13641],{"className":13642,"style":1012},[489],[123,13644,13646],{"className":13645},[494,495,496,497],[123,13647,13649],{"className":13648},[186,497],[123,13650,11453],{"className":13651,"style":11545},[186,504,497],[123,13653,13654,13657],{"style":1046},[123,13655],{"className":13656,"style":1012},[489],[123,13658],{"className":13659,"style":1054},[1053],[123,13661,13662,13665],{"style":1057},[123,13663],{"className":13664,"style":1012},[489],[123,13666,13668],{"className":13667},[494,495,496,497],[123,13669,13671],{"className":13670},[186,497],[123,13672,690],{"className":13673},[186,497],[123,13675,525],{"className":13676},[524],[123,13678,13680],{"className":13679},[477],[123,13681,13683],{"className":13682,"style":532},[481],[123,13684],{},[123,13686],{"className":13687},[538,992],[123,13689],{"className":13690,"style":195},[194],[123,13692,13591],{"className":13693},[8039],[123,13695],{"className":13696,"style":13697},[194],"margin-right:-0.1667em;",[123,13699],{"className":13700,"style":195},[194],[123,13702,13704,13707,13761],{"className":13703},[2195],[123,13705,424],{"className":13706,"style":460},[458,459],[123,13708,13710],{"className":13709},[186,11549],[123,13711,13713,13752],{"className":13712},[472,473],[123,13714,13716,13749],{"className":13715},[477],[123,13717,13720,13734],{"className":13718,"style":13719},[481],"height:0.8003em;",[123,13721,13724,13727],{"className":13722,"style":13723},[11563],"top:-3em;",[123,13725],{"className":13726,"style":1012},[489],[123,13728,13731],{"className":13729,"style":13730},[186],"padding-left:0.833em;",[123,13732,6365],{"className":13733},[186,504],[123,13735,13737,13740],{"style":13736},"top:-2.7603em;",[123,13738],{"className":13739,"style":1012},[489],[123,13741,13744],{"className":13742,"style":13743},[11600],"min-width:0.853em;height:1.08em;",[11603,13745,13747],{"xmlns":11605,"width":11606,"height":13746,"viewBox":11608,"preserveAspectRatio":11609},"1.08em",[11611,13748],{"d":11613},[123,13750,525],{"className":13751},[524],[123,13753,13755],{"className":13754},[477],[123,13756,13759],{"className":13757,"style":13758},[481],"height:0.2397em;",[123,13760],{},[123,13762,438],{"className":13763,"style":460},[538,459],", the arcsine function in disguise. And both stories are secretly powered by the same number, the central binomial coefficient ",[123,13766,13768,13794],{"className":13767},[126],[123,13769,13771],{"className":13770},[130],[132,13772,13773],{"xmlns":134},[136,13774,13775,13791],{},[139,13776,13777,13779,13789],{},[145,13778,424],{"fence":147},[426,13780,13781,13787],{"linethickness":428},[139,13782,13783,13785],{},[142,13784,690],{},[430,13786,432],{},[430,13788,432],{},[145,13790,438],{"fence":147},[166,13792,13793],{"encoding":168},"\\binom{2n}{n}",[123,13795,13797],{"className":13796,"ariaHidden":147},[173],[123,13798,13800,13803],{"className":13799},[177],[123,13801],{"className":13802,"style":710},[181],[123,13804,13806,13812,13866],{"className":13805},[186],[123,13807,13809],{"className":13808,"style":460},[458,459],[123,13810,424],{"className":13811},[464,465],[123,13813,13815],{"className":13814},[426],[123,13816,13818,13858],{"className":13817},[472,473],[123,13819,13821,13855],{"className":13820},[477],[123,13822,13824,13838],{"className":13823,"style":732},[481],[123,13825,13826,13829],{"style":485},[123,13827],{"className":13828,"style":490},[489],[123,13830,13832],{"className":13831},[494,495,496,497],[123,13833,13835],{"className":13834},[186,497],[123,13836,432],{"className":13837},[186,504,497],[123,13839,13840,13843],{"style":508},[123,13841],{"className":13842,"style":490},[489],[123,13844,13846],{"className":13845},[494,495,496,497],[123,13847,13849,13852],{"className":13848},[186,497],[123,13850,690],{"className":13851},[186,497],[123,13853,432],{"className":13854},[186,504,497],[123,13856,525],{"className":13857},[524],[123,13859,13861],{"className":13860},[477],[123,13862,13864],{"className":13863,"style":532},[481],[123,13865],{},[123,13867,13869],{"className":13868,"style":460},[538,459],[123,13870,438],{"className":13871},[464,465],". Randomness keeps reusing its favorite parts.",[34,13874,13876],{"id":13875},"what-the-central-limit-theorem-doesnt-tell-us","What the Central Limit Theorem doesn't tell us",[30,13878,13879,13880],{},"At this point a careful reader feels a contradiction coming. We said the final score follows a tidy bell curve centered at a tie. Now we're saying the lead is almost never evenly shared. ",[42,13881,13882],{},"Is the Central Limit Theorem wrong?",[30,13884,13885],{},"No. The two laws are answering two completely different questions about the same coin. This is the journey-versus-destination split from the very beginning, made precise:",[2632,13887,13888,13898],{},[2635,13889,13890],{},[2638,13891,13892,13895],{},[2641,13893,13894],{"align":2898},"What we measure",[2641,13896,13897],{"align":2898},"Which law governs it",[2735,13899,13900,13908,13916,13923],{},[2638,13901,13902,13905],{},[2740,13903,13904],{"align":2898},"Final score (where the walk ends)",[2740,13906,13907],{"align":2898},"Central Limit Theorem — bell curve",[2638,13909,13910,13913],{},[2740,13911,13912],{"align":2898},"Fraction of time in the lead",[2740,13914,13915],{"align":2898},"Arcsine law — U-shape",[2638,13917,13918,13921],{},[2740,13919,13920],{"align":2898},"Time the maximum lead occurs",[2740,13922,13915],{"align":2898},[2638,13924,13925,13928],{},[2740,13926,13927],{"align":2898},"Moment of the last tie",[2740,13929,13915],{"align":2898},[30,13931,13932,13933,13936],{},"Same process. Different observables. Different answers. The coin can finish almost level (the bell curve, about the destination) ",[47,13934,13935],{},"and"," spend nearly the whole game lopsided (the arcsine law, about the journey). There is no contradiction — only two truths that live in the gap we marked out at the start.",[34,13938,13940],{"id":13939},"a-final-reflection-on-randomness","A final reflection on randomness",[30,13942,13943,13944,13946],{},"Why does the U-shape feel so wrong to us? Because we are storytellers. We don't experience a year, or a game, or a life, as a single final score. We live it moment by moment, as a ",[47,13945,11611],{},". And a path that lingers on one side for ages reads to us like momentum, like a streak, like a hot hand, like destiny — even when it is nothing but a fair coin doing exactly what fair coins do.",[30,13948,13949,13950,13953,13954,13959,13960,13963],{},"Our gut is unreliable at ",[47,13951,13952],{},"both"," ends of the spectrum. We badly underestimate how long an ordinary fair coin will stay one-sided — and we just as badly overestimate the truly spectacular streaks. As Evelyn Lamb explains in ",[2779,13955,13958],{"href":13956,"rel":13957},"https:\u002F\u002Fwww.scientificamerican.com\u002Fblog\u002Froots-of-unity\u002Fhas-anyone-ever-flipped-heads-76-times-in-a-row\u002F",[2783],"a lovely piece on whether anyone has ever flipped 76 heads in a row",", runs that ",[47,13961,13962],{},"feel"," like they should turn up every now and then are so astronomically unlikely that, across all the coins ever flipped, they essentially never have.",[30,13965,13966,13967,13970],{},"The Central Limit Theorem promises that randomness balances out in the long run. The arcsine laws add a quieter, stranger warning: ",[42,13968,13969],{},"that balancing can stay completely invisible while it is happening."," The books only even out when you step far enough back to see the whole thing at once.",[30,13972,13973],{},"So the next time a fair process looks unfair — a coin on a streak, a bad season where nothing goes your way — remember the molecule in the river. It is not broken. It is not rigged. It simply never promised to stay in the middle. Life comes in waves and as Einstein said, \"Life is like a bicycle; to keep your balance you must keep moving\".",[30,13975,10620,13976,13979],{},[2779,13977,10625],{"href":10623,"rel":13978},[2783]," and watch a fair coin look unfair and how streaks of repeated results are much more likely than you might expect.",{"title":105,"searchDepth":10628,"depth":10628,"links":13981},[13982,13983,13984,13985,13986,13987,13988],{"id":10701,"depth":10628,"text":10702},{"id":10832,"depth":10628,"text":10833},{"id":10945,"depth":10628,"text":10946},{"id":11042,"depth":10628,"text":11043},{"id":11353,"depth":10628,"text":11354},{"id":13875,"depth":10628,"text":13876},{"id":13939,"depth":10628,"text":13940},"2026-06-06T10:00:00Z","A fair coin tends to finish near a tie — yet it spends most of its life with one side in the lead. The story of random walks, Brownian motion, and the arcsine laws.",{"type":23,"value":13992},[13993,13995],[26,13994,10669],{"id":10674},[30,13996,10677],{},"advanced",{},"\u002Fblog\u002Ffair-coin-looks-unfair.en",{"title":10669,"description":13990},"fair-coin-random-walk-arcsine-law","blog\u002Ffair-coin-looks-unfair.en",[10784,10698,14004,14005,14006,14007,14008,14009,14010,14011],"central limit theorem","brownian motion","bell curve","probability distribution","galton board","fair coin","fairness","streaks","jHGCHWNIkprN_fcpAaAnZCGoNq8HeAPshCCJZ547QS4",{"id":14014,"title":14015,"_locale":8,"author":3,"body":14016,"date":26619,"description":26620,"excerpt":26621,"extension":10652,"level":13997,"meta":26767,"navigation":10655,"path":26768,"published":10655,"seo":26769,"series":15,"seriesOrder":10634,"slug":14020,"stem":26770,"tags":26771,"__hash__":26775},"blog\u002Fblog\u002Fpi-coin-flips-catalan-numbers.en.md","Flipping a Coin Until You Get Pi",{"type":23,"value":14017,"toc":26605},[14018,14021,14136,14173,14182,14185,14204,14207,14488,14492,14507,14511,14514,14552,14772,14776,14783,14828,15173,15177,15184,15191,15194,15200,15275,15399,15402,15408,15527,15564,15702,16049,16053,16059,16063,17430,17433,17853,17950,17953,17957,18091,18120,18123,18154,18422,18725,18728,19237,19743,20007,20014,20018,20219,20223,20293,21034,21093,21520,21523,22853,23007,23011,23014,23274,23537,24050,24053,24824,24831,25329,25418,25472,25840,25952,26092,26095,26403,26448,26451,26487,26492,26494,26497,26528,26560,26563,26595,26598],[26,14019,14015],{"id":14020},"flipping-a-coin-until-you-get-pi",[30,14022,14023,14024,14029,14030,14035,14036,827],{},"Matt Parker, from the ",[2779,14025,14028],{"href":14026,"rel":14027},"https:\u002F\u002Fwww.youtube.com\u002F@standupmaths",[2783],"Stand-up Maths"," channel, recently published ",[2779,14031,14034],{"href":14032,"rel":14033},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=kahGSss6SsU",[2783],"a video"," with a premise as simple to state as it is hard to believe: if you flip a coin repeatedly until heads outnumber tails, and compute the average ratio of heads to total flips... the result is ",[123,14037,14039,14057],{"className":14038},[126],[123,14040,14042],{"className":14041},[130],[132,14043,14044],{"xmlns":134},[136,14045,14046,14054],{},[139,14047,14048],{},[426,14049,14050,14052],{},[430,14051,11453],{},[142,14053,151],{},[166,14055,14056],{"encoding":168},"\\frac{\\pi}{4}",[123,14058,14060],{"className":14059,"ariaHidden":147},[173],[123,14061,14063,14067],{"className":14062},[177],[123,14064],{"className":14065,"style":14066},[181],"height:1.0404em;vertical-align:-0.345em;",[123,14068,14070,14073,14133],{"className":14069},[186],[123,14071],{"className":14072},[458,992],[123,14074,14076],{"className":14075},[426],[123,14077,14079,14125],{"className":14078},[472,473],[123,14080,14082,14122],{"className":14081},[477],[123,14083,14086,14100,14108],{"className":14084,"style":14085},[481],"height:0.6954em;",[123,14087,14088,14091],{"style":1008},[123,14089],{"className":14090,"style":1012},[489],[123,14092,14094],{"className":14093},[494,495,496,497],[123,14095,14097],{"className":14096},[186,497],[123,14098,151],{"className":14099},[186,497],[123,14101,14102,14105],{"style":1046},[123,14103],{"className":14104,"style":1012},[489],[123,14106],{"className":14107,"style":1054},[1053],[123,14109,14110,14113],{"style":1057},[123,14111],{"className":14112,"style":1012},[489],[123,14114,14116],{"className":14115},[494,495,496,497],[123,14117,14119],{"className":14118},[186,497],[123,14120,11453],{"className":14121,"style":11545},[186,504,497],[123,14123,525],{"className":14124},[524],[123,14126,14128],{"className":14127},[477],[123,14129,14131],{"className":14130,"style":532},[481],[123,14132],{},[123,14134],{"className":14135},[538,992],[30,14137,14138,14139,14168,14169,14172],{},"Yes, ",[123,14140,14142,14156],{"className":14141},[126],[123,14143,14145],{"className":14144},[130],[132,14146,14147],{"xmlns":134},[136,14148,14149,14153],{},[139,14150,14151],{},[430,14152,11453],{},[166,14154,14155],{"encoding":168},"\\pi",[123,14157,14159],{"className":14158,"ariaHidden":147},[173],[123,14160,14162,14165],{"className":14161},[177],[123,14163],{"className":14164,"style":572},[181],[123,14166,11453],{"className":14167,"style":11545},[186,504],". ",[47,14170,14171],{},"The number that belongs to circles",". Hiding inside coin flips.",[30,14174,14175,14176,14181],{},"In our ",[2779,14177,14180],{"href":14178,"rel":14179},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fpascals-triangle-combinatorics-and-coin-flips\u002F",[2783],"previous article on Pascal's triangle"," we saw how a triangle built from simple addition contained, within its rows, the seeds of combinatorics, algebra, and probability. Today we are going to pull on that same thread. And it will take us somewhere no one expected.",[30,14183,14184],{},"The rules of the experiment are straightforward:",[72,14186,14187,14190,14197],{},[75,14188,14189],{},"Flip a fair coin repeatedly.",[75,14191,14192,14193,14196],{},"Stop ",[42,14194,14195],{},"the moment"," the number of heads exceeds the number of tails.",[75,14198,14199,14200,14203],{},"Record the ",[42,14201,14202],{},"ratio of heads"," to total flips for that sequence.",[30,14205,14206],{},"The question is: if we repeat this process infinitely many times and compute the average of all those ratios... what do we get?",[30,14208,14209,14210,14330,14331,14429,14430,8525,14459,14487],{},"Before answering, let's think about what range that ratio can take. In the best case, the very first flip is heads: we stop immediately and the ratio is ",[123,14211,14213,14235],{"className":14212},[126],[123,14214,14216],{"className":14215},[130],[132,14217,14218],{"xmlns":134},[136,14219,14220,14232],{},[139,14221,14222,14228,14230],{},[426,14223,14224,14226],{},[142,14225,80],{},[142,14227,80],{},[145,14229,695],{},[142,14231,80],{},[166,14233,14234],{"encoding":168},"\\frac{1}{1} = 1",[123,14236,14238,14321],{"className":14237,"ariaHidden":147},[173],[123,14239,14241,14244,14312,14315,14318],{"className":14240},[177],[123,14242],{"className":14243,"style":1751},[181],[123,14245,14247,14250,14309],{"className":14246},[186],[123,14248],{"className":14249},[458,992],[123,14251,14253],{"className":14252},[426],[123,14254,14256,14301],{"className":14255},[472,473],[123,14257,14259,14298],{"className":14258},[477],[123,14260,14262,14276,14284],{"className":14261,"style":1678},[481],[123,14263,14264,14267],{"style":1008},[123,14265],{"className":14266,"style":1012},[489],[123,14268,14270],{"className":14269},[494,495,496,497],[123,14271,14273],{"className":14272},[186,497],[123,14274,80],{"className":14275},[186,497],[123,14277,14278,14281],{"style":1046},[123,14279],{"className":14280,"style":1012},[489],[123,14282],{"className":14283,"style":1054},[1053],[123,14285,14286,14289],{"style":1057},[123,14287],{"className":14288,"style":1012},[489],[123,14290,14292],{"className":14291},[494,495,496,497],[123,14293,14295],{"className":14294},[186,497],[123,14296,80],{"className":14297},[186,497],[123,14299,525],{"className":14300},[524],[123,14302,14304],{"className":14303},[477],[123,14305,14307],{"className":14306,"style":532},[481],[123,14308],{},[123,14310],{"className":14311},[538,992],[123,14313],{"className":14314,"style":781},[194],[123,14316,695],{"className":14317},[785],[123,14319],{"className":14320,"style":781},[194],[123,14322,14324,14327],{"className":14323},[177],[123,14325],{"className":14326,"style":252},[181],[123,14328,80],{"className":14329},[186]," (100% heads). In the worst case, we need thousands of flips before heads finally edges ahead by one, and the ratio will be barely above ",[123,14332,14334,14352],{"className":14333},[126],[123,14335,14337],{"className":14336},[130],[132,14338,14339],{"xmlns":134},[136,14340,14341,14349],{},[139,14342,14343],{},[426,14344,14345,14347],{},[142,14346,80],{},[142,14348,690],{},[166,14350,14351],{"encoding":168},"\\frac{1}{2}",[123,14353,14355],{"className":14354,"ariaHidden":147},[173],[123,14356,14358,14361],{"className":14357},[177],[123,14359],{"className":14360,"style":1751},[181],[123,14362,14364,14367,14426],{"className":14363},[186],[123,14365],{"className":14366},[458,992],[123,14368,14370],{"className":14369},[426],[123,14371,14373,14418],{"className":14372},[472,473],[123,14374,14376,14415],{"className":14375},[477],[123,14377,14379,14393,14401],{"className":14378,"style":1678},[481],[123,14380,14381,14384],{"style":1008},[123,14382],{"className":14383,"style":1012},[489],[123,14385,14387],{"className":14386},[494,495,496,497],[123,14388,14390],{"className":14389},[186,497],[123,14391,690],{"className":14392},[186,497],[123,14394,14395,14398],{"style":1046},[123,14396],{"className":14397,"style":1012},[489],[123,14399],{"className":14400,"style":1054},[1053],[123,14402,14403,14406],{"style":1057},[123,14404],{"className":14405,"style":1012},[489],[123,14407,14409],{"className":14408},[494,495,496,497],[123,14410,14412],{"className":14411},[186,497],[123,14413,80],{"className":14414},[186,497],[123,14416,525],{"className":14417},[524],[123,14419,14421],{"className":14420},[477],[123,14422,14424],{"className":14423,"style":532},[481],[123,14425],{},[123,14427],{"className":14428},[538,992],". So the average must land somewhere between ",[123,14431,14433,14447],{"className":14432},[126],[123,14434,14436],{"className":14435},[130],[132,14437,14438],{"xmlns":134},[136,14439,14440,14445],{},[139,14441,14442],{},[142,14443,14444],{},"0.5",[166,14446,14444],{"encoding":168},[123,14448,14450],{"className":14449,"ariaHidden":147},[173],[123,14451,14453,14456],{"className":14452},[177],[123,14454],{"className":14455,"style":252},[181],[123,14457,14444],{"className":14458},[186],[123,14460,14462,14475],{"className":14461},[126],[123,14463,14465],{"className":14464},[130],[132,14466,14467],{"xmlns":134},[136,14468,14469,14473],{},[139,14470,14471],{},[142,14472,80],{},[166,14474,80],{"encoding":168},[123,14476,14478],{"className":14477,"ariaHidden":147},[173],[123,14479,14481,14484],{"className":14480},[177],[123,14482],{"className":14483,"style":252},[181],[123,14485,80],{"className":14486},[186],". Let's find out exactly where.",[34,14489,14491],{"id":14490},"building-the-sequences-step-by-step","Building the sequences, step by step",[30,14493,14494,14495,14498,14499,14502,14503,827],{},"Let's construct the possible sequences starting from the shortest ones. We will use ",[42,14496,14497],{},"O"," for heads and ",[42,14500,14501],{},"X"," for tails — the same notation we use in ",[2779,14504,14506],{"href":10623,"rel":14505},[2783],"our coin flip tool",[395,14508,14510],{"id":14509},"length-1","Length 1",[30,14512,14513],{},"The shortest possible sequence: flip once, get heads.",[30,14515,14517],{"className":14516},[409],[123,14518,14520,14535],{"className":14519},[126],[123,14521,14523],{"className":14522},[130],[132,14524,14525],{"xmlns":134},[136,14526,14527,14532],{},[139,14528,14529],{},[1901,14530,14497],{"mathvariant":14531},"bold",[166,14533,14534],{"encoding":168},"\\textbf{O}",[123,14536,14538],{"className":14537,"ariaHidden":147},[173],[123,14539,14541,14545],{"className":14540},[177],[123,14542],{"className":14543,"style":14544},[181],"height:0.6861em;",[123,14546,14548],{"className":14547},[186,104],[123,14549,14497],{"className":14550},[186,14551],"textbf",[30,14553,14554,14555,14674,14675,827],{},"We stop. We have 1 heads and 0 tails. The ratio is ",[123,14556,14558,14579],{"className":14557},[126],[123,14559,14561],{"className":14560},[130],[132,14562,14563],{"xmlns":134},[136,14564,14565,14577],{},[139,14566,14567,14573,14575],{},[426,14568,14569,14571],{},[142,14570,80],{},[142,14572,80],{},[145,14574,695],{},[142,14576,80],{},[166,14578,14234],{"encoding":168},[123,14580,14582,14665],{"className":14581,"ariaHidden":147},[173],[123,14583,14585,14588,14656,14659,14662],{"className":14584},[177],[123,14586],{"className":14587,"style":1751},[181],[123,14589,14591,14594,14653],{"className":14590},[186],[123,14592],{"className":14593},[458,992],[123,14595,14597],{"className":14596},[426],[123,14598,14600,14645],{"className":14599},[472,473],[123,14601,14603,14642],{"className":14602},[477],[123,14604,14606,14620,14628],{"className":14605,"style":1678},[481],[123,14607,14608,14611],{"style":1008},[123,14609],{"className":14610,"style":1012},[489],[123,14612,14614],{"className":14613},[494,495,496,497],[123,14615,14617],{"className":14616},[186,497],[123,14618,80],{"className":14619},[186,497],[123,14621,14622,14625],{"style":1046},[123,14623],{"className":14624,"style":1012},[489],[123,14626],{"className":14627,"style":1054},[1053],[123,14629,14630,14633],{"style":1057},[123,14631],{"className":14632,"style":1012},[489],[123,14634,14636],{"className":14635},[494,495,496,497],[123,14637,14639],{"className":14638},[186,497],[123,14640,80],{"className":14641},[186,497],[123,14643,525],{"className":14644},[524],[123,14646,14648],{"className":14647},[477],[123,14649,14651],{"className":14650,"style":532},[481],[123,14652],{},[123,14654],{"className":14655},[538,992],[123,14657],{"className":14658,"style":781},[194],[123,14660,695],{"className":14661},[785],[123,14663],{"className":14664,"style":781},[194],[123,14666,14668,14671],{"className":14667},[177],[123,14669],{"className":14670,"style":252},[181],[123,14672,80],{"className":14673},[186]," and the probability of this happening is ",[123,14676,14678,14695],{"className":14677},[126],[123,14679,14681],{"className":14680},[130],[132,14682,14683],{"xmlns":134},[136,14684,14685,14693],{},[139,14686,14687],{},[426,14688,14689,14691],{},[142,14690,80],{},[142,14692,690],{},[166,14694,14351],{"encoding":168},[123,14696,14698],{"className":14697,"ariaHidden":147},[173],[123,14699,14701,14704],{"className":14700},[177],[123,14702],{"className":14703,"style":1751},[181],[123,14705,14707,14710,14769],{"className":14706},[186],[123,14708],{"className":14709},[458,992],[123,14711,14713],{"className":14712},[426],[123,14714,14716,14761],{"className":14715},[472,473],[123,14717,14719,14758],{"className":14718},[477],[123,14720,14722,14736,14744],{"className":14721,"style":1678},[481],[123,14723,14724,14727],{"style":1008},[123,14725],{"className":14726,"style":1012},[489],[123,14728,14730],{"className":14729},[494,495,496,497],[123,14731,14733],{"className":14732},[186,497],[123,14734,690],{"className":14735},[186,497],[123,14737,14738,14741],{"style":1046},[123,14739],{"className":14740,"style":1012},[489],[123,14742],{"className":14743,"style":1054},[1053],[123,14745,14746,14749],{"style":1057},[123,14747],{"className":14748,"style":1012},[489],[123,14750,14752],{"className":14751},[494,495,496,497],[123,14753,14755],{"className":14754},[186,497],[123,14756,80],{"className":14757},[186,497],[123,14759,525],{"className":14760},[524],[123,14762,14764],{"className":14763},[477],[123,14765,14767],{"className":14766,"style":532},[481],[123,14768],{},[123,14770],{"className":14771},[538,992],[395,14773,14775],{"id":14774},"length-3","Length 3",[30,14777,14778,14779,14782],{},"If the first flip is tails, we cannot stop. And with just two flips, the best we can achieve is a tie (1 heads, 1 tails) — a tie is not enough, we need heads to ",[47,14780,14781],{},"exceed"," tails. So the next possible sequence has length 3:",[30,14784,14786],{"className":14785},[409],[123,14787,14789,14812],{"className":14788},[126],[123,14790,14792],{"className":14791},[130],[132,14793,14794],{"xmlns":134},[136,14795,14796,14809],{},[139,14797,14798,14800,14803,14805,14807],{},[1901,14799,14501],{"mathvariant":14531},[1901,14801,14802],{}," ",[1901,14804,14497],{"mathvariant":14531},[1901,14806,14802],{},[1901,14808,14497],{"mathvariant":14531},[166,14810,14811],{"encoding":168},"\\textbf{X O O}",[123,14813,14815],{"className":14814,"ariaHidden":147},[173],[123,14816,14818,14821],{"className":14817},[177],[123,14819],{"className":14820,"style":14544},[181],[123,14822,14824],{"className":14823},[186,104],[123,14825,14827],{"className":14826},[186,14551],"X O O",[30,14829,14830,14831,14929,14930,15172],{},"Tails, heads, heads. The third flip gives us 2 heads versus 1 tails. Ratio: ",[123,14832,14834,14852],{"className":14833},[126],[123,14835,14837],{"className":14836},[130],[132,14838,14839],{"xmlns":134},[136,14840,14841,14849],{},[139,14842,14843],{},[426,14844,14845,14847],{},[142,14846,690],{},[142,14848,271],{},[166,14850,14851],{"encoding":168},"\\frac{2}{3}",[123,14853,14855],{"className":14854,"ariaHidden":147},[173],[123,14856,14858,14861],{"className":14857},[177],[123,14859],{"className":14860,"style":1751},[181],[123,14862,14864,14867,14926],{"className":14863},[186],[123,14865],{"className":14866},[458,992],[123,14868,14870],{"className":14869},[426],[123,14871,14873,14918],{"className":14872},[472,473],[123,14874,14876,14915],{"className":14875},[477],[123,14877,14879,14893,14901],{"className":14878,"style":1678},[481],[123,14880,14881,14884],{"style":1008},[123,14882],{"className":14883,"style":1012},[489],[123,14885,14887],{"className":14886},[494,495,496,497],[123,14888,14890],{"className":14889},[186,497],[123,14891,271],{"className":14892},[186,497],[123,14894,14895,14898],{"style":1046},[123,14896],{"className":14897,"style":1012},[489],[123,14899],{"className":14900,"style":1054},[1053],[123,14902,14903,14906],{"style":1057},[123,14904],{"className":14905,"style":1012},[489],[123,14907,14909],{"className":14908},[494,495,496,497],[123,14910,14912],{"className":14911},[186,497],[123,14913,690],{"className":14914},[186,497],[123,14916,525],{"className":14917},[524],[123,14919,14921],{"className":14920},[477],[123,14922,14924],{"className":14923,"style":532},[481],[123,14925],{},[123,14927],{"className":14928},[538,992]," (2 heads out of 3 flips). Probability: ",[123,14931,14933,14969],{"className":14932},[126],[123,14934,14936],{"className":14935},[130],[132,14937,14938],{"xmlns":134},[136,14939,14940,14966],{},[139,14941,14942,14958,14960],{},[2520,14943,14944,14956],{},[139,14945,14946,14948,14954],{},[145,14947,424],{"fence":147},[426,14949,14950,14952],{},[142,14951,80],{},[142,14953,690],{},[145,14955,438],{"fence":147},[142,14957,271],{},[145,14959,695],{},[426,14961,14962,14964],{},[142,14963,80],{},[142,14965,2427],{},[166,14967,14968],{"encoding":168},"\\left(\\frac{1}{2}\\right)^3 = \\frac{1}{8}",[123,14970,14972,15098],{"className":14971,"ariaHidden":147},[173],[123,14973,14975,14979,15089,15092,15095],{"className":14974},[177],[123,14976],{"className":14977,"style":14978},[181],"height:1.404em;vertical-align:-0.35em;",[123,14980,14982,15065],{"className":14981},[2195],[123,14983,14985,14991,15059],{"className":14984},[2195],[123,14986,14988],{"className":14987,"style":460},[458,459],[123,14989,424],{"className":14990},[464,465],[123,14992,14994,14997,15056],{"className":14993},[186],[123,14995],{"className":14996},[458,992],[123,14998,15000],{"className":14999},[426],[123,15001,15003,15048],{"className":15002},[472,473],[123,15004,15006,15045],{"className":15005},[477],[123,15007,15009,15023,15031],{"className":15008,"style":1678},[481],[123,15010,15011,15014],{"style":1008},[123,15012],{"className":15013,"style":1012},[489],[123,15015,15017],{"className":15016},[494,495,496,497],[123,15018,15020],{"className":15019},[186,497],[123,15021,690],{"className":15022},[186,497],[123,15024,15025,15028],{"style":1046},[123,15026],{"className":15027,"style":1012},[489],[123,15029],{"className":15030,"style":1054},[1053],[123,15032,15033,15036],{"style":1057},[123,15034],{"className":15035,"style":1012},[489],[123,15037,15039],{"className":15038},[494,495,496,497],[123,15040,15042],{"className":15041},[186,497],[123,15043,80],{"className":15044},[186,497],[123,15046,525],{"className":15047},[524],[123,15049,15051],{"className":15050},[477],[123,15052,15054],{"className":15053,"style":532},[481],[123,15055],{},[123,15057],{"className":15058},[538,992],[123,15060,15062],{"className":15061,"style":460},[538,459],[123,15063,438],{"className":15064},[464,465],[123,15066,15068],{"className":15067},[2548],[123,15069,15071],{"className":15070},[472],[123,15072,15074],{"className":15073},[477],[123,15075,15078],{"className":15076,"style":15077},[481],"height:1.054em;",[123,15079,15080,15083],{"style":9203},[123,15081],{"className":15082,"style":490},[489],[123,15084,15086],{"className":15085},[494,495,496,497],[123,15087,271],{"className":15088},[186,497],[123,15090],{"className":15091,"style":781},[194],[123,15093,695],{"className":15094},[785],[123,15096],{"className":15097,"style":781},[194],[123,15099,15101,15104],{"className":15100},[177],[123,15102],{"className":15103,"style":1751},[181],[123,15105,15107,15110,15169],{"className":15106},[186],[123,15108],{"className":15109},[458,992],[123,15111,15113],{"className":15112},[426],[123,15114,15116,15161],{"className":15115},[472,473],[123,15117,15119,15158],{"className":15118},[477],[123,15120,15122,15136,15144],{"className":15121,"style":1678},[481],[123,15123,15124,15127],{"style":1008},[123,15125],{"className":15126,"style":1012},[489],[123,15128,15130],{"className":15129},[494,495,496,497],[123,15131,15133],{"className":15132},[186,497],[123,15134,2427],{"className":15135},[186,497],[123,15137,15138,15141],{"style":1046},[123,15139],{"className":15140,"style":1012},[489],[123,15142],{"className":15143,"style":1054},[1053],[123,15145,15146,15149],{"style":1057},[123,15147],{"className":15148,"style":1012},[489],[123,15150,15152],{"className":15151},[494,495,496,497],[123,15153,15155],{"className":15154},[186,497],[123,15156,80],{"className":15157},[186,497],[123,15159,525],{"className":15160},[524],[123,15162,15164],{"className":15163},[477],[123,15165,15167],{"className":15166,"style":532},[481],[123,15168],{},[123,15170],{"className":15171},[538,992],". And it is the only possible sequence of this length.",[395,15174,15176],{"id":15175},"length-5","Length 5",[30,15178,15179,15180,15183],{},"This is where things get interesting. We need 3 heads and 2 tails, and the fifth flip (the one that stops us) must be heads. Moreover — and this is the key constraint — ",[42,15181,15182],{},"at no point before the end can we have had more heads than tails",", because if that had happened, we would have stopped earlier.",[30,15185,15186,15187,15190],{},"It is not enough for the sequence to end with more heads than tails; it must be ",[42,15188,15189],{},"the first time"," it happens.",[30,15192,15193],{},"How many sequences satisfy these conditions? Only two:",[99,15195,15198],{"className":15196,"code":15197,"language":104,"meta":105},[102],"X X O O O\nX O X O O\n",[107,15199,15197],{"__ignoreMap":105},[30,15201,15202,15203,15206,15207,15240,15241,15274],{},"Let's check the first: ",[42,15204,15205],{},"XXOOO",". Counting O as ",[123,15208,15210,15225],{"className":15209},[126],[123,15211,15213],{"className":15212},[130],[132,15214,15215],{"xmlns":134},[136,15216,15217,15223],{},[139,15218,15219,15221],{},[145,15220,274],{},[142,15222,80],{},[166,15224,10724],{"encoding":168},[123,15226,15228],{"className":15227,"ariaHidden":147},[173],[123,15229,15231,15234,15237],{"className":15230},[177],[123,15232],{"className":15233,"style":289},[181],[123,15235,274],{"className":15236},[186],[123,15238,80],{"className":15239},[186]," and X as ",[123,15242,15244,15259],{"className":15243},[126],[123,15245,15247],{"className":15246},[130],[132,15248,15249],{"xmlns":134},[136,15250,15251,15257],{},[139,15252,15253,15255],{},[145,15254,885],{},[142,15256,80],{},[166,15258,10892],{"encoding":168},[123,15260,15262],{"className":15261,"ariaHidden":147},[173],[123,15263,15265,15268,15271],{"className":15264},[177],[123,15266],{"className":15267,"style":289},[181],[123,15269,885],{"className":15270},[186],[123,15272,80],{"className":15273},[186],", the running totals are:",[30,15276,15278],{"className":15277},[409],[123,15279,15281,15327],{"className":15280},[126],[123,15282,15284],{"className":15283},[130],[132,15285,15286],{"xmlns":134},[136,15287,15288,15324],{},[139,15289,15290,15292,15294,15296,15298,15300,15302,15304,15306,15308,15310,15312,15314,15316,15318,15320,15322],{},[145,15291,885],{},[142,15293,80],{},[145,15295,148],{"separator":147},[1901,15297,1903],{},[145,15299,885],{},[142,15301,690],{},[145,15303,148],{"separator":147},[1901,15305,1903],{},[145,15307,885],{},[142,15309,80],{},[145,15311,148],{"separator":147},[1901,15313,1903],{},[142,15315,1349],{},[145,15317,148],{"separator":147},[1901,15319,1903],{},[145,15321,274],{},[142,15323,80],{},[166,15325,15326],{"encoding":168},"-1,\\; -2,\\; -1,\\; 0,\\; +1",[123,15328,15330],{"className":15329,"ariaHidden":147},[173],[123,15331,15333,15336,15339,15342,15345,15348,15351,15354,15357,15360,15363,15366,15369,15372,15375,15378,15381,15384,15387,15390,15393,15396],{"className":15332},[177],[123,15334],{"className":15335,"style":182},[181],[123,15337,885],{"className":15338},[186],[123,15340,80],{"className":15341},[186],[123,15343,148],{"className":15344},[190],[123,15346],{"className":15347,"style":781},[194],[123,15349],{"className":15350,"style":195},[194],[123,15352,885],{"className":15353},[186],[123,15355,690],{"className":15356},[186],[123,15358,148],{"className":15359},[190],[123,15361],{"className":15362,"style":781},[194],[123,15364],{"className":15365,"style":195},[194],[123,15367,885],{"className":15368},[186],[123,15370,80],{"className":15371},[186],[123,15373,148],{"className":15374},[190],[123,15376],{"className":15377,"style":781},[194],[123,15379],{"className":15380,"style":195},[194],[123,15382,1349],{"className":15383},[186],[123,15385,148],{"className":15386},[190],[123,15388],{"className":15389,"style":781},[194],[123,15391],{"className":15392,"style":195},[194],[123,15394,274],{"className":15395},[186],[123,15397,80],{"className":15398},[186],[30,15400,15401],{},"Never exceeds zero until the fifth flip. ✅",[30,15403,15404,15405,827],{},"Now the second: ",[42,15406,15407],{},"XOXOO",[30,15409,15411],{"className":15410},[409],[123,15412,15414,15458],{"className":15413},[126],[123,15415,15417],{"className":15416},[130],[132,15418,15419],{"xmlns":134},[136,15420,15421,15455],{},[139,15422,15423,15425,15427,15429,15431,15433,15435,15437,15439,15441,15443,15445,15447,15449,15451,15453],{},[145,15424,885],{},[142,15426,80],{},[145,15428,148],{"separator":147},[1901,15430,1903],{},[142,15432,1349],{},[145,15434,148],{"separator":147},[1901,15436,1903],{},[145,15438,885],{},[142,15440,80],{},[145,15442,148],{"separator":147},[1901,15444,1903],{},[142,15446,1349],{},[145,15448,148],{"separator":147},[1901,15450,1903],{},[145,15452,274],{},[142,15454,80],{},[166,15456,15457],{"encoding":168},"-1,\\; 0,\\; -1,\\; 0,\\; +1",[123,15459,15461],{"className":15460,"ariaHidden":147},[173],[123,15462,15464,15467,15470,15473,15476,15479,15482,15485,15488,15491,15494,15497,15500,15503,15506,15509,15512,15515,15518,15521,15524],{"className":15463},[177],[123,15465],{"className":15466,"style":182},[181],[123,15468,885],{"className":15469},[186],[123,15471,80],{"className":15472},[186],[123,15474,148],{"className":15475},[190],[123,15477],{"className":15478,"style":781},[194],[123,15480],{"className":15481,"style":195},[194],[123,15483,1349],{"className":15484},[186],[123,15486,148],{"className":15487},[190],[123,15489],{"className":15490,"style":781},[194],[123,15492],{"className":15493,"style":195},[194],[123,15495,885],{"className":15496},[186],[123,15498,80],{"className":15499},[186],[123,15501,148],{"className":15502},[190],[123,15504],{"className":15505,"style":781},[194],[123,15507],{"className":15508,"style":195},[194],[123,15510,1349],{"className":15511},[186],[123,15513,148],{"className":15514},[190],[123,15516],{"className":15517,"style":781},[194],[123,15519],{"className":15520,"style":195},[194],[123,15522,274],{"className":15523},[186],[123,15525,80],{"className":15526},[186],[30,15528,15529,15530,15563],{},"It returns to zero at the second and fourth flip, but never exceeds it. This is the first time it reaches ",[123,15531,15533,15548],{"className":15532},[126],[123,15534,15536],{"className":15535},[130],[132,15537,15538],{"xmlns":134},[136,15539,15540,15546],{},[139,15541,15542,15544],{},[145,15543,274],{},[142,15545,80],{},[166,15547,10724],{"encoding":168},[123,15549,15551],{"className":15550,"ariaHidden":147},[173],[123,15552,15554,15557,15560],{"className":15553},[177],[123,15555],{"className":15556,"style":289},[181],[123,15558,274],{"className":15559},[186],[123,15561,80],{"className":15562},[186],". ✅",[30,15565,15566,15567,15570,15571,15667,15668,15701],{},"What about ",[42,15568,15569],{},"OXXOO","? Let's see: ",[123,15572,15574,15610],{"className":15573},[126],[123,15575,15577],{"className":15576},[130],[132,15578,15579],{"xmlns":134},[136,15580,15581,15607],{},[139,15582,15583,15585,15587,15589,15591,15593,15595,15597,15599,15601,15603,15605],{},[145,15584,274],{},[142,15586,80],{},[145,15588,148],{"separator":147},[142,15590,1349],{},[145,15592,148],{"separator":147},[145,15594,885],{},[142,15596,80],{},[145,15598,148],{"separator":147},[142,15600,1349],{},[145,15602,148],{"separator":147},[145,15604,274],{},[142,15606,80],{},[166,15608,15609],{"encoding":168},"+1, 0, -1, 0, +1",[123,15611,15613],{"className":15612,"ariaHidden":147},[173],[123,15614,15616,15619,15622,15625,15628,15631,15634,15637,15640,15643,15646,15649,15652,15655,15658,15661,15664],{"className":15615},[177],[123,15617],{"className":15618,"style":182},[181],[123,15620,274],{"className":15621},[186],[123,15623,80],{"className":15624},[186],[123,15626,148],{"className":15627},[190],[123,15629],{"className":15630,"style":195},[194],[123,15632,1349],{"className":15633},[186],[123,15635,148],{"className":15636},[190],[123,15638],{"className":15639,"style":195},[194],[123,15641,885],{"className":15642},[186],[123,15644,80],{"className":15645},[186],[123,15647,148],{"className":15648},[190],[123,15650],{"className":15651,"style":195},[194],[123,15653,1349],{"className":15654},[186],[123,15656,148],{"className":15657},[190],[123,15659],{"className":15660,"style":195},[194],[123,15662,274],{"className":15663},[186],[123,15665,80],{"className":15666},[186],". The first flip already puts us at ",[123,15669,15671,15686],{"className":15670},[126],[123,15672,15674],{"className":15673},[130],[132,15675,15676],{"xmlns":134},[136,15677,15678,15684],{},[139,15679,15680,15682],{},[145,15681,274],{},[142,15683,80],{},[166,15685,10724],{"encoding":168},[123,15687,15689],{"className":15688,"ariaHidden":147},[173],[123,15690,15692,15695,15698],{"className":15691},[177],[123,15693],{"className":15694,"style":289},[181],[123,15696,274],{"className":15697},[186],[123,15699,80],{"className":15700},[186],": we would have stopped right there. Not valid. ❌",[30,15703,15704,15705,15708,15709,15949,15950,16048],{},"So there are exactly ",[42,15706,15707],{},"2 sequences"," of length 5, each with probability ",[123,15710,15712,15748],{"className":15711},[126],[123,15713,15715],{"className":15714},[130],[132,15716,15717],{"xmlns":134},[136,15718,15719,15745],{},[139,15720,15721,15737,15739],{},[2520,15722,15723,15735],{},[139,15724,15725,15727,15733],{},[145,15726,424],{"fence":147},[426,15728,15729,15731],{},[142,15730,80],{},[142,15732,690],{},[145,15734,438],{"fence":147},[142,15736,2760],{},[145,15738,695],{},[426,15740,15741,15743],{},[142,15742,80],{},[142,15744,2763],{},[166,15746,15747],{"encoding":168},"\\left(\\frac{1}{2}\\right)^5 = \\frac{1}{32}",[123,15749,15751,15875],{"className":15750,"ariaHidden":147},[173],[123,15752,15754,15757,15866,15869,15872],{"className":15753},[177],[123,15755],{"className":15756,"style":14978},[181],[123,15758,15760,15843],{"className":15759},[2195],[123,15761,15763,15769,15837],{"className":15762},[2195],[123,15764,15766],{"className":15765,"style":460},[458,459],[123,15767,424],{"className":15768},[464,465],[123,15770,15772,15775,15834],{"className":15771},[186],[123,15773],{"className":15774},[458,992],[123,15776,15778],{"className":15777},[426],[123,15779,15781,15826],{"className":15780},[472,473],[123,15782,15784,15823],{"className":15783},[477],[123,15785,15787,15801,15809],{"className":15786,"style":1678},[481],[123,15788,15789,15792],{"style":1008},[123,15790],{"className":15791,"style":1012},[489],[123,15793,15795],{"className":15794},[494,495,496,497],[123,15796,15798],{"className":15797},[186,497],[123,15799,690],{"className":15800},[186,497],[123,15802,15803,15806],{"style":1046},[123,15804],{"className":15805,"style":1012},[489],[123,15807],{"className":15808,"style":1054},[1053],[123,15810,15811,15814],{"style":1057},[123,15812],{"className":15813,"style":1012},[489],[123,15815,15817],{"className":15816},[494,495,496,497],[123,15818,15820],{"className":15819},[186,497],[123,15821,80],{"className":15822},[186,497],[123,15824,525],{"className":15825},[524],[123,15827,15829],{"className":15828},[477],[123,15830,15832],{"className":15831,"style":532},[481],[123,15833],{},[123,15835],{"className":15836},[538,992],[123,15838,15840],{"className":15839,"style":460},[538,459],[123,15841,438],{"className":15842},[464,465],[123,15844,15846],{"className":15845},[2548],[123,15847,15849],{"className":15848},[472],[123,15850,15852],{"className":15851},[477],[123,15853,15855],{"className":15854,"style":15077},[481],[123,15856,15857,15860],{"style":9203},[123,15858],{"className":15859,"style":490},[489],[123,15861,15863],{"className":15862},[494,495,496,497],[123,15864,2760],{"className":15865},[186,497],[123,15867],{"className":15868,"style":781},[194],[123,15870,695],{"className":15871},[785],[123,15873],{"className":15874,"style":781},[194],[123,15876,15878,15881],{"className":15877},[177],[123,15879],{"className":15880,"style":1751},[181],[123,15882,15884,15887,15946],{"className":15883},[186],[123,15885],{"className":15886},[458,992],[123,15888,15890],{"className":15889},[426],[123,15891,15893,15938],{"className":15892},[472,473],[123,15894,15896,15935],{"className":15895},[477],[123,15897,15899,15913,15921],{"className":15898,"style":1678},[481],[123,15900,15901,15904],{"style":1008},[123,15902],{"className":15903,"style":1012},[489],[123,15905,15907],{"className":15906},[494,495,496,497],[123,15908,15910],{"className":15909},[186,497],[123,15911,2763],{"className":15912},[186,497],[123,15914,15915,15918],{"style":1046},[123,15916],{"className":15917,"style":1012},[489],[123,15919],{"className":15920,"style":1054},[1053],[123,15922,15923,15926],{"style":1057},[123,15924],{"className":15925,"style":1012},[489],[123,15927,15929],{"className":15928},[494,495,496,497],[123,15930,15932],{"className":15931},[186,497],[123,15933,80],{"className":15934},[186,497],[123,15936,525],{"className":15937},[524],[123,15939,15941],{"className":15940},[477],[123,15942,15944],{"className":15943,"style":532},[481],[123,15945],{},[123,15947],{"className":15948},[538,992]," and ratio ",[123,15951,15953,15971],{"className":15952},[126],[123,15954,15956],{"className":15955},[130],[132,15957,15958],{"xmlns":134},[136,15959,15960,15968],{},[139,15961,15962],{},[426,15963,15964,15966],{},[142,15965,271],{},[142,15967,2760],{},[166,15969,15970],{"encoding":168},"\\frac{3}{5}",[123,15972,15974],{"className":15973,"ariaHidden":147},[173],[123,15975,15977,15980],{"className":15976},[177],[123,15978],{"className":15979,"style":1751},[181],[123,15981,15983,15986,16045],{"className":15982},[186],[123,15984],{"className":15985},[458,992],[123,15987,15989],{"className":15988},[426],[123,15990,15992,16037],{"className":15991},[472,473],[123,15993,15995,16034],{"className":15994},[477],[123,15996,15998,16012,16020],{"className":15997,"style":1678},[481],[123,15999,16000,16003],{"style":1008},[123,16001],{"className":16002,"style":1012},[489],[123,16004,16006],{"className":16005},[494,495,496,497],[123,16007,16009],{"className":16008},[186,497],[123,16010,2760],{"className":16011},[186,497],[123,16013,16014,16017],{"style":1046},[123,16015],{"className":16016,"style":1012},[489],[123,16018],{"className":16019,"style":1054},[1053],[123,16021,16022,16025],{"style":1057},[123,16023],{"className":16024,"style":1012},[489],[123,16026,16028],{"className":16027},[494,495,496,497],[123,16029,16031],{"className":16030},[186,497],[123,16032,271],{"className":16033},[186,497],[123,16035,525],{"className":16036},[524],[123,16038,16040],{"className":16039},[477],[123,16041,16043],{"className":16042,"style":532},[481],[123,16044],{},[123,16046],{"className":16047},[538,992]," (3 heads out of 5 flips).",[395,16050,16052],{"id":16051},"length-7","Length 7",[30,16054,16055,16056,827],{},"The same reasoning applies to sequences of length 7 (4 heads, 3 tails). If you enumerate them all — and trust me, it takes a while — you find exactly ",[42,16057,16058],{},"5 valid sequences",[395,16060,16062],{"id":16061},"the-summary-table","The summary table",[2632,16064,16065,16137],{},[2635,16066,16067],{},[2638,16068,16069,16128,16131,16134],{},[2641,16070,16071,16072,438],{"align":2643},"Length 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sequences",[2641,16135,16136],{"align":2643},"Probability of each",[2735,16138,16139,16396,16653,16909,17169],{},[2638,16140,16141,16195,16295,16297],{},[2740,16142,16143,16144,438],{"align":2643},"1 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(",[123,17175,17177,17195],{"className":17176},[126],[123,17178,17180],{"className":17179},[130],[132,17181,17182],{"xmlns":134},[136,17183,17184,17192],{},[139,17185,17186,17188,17190],{},[430,17187,432],{},[145,17189,695],{},[142,17191,151],{},[166,17193,17194],{"encoding":168},"n=4",[123,17196,17198,17216],{"className":17197,"ariaHidden":147},[173],[123,17199,17201,17204,17207,17210,17213],{"className":17200},[177],[123,17202],{"className":17203,"style":572},[181],[123,17205,432],{"className":17206},[186,504],[123,17208],{"className":17209,"style":781},[194],[123,17211,695],{"className":17212},[785],[123,17214],{"className":17215,"style":781},[194],[123,17217,17219,17222],{"className":17218},[177],[123,17220],{"className":17221,"style":252},[181],[123,17223,151],{"className":17224},[186],[2740,17226,17227],{"align":2643},[123,17228,17230,17249],{"className":17229},[126],[123,17231,17233],{"className":17232},[130],[132,17234,17235],{"xmlns":134},[136,17236,17237,17246],{},[139,17238,17239],{},[426,17240,17241,17243],{},[142,17242,2760],{},[142,17244,17245],{},"9",[166,17247,17248],{"encoding":168},"\\frac{5}{9}",[123,17250,17252],{"className":17251,"ariaHidden":147},[173],[123,17253,17255,17258],{"className":17254},[177],[123,17256],{"className":17257,"style":1751},[181],[123,17259,17261,17264,17323],{"className":17260},[186],[123,17262],{"className":17263},[458,992],[123,17265,17267],{"className":17266},[426],[123,17268,17270,17315],{"className":17269},[472,473],[123,17271,17273,17312],{"className":17272},[477],[123,17274,17276,17290,17298],{"className":17275,"style":1678},[481],[123,17277,17278,17281],{"style":1008},[123,17279],{"className":17280,"style":1012},[489],[123,17282,17284],{"className":17283},[494,495,496,497],[123,17285,17287],{"className":17286},[186,497],[123,17288,17245],{"className":17289},[186,497],[123,17291,17292,17295],{"style":1046},[123,17293],{"className":17294,"style":1012},[489],[123,17296],{"className":17297,"style":1054},[1053],[123,17299,17300,17303],{"style":1057},[123,17301],{"className":17302,"style":1012},[489],[123,17304,17306],{"className":17305},[494,495,496,497],[123,17307,17309],{"className":17308},[186,497],[123,17310,2760],{"className":17311},[186,497],[123,17313,525],{"className":17314},[524],[123,17316,17318],{"className":17317},[477],[123,17319,17321],{"className":17320,"style":532},[481],[123,17322],{},[123,17324],{"className":17325},[538,992],[2740,17327,17328],{"align":2643},"14",[2740,17330,17331],{"align":2643},[123,17332,17334,17353],{"className":17333},[126],[123,17335,17337],{"className":17336},[130],[132,17338,17339],{"xmlns":134},[136,17340,17341,17350],{},[139,17342,17343],{},[426,17344,17345,17347],{},[142,17346,80],{},[142,17348,17349],{},"512",[166,17351,17352],{"encoding":168},"\\frac{1}{512}",[123,17354,17356],{"className":17355,"ariaHidden":147},[173],[123,17357,17359,17362],{"className":17358},[177],[123,17360],{"className":17361,"style":1751},[181],[123,17363,17365,17368,17427],{"className":17364},[186],[123,17366],{"className":17367},[458,992],[123,17369,17371],{"className":17370},[426],[123,17372,17374,17419],{"className":17373},[472,473],[123,17375,17377,17416],{"className":17376},[477],[123,17378,17380,17394,17402],{"className":17379,"style":1678},[481],[123,17381,17382,17385],{"style":1008},[123,17383],{"className":17384,"style":1012},[489],[123,17386,17388],{"className":17387},[494,495,496,497],[123,17389,17391],{"className":17390},[186,497],[123,17392,17349],{"className":17393},[186,497],[123,17395,17396,17399],{"style":1046},[123,17397],{"className":17398,"style":1012},[489],[123,17400],{"className":17401,"style":1054},[1053],[123,17403,17404,17407],{"style":1057},[123,17405],{"className":17406,"style":1012},[489],[123,17408,17410],{"className":17409},[494,495,496,497],[123,17411,17413],{"className":17412},[186,497],[123,17414,80],{"className":17415},[186,497],[123,17417,525],{"className":17418},[524],[123,17420,17422],{"className":17421},[477],[123,17423,17425],{"className":17424,"style":532},[481],[123,17426],{},[123,17428],{"className":17429},[538,992],[30,17431,17432],{},"Notice the patterns:",[6338,17434,17435,17494,17624],{},[75,17436,17437,17438,827],{},"The length is always odd: ",[123,17439,17441,17461],{"className":17440},[126],[123,17442,17444],{"className":17443},[130],[132,17445,17446],{"xmlns":134},[136,17447,17448,17458],{},[139,17449,17450,17452,17454,17456],{},[142,17451,690],{},[430,17453,432],{},[145,17455,274],{},[142,17457,80],{},[166,17459,17460],{"encoding":168},"2n + 1",[123,17462,17464,17485],{"className":17463,"ariaHidden":147},[173],[123,17465,17467,17470,17473,17476,17479,17482],{"className":17466},[177],[123,17468],{"className":17469,"style":289},[181],[123,17471,690],{"className":17472},[186],[123,17474,432],{"className":17475},[186,504],[123,17477],{"className":17478,"style":296},[194],[123,17480,274],{"className":17481},[300],[123,17483],{"className":17484,"style":296},[194],[123,17486,17488,17491],{"className":17487},[177],[123,17489],{"className":17490,"style":252},[181],[123,17492,80],{"className":17493},[186],[75,17495,17496,17497,827],{},"The ratio of heads is always ",[123,17498,17500,17532],{"className":17499},[126],[123,17501,17503],{"className":17502},[130],[132,17504,17505],{"xmlns":134},[136,17506,17507,17529],{},[139,17508,17509],{},[426,17510,17511,17519],{},[139,17512,17513,17515,17517],{},[430,17514,432],{},[145,17516,274],{},[142,17518,80],{},[139,17520,17521,17523,17525,17527],{},[142,17522,690],{},[430,17524,432],{},[145,17526,274],{},[142,17528,80],{},[166,17530,17531],{"encoding":168},"\\frac{n+1}{2n+1}",[123,17533,17535],{"className":17534,"ariaHidden":147},[173],[123,17536,17538,17541],{"className":17537},[177],[123,17539],{"className":17540,"style":1659},[181],[123,17542,17544,17547,17621],{"className":17543},[186],[123,17545],{"className":17546},[458,992],[123,17548,17550],{"className":17549},[426],[123,17551,17553,17613],{"className":17552},[472,473],[123,17554,17556,17610],{"className":17555},[477],[123,17557,17559,17582,17590],{"className":17558,"style":1678},[481],[123,17560,17561,17564],{"style":1008},[123,17562],{"className":17563,"style":1012},[489],[123,17565,17567],{"className":17566},[494,495,496,497],[123,17568,17570,17573,17576,17579],{"className":17569},[186,497],[123,17571,690],{"className":17572},[186,497],[123,17574,432],{"className":17575},[186,504,497],[123,17577,274],{"className":17578},[300,497],[123,17580,80],{"className":17581},[186,497],[123,17583,17584,17587],{"style":1046},[123,17585],{"className":17586,"style":1012},[489],[123,17588],{"className":17589,"style":1054},[1053],[123,17591,17592,17595],{"style":1057},[123,17593],{"className":17594,"style":1012},[489],[123,17596,17598],{"className":17597},[494,495,496,497],[123,17599,17601,17604,17607],{"className":17600},[186,497],[123,17602,432],{"className":17603},[186,504,497],[123,17605,274],{"className":17606},[300,497],[123,17608,80],{"className":17609},[186,497],[123,17611,525],{"className":17612},[524],[123,17614,17616],{"className":17615},[477],[123,17617,17619],{"className":17618,"style":1730},[481],[123,17620],{},[123,17622],{"className":17623},[538,992],[75,17625,17626,17627,17796,17797,17852],{},"The probability of each individual sequence is ",[123,17628,17630,17666],{"className":17629},[126],[123,17631,17633],{"className":17632},[130],[132,17634,17635],{"xmlns":134},[136,17636,17637,17663],{},[139,17638,17639],{},[2520,17640,17641,17653],{},[139,17642,17643,17645,17651],{},[145,17644,424],{"fence":147},[426,17646,17647,17649],{},[142,17648,80],{},[142,17650,690],{},[145,17652,438],{"fence":147},[139,17654,17655,17657,17659,17661],{},[142,17656,690],{},[430,17658,432],{},[145,17660,274],{},[142,17662,80],{},[166,17664,17665],{"encoding":168},"\\left(\\frac{1}{2}\\right)^{2n+1}",[123,17667,17669],{"className":17668,"ariaHidden":147},[173],[123,17670,17672,17675],{"className":17671},[177],[123,17673],{"className":17674,"style":14978},[181],[123,17676,17678,17761],{"className":17677},[2195],[123,17679,17681,17687,17755],{"className":17680},[2195],[123,17682,17684],{"className":17683,"style":460},[458,459],[123,17685,424],{"className":17686},[464,465],[123,17688,17690,17693,17752],{"className":17689},[186],[123,17691],{"className":17692},[458,992],[123,17694,17696],{"className":17695},[426],[123,17697,17699,17744],{"className":17698},[472,473],[123,17700,17702,17741],{"className":17701},[477],[123,17703,17705,17719,17727],{"className":17704,"style":1678},[481],[123,17706,17707,17710],{"style":1008},[123,17708],{"className":17709,"style":1012},[489],[123,17711,17713],{"className":17712},[494,495,496,497],[123,17714,17716],{"className":17715},[186,497],[123,17717,690],{"className":17718},[186,497],[123,17720,17721,17724],{"style":1046},[123,17722],{"className":17723,"style":1012},[489],[123,17725],{"className":17726,"style":1054},[1053],[123,17728,17729,17732],{"style":1057},[123,17730],{"className":17731,"style":1012},[489],[123,17733,17735],{"className":17734},[494,495,496,497],[123,17736,17738],{"className":17737},[186,497],[123,17739,80],{"className":17740},[186,497],[123,17742,525],{"className":17743},[524],[123,17745,17747],{"className":17746},[477],[123,17748,17750],{"className":17749,"style":532},[481],[123,17751],{},[123,17753],{"className":17754},[538,992],[123,17756,17758],{"className":17757,"style":460},[538,459],[123,17759,438],{"className":17760},[464,465],[123,17762,17764],{"className":17763},[2548],[123,17765,17767],{"className":17766},[472],[123,17768,17770],{"className":17769},[477],[123,17771,17773],{"className":17772,"style":15077},[481],[123,17774,17775,17778],{"style":9203},[123,17776],{"className":17777,"style":490},[489],[123,17779,17781],{"className":17780},[494,495,496,497],[123,17782,17784,17787,17790,17793],{"className":17783},[186,497],[123,17785,690],{"className":17786},[186,497],[123,17788,432],{"className":17789},[186,504,497],[123,17791,274],{"className":17792},[300,497],[123,17794,80],{"className":17795},[186,497]," — each of the ",[123,17798,17800,17819],{"className":17799},[126],[123,17801,17803],{"className":17802},[130],[132,17804,17805],{"xmlns":134},[136,17806,17807,17817],{},[139,17808,17809,17811,17813,17815],{},[142,17810,690],{},[430,17812,432],{},[145,17814,274],{},[142,17816,80],{},[166,17818,16094],{"encoding":168},[123,17820,17822,17843],{"className":17821,"ariaHidden":147},[173],[123,17823,17825,17828,17831,17834,17837,17840],{"className":17824},[177],[123,17826],{"className":17827,"style":289},[181],[123,17829,690],{"className":17830},[186],[123,17832,432],{"className":17833},[186,504],[123,17835],{"className":17836,"style":296},[194],[123,17838,274],{"className":17839},[300],[123,17841],{"className":17842,"style":296},[194],[123,17844,17846,17849],{"className":17845},[177],[123,17847],{"className":17848,"style":252},[181],[123,17850,80],{"className":17851},[186]," flips has a one-in-two chance.",[30,17854,17855,17856],{},"But the column that really matters is the number of valid sequences: ",[123,17857,17859,17893],{"className":17858},[126],[123,17860,17862],{"className":17861},[130],[132,17863,17864],{"xmlns":134},[136,17865,17866,17890],{},[139,17867,17868,17870,17872,17874,17876,17878,17880,17882,17884,17886,17888],{},[142,17869,80],{},[145,17871,148],{"separator":147},[142,17873,80],{},[145,17875,148],{"separator":147},[142,17877,690],{},[145,17879,148],{"separator":147},[142,17881,2760],{},[145,17883,148],{"separator":147},[142,17885,17328],{},[145,17887,148],{"separator":147},[145,17889,1938],{},[166,17891,17892],{"encoding":168},"1, 1, 2, 5, 14, \\ldots",[123,17894,17896],{"className":17895,"ariaHidden":147},[173],[123,17897,17899,17902,17905,17908,17911,17914,17917,17920,17923,17926,17929,17932,17935,17938,17941,17944,17947],{"className":17898},[177],[123,17900],{"className":17901,"style":182},[181],[123,17903,80],{"className":17904},[186],[123,17906,148],{"className":17907},[190],[123,17909],{"className":17910,"style":195},[194],[123,17912,80],{"className":17913},[186],[123,17915,148],{"className":17916},[190],[123,17918],{"className":17919,"style":195},[194],[123,17921,690],{"className":17922},[186],[123,17924,148],{"className":17925},[190],[123,17927],{"className":17928,"style":195},[194],[123,17930,2760],{"className":17931},[186],[123,17933,148],{"className":17934},[190],[123,17936],{"className":17937,"style":195},[194],[123,17939,17328],{"className":17940},[186],[123,17942,148],{"className":17943},[190],[123,17945],{"className":17946,"style":195},[194],[123,17948,1938],{"className":17949},[2195],[30,17951,17952],{},"Ring a bell?",[34,17954,17956],{"id":17955},"from-manual-counting-to-the-catalan-numbers","From manual counting to the Catalan numbers",[30,17958,17959,17960,18082,18083,18090],{},"Those numbers — ",[123,17961,17963,18007],{"className":17962},[126],[123,17964,17966],{"className":17965},[130],[132,17967,17968],{"xmlns":134},[136,17969,17970,18004],{},[139,17971,17972,17974,17976,17978,17980,17982,17984,17986,17988,17990,17992,17995,17997,18000,18002],{},[142,17973,80],{},[145,17975,148],{"separator":147},[142,17977,80],{},[145,17979,148],{"separator":147},[142,17981,690],{},[145,17983,148],{"separator":147},[142,17985,2760],{},[145,17987,148],{"separator":147},[142,17989,17328],{},[145,17991,148],{"separator":147},[142,17993,17994],{},"42",[145,17996,148],{"separator":147},[142,17998,17999],{},"132",[145,18001,148],{"separator":147},[145,18003,1938],{},[166,18005,18006],{"encoding":168},"1, 1, 2, 5, 14, 42, 132, \\ldots",[123,18008,18010],{"className":18009,"ariaHidden":147},[173],[123,18011,18013,18016,18019,18022,18025,18028,18031,18034,18037,18040,18043,18046,18049,18052,18055,18058,18061,18064,18067,18070,18073,18076,18079],{"className":18012},[177],[123,18014],{"className":18015,"style":182},[181],[123,18017,80],{"className":18018},[186],[123,18020,148],{"className":18021},[190],[123,18023],{"className":18024,"style":195},[194],[123,18026,80],{"className":18027},[186],[123,18029,148],{"className":18030},[190],[123,18032],{"className":18033,"style":195},[194],[123,18035,690],{"className":18036},[186],[123,18038,148],{"className":18039},[190],[123,18041],{"className":18042,"style":195},[194],[123,18044,2760],{"className":18045},[186],[123,18047,148],{"className":18048},[190],[123,18050],{"className":18051,"style":195},[194],[123,18053,17328],{"className":18054},[186],[123,18056,148],{"className":18057},[190],[123,18059],{"className":18060,"style":195},[194],[123,18062,17994],{"className":18063},[186],[123,18065,148],{"className":18066},[190],[123,18068],{"className":18069,"style":195},[194],[123,18071,17999],{"className":18072},[186],[123,18074,148],{"className":18075},[190],[123,18077],{"className":18078,"style":195},[194],[123,18080,1938],{"className":18081},[2195]," — have a name. They are the ",[2779,18084,18087],{"href":18085,"rel":18086},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCatalan_number",[2783],[42,18088,18089],{},"Catalan numbers",", one of the most ubiquitous sequences in all of combinatorics.",[30,18092,18093,18094,18101,18102,18105,18106,18109,18110,18115,18116,18119],{},"Their history is itself a story of unexpected connections. The Mongolian mathematician ",[2779,18095,18098],{"href":18096,"rel":18097},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMinggatu",[2783],[42,18099,18100],{},"Mingantu"," (c. 1692–1763), working in China, was the first known person to use them — around the 1730s, in trigonometric series included in his book ",[47,18103,18104],{},"Ge Yuan Mi Lu Jie Fa",". Two decades later, in 1751, ",[42,18107,18108],{},"Leonhard Euler"," ",[2779,18111,18114],{"href":18112,"rel":18113},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCatalan_number#History",[2783],"independently discovered"," the same sequence in a letter to Goldbach, this time counting the number of ways to cut a polygon into triangles. And nearly a century after Mingantu, the Franco-Belgian mathematician ",[42,18117,18118],{},"Eugène Charles Catalan"," (1814–1894) found them again in 1838, while counting the number of ways to parenthesize a product of factors. The sequence bears his name — coined by John Riordan in the 20th century — even though the numbers had been surfacing, in different guises, for over a hundred years before him.",[30,18121,18122],{},"The same sequence, discovered independently across three corners of the world and three centuries, each time in a completely different context. If that pattern sounds familiar... keep reading.",[30,18124,10594,18125,18153],{},[123,18126,18128,18141],{"className":18127},[126],[123,18129,18131],{"className":18130},[130],[132,18132,18133],{"xmlns":134},[136,18134,18135,18139],{},[139,18136,18137],{},[430,18138,432],{},[166,18140,432],{"encoding":168},[123,18142,18144],{"className":18143,"ariaHidden":147},[173],[123,18145,18147,18150],{"className":18146},[177],[123,18148],{"className":18149,"style":572},[181],[123,18151,432],{"className":18152},[186,504],"-th Catalan number is defined as:",[30,18155,18157],{"className":18156},[409],[123,18158,18160,18209],{"className":18159},[126],[123,18161,18163],{"className":18162},[130],[132,18164,18165],{"xmlns":134},[136,18166,18167,18206],{},[139,18168,18169,18176,18178,18190],{},[11964,18170,18171,18174],{},[430,18172,18173],{},"C",[430,18175,432],{},[145,18177,695],{},[426,18179,18180,18182],{},[142,18181,80],{},[139,18183,18184,18186,18188],{},[430,18185,432],{},[145,18187,274],{},[142,18189,80],{},[139,18191,18192,18194,18204],{},[145,18193,424],{"fence":147},[426,18195,18196,18202],{"linethickness":428},[139,18197,18198,18200],{},[142,18199,690],{},[430,18201,432],{},[430,18203,432],{},[145,18205,438],{"fence":147},[166,18207,18208],{"encoding":168},"C_n = \\frac{1}{n+1}\\binom{2n}{n}",[123,18210,18212,18272],{"className":18211,"ariaHidden":147},[173],[123,18213,18215,18219,18263,18266,18269],{"className":18214},[177],[123,18216],{"className":18217,"style":18218},[181],"height:0.8333em;vertical-align:-0.15em;",[123,18220,18222,18226],{"className":18221},[186],[123,18223,18173],{"className":18224,"style":18225},[186,504],"margin-right:0.0715em;",[123,18227,18229],{"className":18228},[2548],[123,18230,18232,18254],{"className":18231},[472,473],[123,18233,18235,18251],{"className":18234},[477],[123,18236,18239],{"className":18237,"style":18238},[481],"height:0.1514em;",[123,18240,18242,18245],{"style":18241},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[123,18243],{"className":18244,"style":490},[489],[123,18246,18248],{"className":18247},[494,495,496,497],[123,18249,432],{"className":18250},[186,504,497],[123,18252,525],{"className":18253},[524],[123,18255,18257],{"className":18256},[477],[123,18258,18261],{"className":18259,"style":18260},[481],"height:0.15em;",[123,18262],{},[123,18264],{"className":18265,"style":781},[194],[123,18267,695],{"className":18268},[785],[123,18270],{"className":18271,"style":781},[194],[123,18273,18275,18279,18353],{"className":18274},[177],[123,18276],{"className":18277,"style":18278},[181],"height:1.2984em;vertical-align:-0.4033em;",[123,18280,18282,18285,18350],{"className":18281},[186],[123,18283],{"className":18284},[458,992],[123,18286,18288],{"className":18287},[426],[123,18289,18291,18342],{"className":18290},[472,473],[123,18292,18294,18339],{"className":18293},[477],[123,18295,18297,18317,18325],{"className":18296,"style":1678},[481],[123,18298,18299,18302],{"style":1008},[123,18300],{"className":18301,"style":1012},[489],[123,18303,18305],{"className":18304},[494,495,496,497],[123,18306,18308,18311,18314],{"className":18307},[186,497],[123,18309,432],{"className":18310},[186,504,497],[123,18312,274],{"className":18313},[300,497],[123,18315,80],{"className":18316},[186,497],[123,18318,18319,18322],{"style":1046},[123,18320],{"className":18321,"style":1012},[489],[123,18323],{"className":18324,"style":1054},[1053],[123,18326,18327,18330],{"style":1057},[123,18328],{"className":18329,"style":1012},[489],[123,18331,18333],{"className":18332},[494,495,496,497],[123,18334,18336],{"className":18335},[186,497],[123,18337,80],{"className":18338},[186,497],[123,18340,525],{"className":18341},[524],[123,18343,18345],{"className":18344},[477],[123,18346,18348],{"className":18347,"style":1730},[481],[123,18349],{},[123,18351],{"className":18352},[538,992],[123,18354,18356,18362,18416],{"className":18355},[186],[123,18357,18359],{"className":18358,"style":460},[458,459],[123,18360,424],{"className":18361},[464,465],[123,18363,18365],{"className":18364},[426],[123,18366,18368,18408],{"className":18367},[472,473],[123,18369,18371,18405],{"className":18370},[477],[123,18372,18374,18388],{"className":18373,"style":732},[481],[123,18375,18376,18379],{"style":485},[123,18377],{"className":18378,"style":490},[489],[123,18380,18382],{"className":18381},[494,495,496,497],[123,18383,18385],{"className":18384},[186,497],[123,18386,432],{"className":18387},[186,504,497],[123,18389,18390,18393],{"style":508},[123,18391],{"className":18392,"style":490},[489],[123,18394,18396],{"className":18395},[494,495,496,497],[123,18397,18399,18402],{"className":18398},[186,497],[123,18400,690],{"className":18401},[186,497],[123,18403,432],{"className":18404},[186,504,497],[123,18406,525],{"className":18407},[524],[123,18409,18411],{"className":18410},[477],[123,18412,18414],{"className":18413,"style":532},[481],[123,18415],{},[123,18417,18419],{"className":18418,"style":460},[538,459],[123,18420,438],{"className":18421},[464,465],[30,18423,2572,18424,18530,18531,576,18564,18592,18593,18595,18596,635,18624,18657,18658,18686,18687,18720,18721,827],{},[123,18425,18427,18452],{"className":18426},[126],[123,18428,18430],{"className":18429},[130],[132,18431,18432],{"xmlns":134},[136,18433,18434,18450],{},[139,18435,18436,18438,18448],{},[145,18437,424],{"fence":147},[426,18439,18440,18446],{"linethickness":428},[139,18441,18442,18444],{},[142,18443,690],{},[430,18445,432],{},[430,18447,432],{},[145,18449,438],{"fence":147},[166,18451,13793],{"encoding":168},[123,18453,18455],{"className":18454,"ariaHidden":147},[173],[123,18456,18458,18461],{"className":18457},[177],[123,18459],{"className":18460,"style":710},[181],[123,18462,18464,18470,18524],{"className":18463},[186],[123,18465,18467],{"className":18466,"style":460},[458,459],[123,18468,424],{"className":18469},[464,465],[123,18471,18473],{"className":18472},[426],[123,18474,18476,18516],{"className":18475},[472,473],[123,18477,18479,18513],{"className":18478},[477],[123,18480,18482,18496],{"className":18481,"style":732},[481],[123,18483,18484,18487],{"style":485},[123,18485],{"className":18486,"style":490},[489],[123,18488,18490],{"className":18489},[494,495,496,497],[123,18491,18493],{"className":18492},[186,497],[123,18494,432],{"className":18495},[186,504,497],[123,18497,18498,18501],{"style":508},[123,18499],{"className":18500,"style":490},[489],[123,18502,18504],{"className":18503},[494,495,496,497],[123,18505,18507,18510],{"className":18506},[186,497],[123,18508,690],{"className":18509},[186,497],[123,18511,432],{"className":18512},[186,504,497],[123,18514,525],{"className":18515},[524],[123,18517,18519],{"className":18518},[477],[123,18520,18522],{"className":18521,"style":532},[481],[123,18523],{},[123,18525,18527],{"className":18526,"style":460},[538,459],[123,18528,438],{"className":18529},[464,465]," — read \"",[123,18532,18534,18549],{"className":18533},[126],[123,18535,18537],{"className":18536},[130],[132,18538,18539],{"xmlns":134},[136,18540,18541,18547],{},[139,18542,18543,18545],{},[142,18544,690],{},[430,18546,432],{},[166,18548,11896],{"encoding":168},[123,18550,18552],{"className":18551,"ariaHidden":147},[173],[123,18553,18555,18558,18561],{"className":18554},[177],[123,18556],{"className":18557,"style":252},[181],[123,18559,690],{"className":18560},[186],[123,18562,432],{"className":18563},[186,504],[123,18565,18567,18580],{"className":18566},[126],[123,18568,18570],{"className":18569},[130],[132,18571,18572],{"xmlns":134},[136,18573,18574,18578],{},[139,18575,18576],{},[430,18577,432],{},[166,18579,432],{"encoding":168},[123,18581,18583],{"className":18582,"ariaHidden":147},[173],[123,18584,18586,18589],{"className":18585},[177],[123,18587],{"className":18588,"style":572},[181],[123,18590,432],{"className":18591},[186,504],"\" — is the ",[42,18594,404],{},": the number of ways to pick ",[123,18597,18599,18612],{"className":18598},[126],[123,18600,18602],{"className":18601},[130],[132,18603,18604],{"xmlns":134},[136,18605,18606,18610],{},[139,18607,18608],{},[430,18609,432],{},[166,18611,432],{"encoding":168},[123,18613,18615],{"className":18614,"ariaHidden":147},[173],[123,18616,18618,18621],{"className":18617},[177],[123,18619],{"className":18620,"style":572},[181],[123,18622,432],{"className":18623},[186,504],[123,18625,18627,18642],{"className":18626},[126],[123,18628,18630],{"className":18629},[130],[132,18631,18632],{"xmlns":134},[136,18633,18634,18640],{},[139,18635,18636,18638],{},[142,18637,690],{},[430,18639,432],{},[166,18641,11896],{"encoding":168},[123,18643,18645],{"className":18644,"ariaHidden":147},[173],[123,18646,18648,18651,18654],{"className":18647},[177],[123,18649],{"className":18650,"style":252},[181],[123,18652,690],{"className":18653},[186],[123,18655,432],{"className":18656},[186,504],". In practice, it answers questions like \"how many ways can I arrange ",[123,18659,18661,18674],{"className":18660},[126],[123,18662,18664],{"className":18663},[130],[132,18665,18666],{"xmlns":134},[136,18667,18668,18672],{},[139,18669,18670],{},[430,18671,432],{},[166,18673,432],{"encoding":168},[123,18675,18677],{"className":18676,"ariaHidden":147},[173],[123,18678,18680,18683],{"className":18679},[177],[123,18681],{"className":18682,"style":572},[181],[123,18684,432],{"className":18685},[186,504]," heads among ",[123,18688,18690,18705],{"className":18689},[126],[123,18691,18693],{"className":18692},[130],[132,18694,18695],{"xmlns":134},[136,18696,18697,18703],{},[139,18698,18699,18701],{},[142,18700,690],{},[430,18702,432],{},[166,18704,11896],{"encoding":168},[123,18706,18708],{"className":18707,"ariaHidden":147},[173],[123,18709,18711,18714,18717],{"className":18710},[177],[123,18712],{"className":18713,"style":252},[181],[123,18715,690],{"className":18716},[186],[123,18718,432],{"className":18719},[186,504]," flips?\" We explored this notation in detail in the ",[2779,18722,18724],{"href":14178,"rel":18723},[2783],"article on Pascal's triangle",[30,18726,18727],{},"Let's verify against our data:",[30,18729,18731],{"className":18730},[409],[123,18732,18734,18809],{"className":18733},[126],[123,18735,18737],{"className":18736},[130],[132,18738,18739],{"xmlns":134},[136,18740,18741,18806],{},[139,18742,18743,18749,18751,18757,18769,18771,18773,18776,18782,18784,18790,18802,18804],{},[11964,18744,18745,18747],{},[430,18746,18173],{},[142,18748,1349],{},[145,18750,695],{},[426,18752,18753,18755],{},[142,18754,80],{},[142,18756,80],{},[139,18758,18759,18761,18767],{},[145,18760,424],{"fence":147},[426,18762,18763,18765],{"linethickness":428},[142,18764,1349],{},[142,18766,1349],{},[145,18768,438],{"fence":147},[145,18770,695],{},[142,18772,80],{},[194,18774],{"width":18775},"2em",[11964,18777,18778,18780],{},[430,18779,18173],{},[142,18781,80],{},[145,18783,695],{},[426,18785,18786,18788],{},[142,18787,80],{},[142,18789,690],{},[139,18791,18792,18794,18800],{},[145,18793,424],{"fence":147},[426,18795,18796,18798],{"linethickness":428},[142,18797,690],{},[142,18799,80],{},[145,18801,438],{"fence":147},[145,18803,695],{},[142,18805,80],{},[166,18807,18808],{"encoding":168},"C_0 = \\frac{1}{1}\\binom{0}{0} = 1 \\qquad C_1 = \\frac{1}{2}\\binom{2}{1} = 1",[123,18810,18812,18868,19017,19079,19228],{"className":18811,"ariaHidden":147},[173],[123,18813,18815,18818,18859,18862,18865],{"className":18814},[177],[123,18816],{"className":18817,"style":18218},[181],[123,18819,18821,18824],{"className":18820},[186],[123,18822,18173],{"className":18823,"style":18225},[186,504],[123,18825,18827],{"className":18826},[2548],[123,18828,18830,18851],{"className":18829},[472,473],[123,18831,18833,18848],{"className":18832},[477],[123,18834,18837],{"className":18835,"style":18836},[481],"height:0.3011em;",[123,18838,18839,18842],{"style":18241},[123,18840],{"className":18841,"style":490},[489],[123,18843,18845],{"className":18844},[494,495,496,497],[123,18846,1349],{"className":18847},[186,497],[123,18849,525],{"className":18850},[524],[123,18852,18854],{"className":18853},[477],[123,18855,18857],{"className":18856,"style":18260},[481],[123,18858],{},[123,18860],{"className":18861,"style":781},[194],[123,18863,695],{"className":18864},[785],[123,18866],{"className":18867,"style":781},[194],[123,18869,18871,18874,18942,19008,19011,19014],{"className":18870},[177],[123,18872],{"className":18873,"style":710},[181],[123,18875,18877,18880,18939],{"className":18876},[186],[123,18878],{"className":18879},[458,992],[123,18881,18883],{"className":18882},[426],[123,18884,18886,18931],{"className":18885},[472,473],[123,18887,18889,18928],{"className":18888},[477],[123,18890,18892,18906,18914],{"className":18891,"style":1678},[481],[123,18893,18894,18897],{"style":1008},[123,18895],{"className":18896,"style":1012},[489],[123,18898,18900],{"className":18899},[494,495,496,497],[123,18901,18903],{"className":18902},[186,497],[123,18904,80],{"className":18905},[186,497],[123,18907,18908,18911],{"style":1046},[123,18909],{"className":18910,"style":1012},[489],[123,18912],{"className":18913,"style":1054},[1053],[123,18915,18916,18919],{"style":1057},[123,18917],{"className":18918,"style":1012},[489],[123,18920,18922],{"className":18921},[494,495,496,497],[123,18923,18925],{"className":18924},[186,497],[123,18926,80],{"className":18927},[186,497],[123,18929,525],{"className":18930},[524],[123,18932,18934],{"className":18933},[477],[123,18935,18937],{"className":18936,"style":532},[481],[123,18938],{},[123,18940],{"className":18941},[538,992],[123,18943,18945,18951,19002],{"className":18944},[186],[123,18946,18948],{"className":18947,"style":460},[458,459],[123,18949,424],{"className":18950},[464,465],[123,18952,18954],{"className":18953},[426],[123,18955,18957,18994],{"className":18956},[472,473],[123,18958,18960,18991],{"className":18959},[477],[123,18961,18963,18977],{"className":18962,"style":732},[481],[123,18964,18965,18968],{"style":485},[123,18966],{"className":18967,"style":490},[489],[123,18969,18971],{"className":18970},[494,495,496,497],[123,18972,18974],{"className":18973},[186,497],[123,18975,1349],{"className":18976},[186,497],[123,18978,18979,18982],{"style":508},[123,18980],{"className":18981,"style":490},[489],[123,18983,18985],{"className":18984},[494,495,496,497],[123,18986,18988],{"className":18987},[186,497],[123,18989,1349],{"className":18990},[186,497],[123,18992,525],{"className":18993},[524],[123,18995,18997],{"className":18996},[477],[123,18998,19000],{"className":18999,"style":532},[481],[123,19001],{},[123,19003,19005],{"className":19004,"style":460},[538,459],[123,19006,438],{"className":19007},[464,465],[123,19009],{"className":19010,"style":781},[194],[123,19012,695],{"className":19013},[785],[123,19015],{"className":19016,"style":781},[194],[123,19018,19020,19023,19026,19030,19070,19073,19076],{"className":19019},[177],[123,19021],{"className":19022,"style":18218},[181],[123,19024,80],{"className":19025},[186],[123,19027],{"className":19028,"style":19029},[194],"margin-right:2em;",[123,19031,19033,19036],{"className":19032},[186],[123,19034,18173],{"className":19035,"style":18225},[186,504],[123,19037,19039],{"className":19038},[2548],[123,19040,19042,19062],{"className":19041},[472,473],[123,19043,19045,19059],{"className":19044},[477],[123,19046,19048],{"className":19047,"style":18836},[481],[123,19049,19050,19053],{"style":18241},[123,19051],{"className":19052,"style":490},[489],[123,19054,19056],{"className":19055},[494,495,496,497],[123,19057,80],{"className":19058},[186,497],[123,19060,525],{"className":19061},[524],[123,19063,19065],{"className":19064},[477],[123,19066,19068],{"className":19067,"style":18260},[481],[123,19069],{},[123,19071],{"className":19072,"style":781},[194],[123,19074,695],{"className":19075},[785],[123,19077],{"className":19078,"style":781},[194],[123,19080,19082,19085,19153,19219,19222,19225],{"className":19081},[177],[123,19083],{"className":19084,"style":710},[181],[123,19086,19088,19091,19150],{"className":19087},[186],[123,19089],{"className":19090},[458,992],[123,19092,19094],{"className":19093},[426],[123,19095,19097,19142],{"className":19096},[472,473],[123,19098,19100,19139],{"className":19099},[477],[123,19101,19103,19117,19125],{"className":19102,"style":1678},[481],[123,19104,19105,19108],{"style":1008},[123,19106],{"className":19107,"style":1012},[489],[123,19109,19111],{"className":19110},[494,495,496,497],[123,19112,19114],{"className":19113},[186,497],[123,19115,690],{"className":19116},[186,497],[123,19118,19119,19122],{"style":1046},[123,19120],{"className":19121,"style":1012},[489],[123,19123],{"className":19124,"style":1054},[1053],[123,19126,19127,19130],{"style":1057},[123,19128],{"className":19129,"style":1012},[489],[123,19131,19133],{"className":19132},[494,495,496,497],[123,19134,19136],{"className":19135},[186,497],[123,19137,80],{"className":19138},[186,497],[123,19140,525],{"className":19141},[524],[123,19143,19145],{"className":19144},[477],[123,19146,19148],{"className":19147,"style":532},[481],[123,19149],{},[123,19151],{"className":19152},[538,992],[123,19154,19156,19162,19213],{"className":19155},[186],[123,19157,19159],{"className":19158,"style":460},[458,459],[123,19160,424],{"className":19161},[464,465],[123,19163,19165],{"className":19164},[426],[123,19166,19168,19205],{"className":19167},[472,473],[123,19169,19171,19202],{"className":19170},[477],[123,19172,19174,19188],{"className":19173,"style":732},[481],[123,19175,19176,19179],{"style":485},[123,19177],{"className":19178,"style":490},[489],[123,19180,19182],{"className":19181},[494,495,496,497],[123,19183,19185],{"className":19184},[186,497],[123,19186,80],{"className":19187},[186,497],[123,19189,19190,19193],{"style":508},[123,19191],{"className":19192,"style":490},[489],[123,19194,19196],{"className":19195},[494,495,496,497],[123,19197,19199],{"className":19198},[186,497],[123,19200,690],{"className":19201},[186,497],[123,19203,525],{"className":19204},[524],[123,19206,19208],{"className":19207},[477],[123,19209,19211],{"className":19210,"style":532},[481],[123,19212],{},[123,19214,19216],{"className":19215,"style":460},[538,459],[123,19217,438],{"className":19218},[464,465],[123,19220],{"className":19221,"style":781},[194],[123,19223,695],{"className":19224},[785],[123,19226],{"className":19227,"style":781},[194],[123,19229,19231,19234],{"className":19230},[177],[123,19232],{"className":19233,"style":252},[181],[123,19235,80],{"className":19236},[186],[30,19238,19240],{"className":19239},[409],[123,19241,19243,19317],{"className":19242},[126],[123,19244,19246],{"className":19245},[130],[132,19247,19248],{"xmlns":134},[136,19249,19250,19314],{},[139,19251,19252,19258,19260,19266,19278,19280,19282,19284,19290,19292,19298,19310,19312],{},[11964,19253,19254,19256],{},[430,19255,18173],{},[142,19257,690],{},[145,19259,695],{},[426,19261,19262,19264],{},[142,19263,80],{},[142,19265,271],{},[139,19267,19268,19270,19276],{},[145,19269,424],{"fence":147},[426,19271,19272,19274],{"linethickness":428},[142,19273,151],{},[142,19275,690],{},[145,19277,438],{"fence":147},[145,19279,695],{},[142,19281,690],{},[194,19283],{"width":18775},[11964,19285,19286,19288],{},[430,19287,18173],{},[142,19289,271],{},[145,19291,695],{},[426,19293,19294,19296],{},[142,19295,80],{},[142,19297,151],{},[139,19299,19300,19302,19308],{},[145,19301,424],{"fence":147},[426,19303,19304,19306],{"linethickness":428},[142,19305,156],{},[142,19307,271],{},[145,19309,438],{"fence":147},[145,19311,695],{},[142,19313,2760],{},[166,19315,19316],{"encoding":168},"C_2 = \\frac{1}{3}\\binom{4}{2} = 2 \\qquad C_3 = \\frac{1}{4}\\binom{6}{3} = 5 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},[481],[123,19610,19611,19614],{"style":1008},[123,19612],{"className":19613,"style":1012},[489],[123,19615,19617],{"className":19616},[494,495,496,497],[123,19618,19620],{"className":19619},[186,497],[123,19621,151],{"className":19622},[186,497],[123,19624,19625,19628],{"style":1046},[123,19626],{"className":19627,"style":1012},[489],[123,19629],{"className":19630,"style":1054},[1053],[123,19632,19633,19636],{"style":1057},[123,19634],{"className":19635,"style":1012},[489],[123,19637,19639],{"className":19638},[494,495,496,497],[123,19640,19642],{"className":19641},[186,497],[123,19643,80],{"className":19644},[186,497],[123,19646,525],{"className":19647},[524],[123,19649,19651],{"className":19650},[477],[123,19652,19654],{"className":19653,"style":532},[481],[123,19655],{},[123,19657],{"className":19658},[538,992],[123,19660,19662,19668,19719],{"className":19661},[186],[123,19663,19665],{"className":19664,"style":460},[458,459],[123,19666,424],{"className":19667},[464,465],[123,19669,19671],{"className":19670},[426],[123,19672,19674,19711],{"className":19673},[472,473],[123,19675,19677,19708],{"className":19676},[477],[123,19678,19680,19694],{"className":19679,"style":732},[481],[123,19681,19682,19685],{"style":485},[123,19683],{"className":19684,"style":490},[489],[123,19686,19688],{"className":19687},[494,495,496,497],[123,19689,19691],{"className":19690},[186,497],[123,19692,271],{"className":19693},[186,497],[123,19695,19696,19699],{"style":508},[123,19697],{"className":19698,"style":490},[489],[123,19700,19702],{"className":19701},[494,495,496,497],[123,19703,19705],{"className":19704},[186,497],[123,19706,156],{"className":19707},[186,497],[123,19709,525],{"className":19710},[524],[123,19712,19714],{"className":19713},[477],[123,19715,19717],{"className":19716,"style":532},[481],[123,19718],{},[123,19720,19722],{"className":19721,"style":460},[538,459],[123,19723,438],{"className":19724},[464,465],[123,19726],{"className":19727,"style":781},[194],[123,19729,695],{"className":19730},[785],[123,19732],{"className":19733,"style":781},[194],[123,19735,19737,19740],{"className":19736},[177],[123,19738],{"className":19739,"style":252},[181],[123,19741,2760],{"className":19742},[186],[30,19744,19746],{"className":19745},[409],[123,19747,19749,19791],{"className":19748},[126],[123,19750,19752],{"className":19751},[130],[132,19753,19754],{"xmlns":134},[136,19755,19756,19788],{},[139,19757,19758,19764,19766,19772,19784,19786],{},[11964,19759,19760,19762],{},[430,19761,18173],{},[142,19763,151],{},[145,19765,695],{},[426,19767,19768,19770],{},[142,19769,80],{},[142,19771,2760],{},[139,19773,19774,19776,19782],{},[145,19775,424],{"fence":147},[426,19777,19778,19780],{"linethickness":428},[142,19779,2427],{},[142,19781,151],{},[145,19783,438],{"fence":147},[145,19785,695],{},[142,19787,17328],{},[166,19789,19790],{"encoding":168},"C_4 = \\frac{1}{5}\\binom{8}{4} = 14",[123,19792,19794,19849,19998],{"className":19793,"ariaHidden":147},[173],[123,19795,19797,19800,19840,19843,19846],{"className":19796},[177],[123,19798],{"className":19799,"style":18218},[181],[123,19801,19803,19806],{"className":19802},[186],[123,19804,18173],{"className":19805,"style":18225},[186,504],[123,19807,19809],{"className":19808},[2548],[123,19810,19812,19832],{"className":19811},[472,473],[123,19813,19815,19829],{"className":19814},[477],[123,19816,19818],{"className":19817,"style":18836},[481],[123,19819,19820,19823],{"style":18241},[123,19821],{"className":19822,"style":490},[489],[123,19824,19826],{"className":19825},[494,495,496,497],[123,19827,151],{"className":19828},[186,497],[123,19830,525],{"className":19831},[524],[123,19833,19835],{"className":19834},[477],[123,19836,19838],{"className":19837,"style":18260},[481],[123,19839],{},[123,19841],{"className":19842,"style":781},[194],[123,19844,695],{"className":19845},[785],[123,19847],{"className":19848,"style":781},[194],[123,19850,19852,19855,19923,19989,19992,19995],{"className":19851},[177],[123,19853],{"className":19854,"style":710},[181],[123,19856,19858,19861,19920],{"className":19857},[186],[123,19859],{"className":19860},[458,992],[123,19862,19864],{"className":19863},[426],[123,19865,19867,19912],{"className":19866},[472,473],[123,19868,19870,19909],{"className":19869},[477],[123,19871,19873,19887,19895],{"className":19872,"style":1678},[481],[123,19874,19875,19878],{"style":1008},[123,19876],{"className":19877,"style":1012},[489],[123,19879,19881],{"className":19880},[494,495,496,497],[123,19882,19884],{"className":19883},[186,497],[123,19885,2760],{"className":19886},[186,497],[123,19888,19889,19892],{"style":1046},[123,19890],{"className":19891,"style":1012},[489],[123,19893],{"className":19894,"style":1054},[1053],[123,19896,19897,19900],{"style":1057},[123,19898],{"className":19899,"style":1012},[489],[123,19901,19903],{"className":19902},[494,495,496,497],[123,19904,19906],{"className":19905},[186,497],[123,19907,80],{"className":19908},[186,497],[123,19910,525],{"className":19911},[524],[123,19913,19915],{"className":19914},[477],[123,19916,19918],{"className":19917,"style":532},[481],[123,19919],{},[123,19921],{"className":19922},[538,992],[123,19924,19926,19932,19983],{"className":19925},[186],[123,19927,19929],{"className":19928,"style":460},[458,459],[123,19930,424],{"className":19931},[464,465],[123,19933,19935],{"className":19934},[426],[123,19936,19938,19975],{"className":19937},[472,473],[123,19939,19941,19972],{"className":19940},[477],[123,19942,19944,19958],{"className":19943,"style":732},[481],[123,19945,19946,19949],{"style":485},[123,19947],{"className":19948,"style":490},[489],[123,19950,19952],{"className":19951},[494,495,496,497],[123,19953,19955],{"className":19954},[186,497],[123,19956,151],{"className":19957},[186,497],[123,19959,19960,19963],{"style":508},[123,19961],{"className":19962,"style":490},[489],[123,19964,19966],{"className":19965},[494,495,496,497],[123,19967,19969],{"className":19968},[186,497],[123,19970,2427],{"className":19971},[186,497],[123,19973,525],{"className":19974},[524],[123,19976,19978],{"className":19977},[477],[123,19979,19981],{"className":19980,"style":532},[481],[123,19982],{},[123,19984,19986],{"className":19985,"style":460},[538,459],[123,19987,438],{"className":19988},[464,465],[123,19990],{"className":19991,"style":781},[194],[123,19993,695],{"className":19994},[785],[123,19996],{"className":19997,"style":781},[194],[123,19999,20001,20004],{"className":20000},[177],[123,20002],{"className":20003,"style":252},[181],[123,20005,17328],{"className":20006},[186],[30,20008,20009,20010,20013],{},"A perfect match. The Catalan numbers count precisely how many flip sequences end ",[42,20011,20012],{},"for the first time"," with more heads than tails at each possible length.",[34,20015,20017],{"id":20016},"the-connection-to-pascals-triangle","The connection to Pascal's triangle",[30,20019,20020,20021,20023,20024,20079,20080,20209,20210,20214,20215,827],{},"There is another way to see why the Catalan numbers appear here. Think of each flip sequence as a ",[42,20022,11611],{}," on a lattice: heads moves one step up, tails moves one step down. The total number of paths of length ",[123,20025,20027,20046],{"className":20026},[126],[123,20028,20030],{"className":20029},[130],[132,20031,20032],{"xmlns":134},[136,20033,20034,20044],{},[139,20035,20036,20038,20040,20042],{},[142,20037,690],{},[430,20039,432],{},[145,20041,274],{},[142,20043,80],{},[166,20045,16094],{"encoding":168},[123,20047,20049,20070],{"className":20048,"ariaHidden":147},[173],[123,20050,20052,20055,20058,20061,20064,20067],{"className":20051},[177],[123,20053],{"className":20054,"style":289},[181],[123,20056,690],{"className":20057},[186],[123,20059,432],{"className":20060},[186,504],[123,20062],{"className":20063,"style":296},[194],[123,20065,274],{"className":20066},[300],[123,20068],{"className":20069,"style":296},[194],[123,20071,20073,20076],{"className":20072},[177],[123,20074],{"className":20075,"style":252},[181],[123,20077,80],{"className":20078},[186]," that end one step above zero is the binomial coefficient ",[123,20081,20083,20119],{"className":20082},[126],[123,20084,20086],{"className":20085},[130],[132,20087,20088],{"xmlns":134},[136,20089,20090,20116],{},[139,20091,20092,20094,20114],{},[145,20093,424],{"fence":147},[426,20095,20096,20106],{"linethickness":428},[139,20097,20098,20100,20102,20104],{},[142,20099,690],{},[430,20101,432],{},[145,20103,274],{},[142,20105,80],{},[139,20107,20108,20110,20112],{},[430,20109,432],{},[145,20111,274],{},[142,20113,80],{},[145,20115,438],{"fence":147},[166,20117,20118],{"encoding":168},"\\binom{2n+1}{n+1}",[123,20120,20122],{"className":20121,"ariaHidden":147},[173],[123,20123,20125,20128],{"className":20124},[177],[123,20126],{"className":20127,"style":18278},[181],[123,20129,20131,20137,20203],{"className":20130},[186],[123,20132,20134],{"className":20133,"style":460},[458,459],[123,20135,424],{"className":20136},[464,465],[123,20138,20140],{"className":20139},[426],[123,20141,20143,20195],{"className":20142},[472,473],[123,20144,20146,20192],{"className":20145},[477],[123,20147,20149,20169],{"className":20148,"style":732},[481],[123,20150,20151,20154],{"style":485},[123,20152],{"className":20153,"style":490},[489],[123,20155,20157],{"className":20156},[494,495,496,497],[123,20158,20160,20163,20166],{"className":20159},[186,497],[123,20161,432],{"className":20162},[186,504,497],[123,20164,274],{"className":20165},[300,497],[123,20167,80],{"className":20168},[186,497],[123,20170,20171,20174],{"style":508},[123,20172],{"className":20173,"style":490},[489],[123,20175,20177],{"className":20176},[494,495,496,497],[123,20178,20180,20183,20186,20189],{"className":20179},[186,497],[123,20181,690],{"className":20182},[186,497],[123,20184,432],{"className":20185},[186,504,497],[123,20187,274],{"className":20188},[300,497],[123,20190,80],{"className":20191},[186,497],[123,20193,525],{"className":20194},[524],[123,20196,20198],{"className":20197},[477],[123,20199,20201],{"className":20200,"style":1730},[481],[123,20202],{},[123,20204,20206],{"className":20205,"style":460},[538,459],[123,20207,438],{"className":20208},[464,465]," — an entry in Pascal's triangle. The Catalan numbers count only the paths that never cross above zero before the final step. They are, quite literally, a filtered selection within the triangle — the same tool we used in the ",[2779,20211,20213],{"href":14178,"rel":20212},[2783],"previous article"," to count coin combinations, now with an additional constraint. If you want to explore this connection in more depth, Matt Parker covers it brilliantly in ",[2779,20216,20218],{"href":14032,"rel":20217},[2783],"his video",[34,20220,20222],{"id":20221},"the-infinite-series","The infinite series",[30,20224,20225,20226,20292],{},"Back to our table. The contribution of length-",[123,20227,20229,20253],{"className":20228},[126],[123,20230,20232],{"className":20231},[130],[132,20233,20234],{"xmlns":134},[136,20235,20236,20250],{},[139,20237,20238,20240,20242,20244,20246,20248],{},[145,20239,424],{"stretchy":868},[142,20241,690],{},[430,20243,432],{},[145,20245,274],{},[142,20247,80],{},[145,20249,438],{"stretchy":868},[166,20251,20252],{"encoding":168},"(2n+1)",[123,20254,20256,20280],{"className":20255,"ariaHidden":147},[173],[123,20257,20259,20262,20265,20268,20271,20274,20277],{"className":20258},[177],[123,20260],{"className":20261,"style":3490},[181],[123,20263,424],{"className":20264},[458],[123,20266,690],{"className":20267},[186],[123,20269,432],{"className":20270},[186,504],[123,20272],{"className":20273,"style":296},[194],[123,20275,274],{"className":20276},[300],[123,20278],{"className":20279,"style":296},[194],[123,20281,20283,20286,20289],{"className":20282},[177],[123,20284],{"className":20285,"style":3490},[181],[123,20287,80],{"className":20288},[186],[123,20290,438],{"className":20291},[538]," sequences to the overall average is:",[30,20294,20296],{"className":20295},[409],[123,20297,20299,20394],{"className":20298},[126],[123,20300,20302],{"className":20301},[130],[132,20303,20304],{"xmlns":134},[136,20305,20306,20391],{},[139,20307,20308,20343,20345,20360,20362],{},[20309,20310,20311,20340],"munder",{},[20309,20312,20313,20337],{},[2520,20314,20315,20327],{},[139,20316,20317,20319,20325],{},[145,20318,424],{"fence":147},[426,20320,20321,20323],{},[142,20322,80],{},[142,20324,690],{},[145,20326,438],{"fence":147},[139,20328,20329,20331,20333,20335],{},[142,20330,690],{},[430,20332,432],{},[145,20334,274],{},[142,20336,80],{},[145,20338,20339],{"stretchy":147},"⏟",[1901,20341,20342],{},"probability",[145,20344,1210],{},[20309,20346,20347,20357],{},[20309,20348,20349,20355],{},[11964,20350,20351,20353],{},[430,20352,18173],{},[430,20354,432],{},[145,20356,20339],{"stretchy":147},[1901,20358,20359],{},"no. of sequences",[145,20361,1210],{},[20309,20363,20364,20388],{},[20309,20365,20366,20386],{},[426,20367,20368,20376],{},[139,20369,20370,20372,20374],{},[430,20371,432],{},[145,20373,274],{},[142,20375,80],{},[139,20377,20378,20380,20382,20384],{},[142,20379,690],{},[430,20381,432],{},[145,20383,274],{},[142,20385,80],{},[145,20387,20339],{"stretchy":147},[1901,20389,20390],{},"heads ratio",[166,20392,20393],{"encoding":168},"\\underbrace{\\left(\\frac{1}{2}\\right)^{2n+1}}_{\\text{probability}} \\times \\underbrace{C_n}_{\\text{no. of sequences}} \\times \\underbrace{\\frac{n+1}{2n+1}}_{\\text{heads ratio}}",[123,20395,20397,20659,20825],{"className":20396,"ariaHidden":147},[173],[123,20398,20400,20404,20650,20653,20656],{"className":20399},[177],[123,20401],{"className":20402,"style":20403},[181],"height:4.0743em;vertical-align:-2.4202em;",[123,20405,20407],{"className":20406},[2195,20309],[123,20408,20410,20641],{"className":20409},[472,473],[123,20411,20413,20638],{"className":20412},[477],[123,20414,20417,20436],{"className":20415,"style":20416},[481],"height:1.654em;",[123,20418,20420,20424],{"style":20419},"top:-1.3699em;",[123,20421],{"className":20422,"style":20423},[489],"height:3.654em;",[123,20425,20427],{"className":20426},[494,495,496,497],[123,20428,20430],{"className":20429},[186,497],[123,20431,20433],{"className":20432},[186,104,497],[123,20434,20342],{"className":20435},[186,497],[123,20437,20439,20442],{"style":20438},"top:-3.654em;",[123,20440],{"className":20441,"style":20423},[489],[123,20443,20445],{"className":20444},[2195,20309],[123,20446,20448,20629],{"className":20447},[472,473],[123,20449,20451,20626],{"className":20450},[477],[123,20452,20454,20498],{"className":20453,"style":20416},[481],[123,20455,20458,20461],{"className":20456,"style":20457},[11563],"top:-2.056em;",[123,20459],{"className":20460,"style":20423},[489],[123,20462,20466,20478,20488],{"className":20463,"style":20465},[20464],"stretchy","height:0.548em;min-width:1.6em;",[123,20467,20471],{"className":20468,"style":20470},[20469],"brace-left","height:0.548em;",[11603,20472,20475],{"xmlns":11605,"width":11606,"height":20473,"viewBox":20474,"preserveAspectRatio":11609},"0.548em","0 0 400000 548",[11611,20476],{"d":20477},"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",[123,20479,20482],{"className":20480,"style":20470},[20481],"brace-center",[11603,20483,20485],{"xmlns":11605,"width":11606,"height":20473,"viewBox":20474,"preserveAspectRatio":20484},"xMidYMin slice",[11611,20486],{"d":20487},"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",[123,20489,20492],{"className":20490,"style":20470},[20491],"brace-right",[11603,20493,20495],{"xmlns":11605,"width":11606,"height":20473,"viewBox":20474,"preserveAspectRatio":20494},"xMaxYMin slice",[11611,20496],{"d":20497},"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",[123,20499,20500,20503],{"style":20438},[123,20501],{"className":20502,"style":20423},[489],[123,20504,20506],{"className":20505},[186],[123,20507,20509,20590],{"className":20508},[2195],[123,20510,20512,20518,20584],{"className":20511},[2195],[123,20513,20515],{"className":20514,"style":460},[458,459],[123,20516,424],{"className":20517},[464,496],[123,20519,20521,20524,20581],{"className":20520},[186],[123,20522],{"className":20523},[458,992],[123,20525,20527],{"className":20526},[426],[123,20528,20530,20572],{"className":20529},[472,473],[123,20531,20533,20569],{"className":20532},[477],[123,20534,20537,20549,20557],{"className":20535,"style":20536},[481],"height:1.3214em;",[123,20538,20540,20543],{"style":20539},"top:-2.314em;",[123,20541],{"className":20542,"style":1012},[489],[123,20544,20546],{"className":20545},[186],[123,20547,690],{"className":20548},[186],[123,20550,20551,20554],{"style":1046},[123,20552],{"className":20553,"style":1012},[489],[123,20555],{"className":20556,"style":1054},[1053],[123,20558,20560,20563],{"style":20559},"top:-3.677em;",[123,20561],{"className":20562,"style":1012},[489],[123,20564,20566],{"className":20565},[186],[123,20567,80],{"className":20568},[186],[123,20570,525],{"className":20571},[524],[123,20573,20575],{"className":20574},[477],[123,20576,20579],{"className":20577,"style":20578},[481],"height:0.686em;",[123,20580],{},[123,20582],{"className":20583},[538,992],[123,20585,20587],{"className":20586,"style":460},[538,459],[123,20588,438],{"className":20589},[464,496],[123,20591,20593],{"className":20592},[2548],[123,20594,20596],{"className":20595},[472],[123,20597,20599],{"className":20598},[477],[123,20600,20602],{"className":20601,"style":20416},[481],[123,20603,20605,20608],{"style":20604},"top:-3.9029em;margin-right:0.05em;",[123,20606],{"className":20607,"style":490},[489],[123,20609,20611],{"className":20610},[494,495,496,497],[123,20612,20614,20617,20620,20623],{"className":20613},[186,497],[123,20615,690],{"className":20616},[186,497],[123,20618,432],{"className":20619},[186,504,497],[123,20621,274],{"className":20622},[300,497],[123,20624,80],{"className":20625},[186,497],[123,20627,525],{"className":20628},[524],[123,20630,20632],{"className":20631},[477],[123,20633,20636],{"className":20634,"style":20635},[481],"height:1.598em;",[123,20637],{},[123,20639,525],{"className":20640},[524],[123,20642,20644],{"className":20643},[477],[123,20645,20648],{"className":20646,"style":20647},[481],"height:2.4202em;",[123,20649],{},[123,20651],{"className":20652,"style":296},[194],[123,20654,1210],{"className":20655},[300],[123,20657],{"className":20658,"style":296},[194],[123,20660,20662,20666,20816,20819,20822],{"className":20661},[177],[123,20663],{"className":20664,"style":20665},[181],"height:2.3035em;vertical-align:-1.6202em;",[123,20667,20669],{"className":20668},[2195,20309],[123,20670,20672,20807],{"className":20671},[472,473],[123,20673,20675,20804],{"className":20674},[477],[123,20676,20678,20696],{"className":20677,"style":3286},[481],[123,20679,20681,20684],{"style":20680},"top:-1.5159em;",[123,20682],{"className":20683,"style":1012},[489],[123,20685,20687],{"className":20686},[494,495,496,497],[123,20688,20690],{"className":20689},[186,497],[123,20691,20693],{"className":20692},[186,104,497],[123,20694,20359],{"className":20695},[186,497],[123,20697,20698,20701],{"style":13723},[123,20699],{"className":20700,"style":1012},[489],[123,20702,20704],{"className":20703},[2195,20309],[123,20705,20707,20795],{"className":20706},[472,473],[123,20708,20710,20792],{"className":20709},[477],[123,20711,20713,20744],{"className":20712,"style":3286},[481],[123,20714,20717,20720],{"className":20715,"style":20716},[11563],"top:-2.202em;",[123,20718],{"className":20719,"style":1012},[489],[123,20721,20723,20730,20737],{"className":20722,"style":20465},[20464],[123,20724,20726],{"className":20725,"style":20470},[20469],[11603,20727,20728],{"xmlns":11605,"width":11606,"height":20473,"viewBox":20474,"preserveAspectRatio":11609},[11611,20729],{"d":20477},[123,20731,20733],{"className":20732,"style":20470},[20481],[11603,20734,20735],{"xmlns":11605,"width":11606,"height":20473,"viewBox":20474,"preserveAspectRatio":20484},[11611,20736],{"d":20487},[123,20738,20740],{"className":20739,"style":20470},[20491],[11603,20741,20742],{"xmlns":11605,"width":11606,"height":20473,"viewBox":20474,"preserveAspectRatio":20494},[11611,20743],{"d":20497},[123,20745,20746,20749],{"style":13723},[123,20747],{"className":20748,"style":1012},[489],[123,20750,20752],{"className":20751},[186],[123,20753,20755,20758],{"className":20754},[186],[123,20756,18173],{"className":20757,"style":18225},[186,504],[123,20759,20761],{"className":20760},[2548],[123,20762,20764,20784],{"className":20763},[472,473],[123,20765,20767,20781],{"className":20766},[477],[123,20768,20770],{"className":20769,"style":18238},[481],[123,20771,20772,20775],{"style":18241},[123,20773],{"className":20774,"style":49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the overall average ratio is the sum of ",[42,21038,21039],{},"all"," these contributions, from ",[123,21042,21044,21062],{"className":21043},[126],[123,21045,21047],{"className":21046},[130],[132,21048,21049],{"xmlns":134},[136,21050,21051,21059],{},[139,21052,21053,21055,21057],{},[430,21054,432],{},[145,21056,695],{},[142,21058,1349],{},[166,21060,21061],{"encoding":168},"n = 0",[123,21063,21065,21083],{"className":21064,"ariaHidden":147},[173],[123,21066,21068,21071,21074,21077,21080],{"className":21067},[177],[123,21069],{"className":21070,"style":572},[181],[123,21072,432],{"className":21073},[186,504],[123,21075],{"className":21076,"style":781},[194],[123,21078,695],{"className":21079},[785],[123,21081],{"className":21082,"style":781},[194],[123,21084,21086,21089],{"className":21085},[177],[123,21087],{"className":21088,"style":252},[181],[123,21090,1349],{"className":21091},[186]," to infinity:",[30,21094,21096],{"className":21095},[409],[123,21097,21099,21176],{"className":21098},[126],[123,21100,21102],{"className":21101},[130],[132,21103,21104],{"xmlns":134},[136,21105,21106,21173],{},[139,21107,21108,21111,21113,21129,21151,21153],{},[1901,21109,21110],{},"Average",[145,21112,695],{},[7906,21114,21115,21117,21125],{},[145,21116,7910],{},[139,21118,21119,21121,21123],{},[430,21120,432],{},[145,21122,695],{},[142,21124,1349],{},[430,21126,21128],{"mathvariant":21127},"normal","∞",[426,21130,21131,21137],{},[11964,21132,21133,21135],{},[430,21134,18173],{},[430,21136,432],{},[2520,21138,21139,21141],{},[142,21140,690],{},[139,21142,21143,21145,21147,21149],{},[142,21144,690],{},[430,21146,432],{},[145,21148,274],{},[142,21150,80],{},[145,21152,878],{},[426,21154,21155,21163],{},[139,21156,21157,21159,21161],{},[430,21158,432],{},[145,21160,274],{},[142,21162,80],{},[139,21164,21165,21167,21169,21171],{},[142,21166,690],{},[430,21168,432],{},[145,21170,274],{},[142,21172,80],{},[166,21174,21175],{"encoding":168},"\\text{Average} = \\sum_{n=0}^{\\infty} \\frac{C_n}{2^{2n+1}} \\cdot \\frac{n+1}{2n+1}",[123,21177,21179,21201,21431],{"className":21178,"ariaHidden":147},[173],[123,21180,21182,21186,21192,21195,21198],{"className":21181},[177],[123,21183],{"className":21184,"style":21185},[181],"height:0.8778em;vertical-align:-0.1944em;",[123,21187,21189],{"className":21188},[186,104],[123,21190,21110],{"className":21191},[186],[123,21193],{"className":21194,"style":781},[194],[123,21196,695],{"className":21197},[785],[123,21199],{"className":21200,"style":781},[194],[123,21202,21204,21208,21271,21274,21422,21425,21428],{"className":21203},[177],[123,21205],{"className":21206,"style":21207},[181],"height:1.2334em;vertical-align:-0.345em;",[123,21209,21211,21214],{"className":21210},[8039],[123,21212,7910],{"className":21213,"style":8045},[8039,8043,8044],[123,21215,21217],{"className":21216},[2548],[123,21218,21220,21263],{"className":21219},[472,473],[123,21221,21223,21260],{"className":21222},[477],[123,21224,21226,21246],{"className":21225,"style":8058},[481],[123,21227,21228,21231],{"style":8061},[123,21229],{"className":21230,"style":490},[489],[123,21232,21234],{"className":21233},[494,495,496,497],[123,21235,21237,21240,21243],{"className":21236},[186,497],[123,21238,432],{"className":21239},[186,504,497],[123,21241,695],{"className":21242},[785,497],[123,21244,1349],{"className":21245},[186,497],[123,21247,21248,21251],{"style":8082},[123,21249],{"className":21250,"style":490},[489],[123,21252,21254],{"className":21253},[494,495,496,497],[123,21255,21257],{"className":21256},[186,497],[123,21258,21128],{"className":21259},[186,497],[123,21261,525],{"className":21262},[524],[123,21264,21266],{"className":21265},[477],[123,21267,21269],{"className":21268,"style":8104},[481],[123,21270],{},[123,21272],{"className":21273,"style":195},[194],[123,21275,21277,21280,21419],{"className":21276},[186],[123,21278],{"className":21279},[458,992],[123,21281,21283],{"className":21282},[426],[123,21284,21286,21411],{"className":21285},[472,473],[123,21287,21289,21408],{"className":21288},[477],[123,21290,21293,21345,21353],{"className":21291,"style":21292},[481],"height:0.8884em;",[123,21294,21295,21298],{"style":1008},[123,21296],{"className":21297,"style":1012},[489],[123,21299,21301],{"className":21300},[494,495,496,497],[123,21302,21304],{"className":21303},[186,497],[123,21305,21307,21310],{"className":21306},[186,497],[123,21308,690],{"className":21309},[186,497],[123,21311,21313],{"className":21312},[2548],[123,21314,21316],{"className":21315},[472],[123,21317,21319],{"className":21318},[477],[123,21320,21322],{"className":21321,"style":3573},[481],[123,21323,21324,21327],{"style":3576},[123,21325],{"className":21326,"style":3580},[489],[123,21328,21330],{"className":21329},[494,3584,465,497],[123,21331,21333,21336,21339,21342],{"className":21332},[186,497],[123,21334,690],{"className":21335},[186,497],[123,21337,432],{"className":21338},[186,504,497],[123,21340,274],{"className":21341},[300,497],[123,21343,80],{"className":21344},[186,497],[123,21346,21347,21350],{"style":1046},[123,21348],{"className":21349,"style":1012},[489],[123,21351],{"className":21352,"style":1054},[1053],[123,21354,21356,21359],{"style":21355},"top:-3.4101em;",[123,21357],{"className":21358,"style":1012},[489],[123,21360,21362],{"className":21361},[494,495,496,497],[123,21363,21365],{"className":21364},[186,497],[123,21366,21368,21371],{"className":21367},[186,497],[123,21369,18173],{"className":21370,"style":18225},[186,504,497],[123,21372,21374],{"className":21373},[2548],[123,21375,21377,21399],{"className":21376},[472,473],[123,21378,21380,21396],{"className":21379},[477],[123,21381,21384],{"className":21382,"style":21383},[481],"height:0.1645em;",[123,21385,21387,21390],{"style":21386},"top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;",[123,21388],{"className":21389,"style":3580},[489],[123,21391,21393],{"className":21392},[494,3584,465,497],[123,21394,432],{"className":21395},[186,504,497],[123,21397,525],{"className":21398},[524],[123,21400,21402],{"className":21401},[477],[123,21403,21406],{"className":21404,"style":21405},[481],"height:0.143em;",[123,21407],{},[123,21409,525],{"className":21410},[524],[123,21412,21414],{"className":21413},[477],[123,21415,21417],{"className":21416,"style":532},[481],[123,21418],{},[123,21420],{"className":21421},[538,992],[123,21423],{"className":21424,"style":296},[194],[123,21426,878],{"className":21427},[300],[123,21429],{"className":21430,"style":296},[194],[123,21432,21434,21437],{"className":21433},[177],[123,21435],{"className":21436,"style":1659},[181],[123,21438,21440,21443,21517],{"className":21439},[186],[123,21441],{"className":21442},[458,992],[123,21444,21446],{"className":21445},[426],[123,21447,21449,21509],{"className":21448},[472,473],[123,21450,21452,21506],{"className":21451},[477],[123,21453,21455,21478,21486],{"className":21454,"style":1678},[481],[123,21456,21457,21460],{"style":1008},[123,21458],{"className":21459,"style":1012},[489],[123,21461,21463],{"className":21462},[494,495,496,497],[123,21464,21466,21469,21472,21475],{"className":21465},[186,497],[123,21467,690],{"className":21468},[186,497],[123,21470,432],{"className":21471},[186,504,497],[123,21473,274],{"className":21474},[300,497],[123,21476,80],{"className":21477},[186,497],[123,21479,21480,21483],{"style":1046},[123,21481],{"className":21482,"style":1012},[489],[123,21484],{"className":21485,"style":1054},[1053],[123,21487,21488,21491],{"style":1057},[123,21489],{"className":21490,"style":1012},[489],[123,21492,21494],{"className":21493},[494,495,496,497],[123,21495,21497,21500,21503],{"className":21496},[186,497],[123,21498,432],{"className":21499},[186,504,497],[123,21501,274],{"className":21502},[300,497],[123,21504,80],{"className":21505},[186,497],[123,21507,525],{"className":21508},[524],[123,21510,21512],{"className":21511},[477],[123,21513,21515],{"className":21514,"style":1730},[481],[123,21516],{},[123,21518],{"className":21519},[538,992],[30,21521,21522],{},"Let's compute the first few terms:",[2632,21524,21525,21565],{},[2635,21526,21527],{},[2638,21528,21529,21559,21562],{},[2641,21530,21531],{"align":2643},[123,21532,21534,21547],{"className":21533},[126],[123,21535,21537],{"className":21536},[130],[132,21538,21539],{"xmlns":134},[136,21540,21541,21545],{},[139,21542,21543],{},[430,21544,432],{},[166,21546,432],{"encoding":168},[123,21548,21550],{"className":21549,"ariaHidden":147},[173],[123,21551,21553,21556],{"className":21552},[177],[123,21554],{"className":21555,"style":572},[181],[123,21557,432],{"className":21558},[186,504],[2641,21560,21561],{"align":2898},"Contribution",[2641,21563,21564],{"align":2898},"Running total",[2735,21566,21567,21768,22040,22311,22582],{},[2638,21568,21569,21571,21737],{},[2740,21570,1349],{"align":2643},[2740,21572,21573],{"align":2898},[123,21574,21576,21606],{"className":21575},[126],[123,21577,21579],{"className":21578},[130],[132,21580,21581],{"xmlns":134},[136,21582,21583,21603],{},[139,21584,21585,21591,21593,21595,21597,21599,21601],{},[426,21586,21587,21589],{},[142,21588,80],{},[142,21590,690],{},[145,21592,1210],{},[142,21594,80],{},[145,21596,1210],{},[142,21598,80],{},[145,21600,695],{},[142,21602,14444],{},[166,21604,21605],{"encoding":168},"\\frac{1}{2} \\times 1 \\times 1 = 0.5",[123,21607,21609,21692,21710,21728],{"className":21608,"ariaHidden":147},[173],[123,21610,21612,21615,21683,21686,21689],{"className":21611},[177],[123,21613],{"className":21614,"style":1751},[181],[123,21616,21618,21621,21680],{"className":21617},[186],[123,21619],{"className":21620},[458,992],[123,21622,21624],{"className":21623},[426],[123,21625,21627,21672],{"className":21626},[472,473],[123,21628,21630,21669],{"className":21629},[477],[123,21631,21633,21647,21655],{"className":21632,"style":1678},[481],[123,21634,21635,21638],{"style":1008},[123,21636],{"className":21637,"style":1012},[489],[123,21639,21641],{"className":21640},[494,495,496,497],[123,21642,21644],{"className":21643},[186,497],[123,21645,690],{"className":21646},[186,497],[123,21648,21649,21652],{"style":1046},[123,21650],{"className":21651,"style":1012},[489],[123,21653],{"className":21654,"style":1054},[1053],[123,21656,21657,21660],{"style":1057},[123,21658],{"className":21659,"style":1012},[489],[123,21661,21663],{"className":21662},[494,495,496,497],[123,21664,21666],{"className":21665},[186,497],[123,21667,80],{"className":21668},[186,497],[123,21670,525],{"className":21671},[524],[123,21673,21675],{"className":21674},[477],[123,21676,21678],{"className":21677,"style":532},[481],[123,21679],{},[123,21681],{"className":21682},[538,992],[123,21684],{"className":21685,"style":296},[194],[123,21687,1210],{"className":21688},[300],[123,21690],{"className":21691,"style":296},[194],[123,21693,21695,21698,21701,21704,21707],{"className":21694},[177],[123,21696],{"className":21697,"style":289},[181],[123,21699,80],{"className":21700},[186],[123,21702],{"className":21703,"style":296},[194],[123,21705,1210],{"className":21706},[300],[123,21708],{"className":21709,"style":296},[194],[123,21711,21713,21716,21719,21722,21725],{"className":21712},[177],[123,21714],{"className":21715,"style":252},[181],[123,21717,80],{"className":21718},[186],[123,21720],{"className":21721,"style":781},[194],[123,21723,695],{"className":21724},[785],[123,21726],{"className":21727,"style":781},[194],[123,21729,21731,21734],{"className":21730},[177],[123,21732],{"className":21733,"style":252},[181],[123,21735,14444],{"className":21736},[186],[2740,21738,21739],{"align":2898},[123,21740,21742,21756],{"className":21741},[126],[123,21743,21745],{"className":21744},[130],[132,21746,21747],{"xmlns":134},[136,21748,21749,21754],{},[139,21750,21751],{},[142,21752,21753],{},"0.5000",[166,21755,21753],{"encoding":168},[123,21757,21759],{"className":21758,"ariaHidden":147},[173],[123,21760,21762,21765],{"className":21761},[177],[123,21763],{"className":21764,"style":252},[181],[123,21766,21753],{"className":21767},[186],[2638,21769,21770,21772,22009],{},[2740,21771,80],{"align":2643},[2740,21773,21774],{"align":2898},[123,21775,21777,21813],{"className":21776},[126],[123,21778,21780],{"className":21779},[130],[132,21781,21782],{"xmlns":134},[136,21783,21784,21810],{},[139,21785,21786,21792,21794,21796,21798,21804,21807],{},[426,21787,21788,21790],{},[142,21789,80],{},[142,21791,2427],{},[145,21793,1210],{},[142,21795,80],{},[145,21797,1210],{},[426,21799,21800,21802],{},[142,21801,690],{},[142,21803,271],{},[145,21805,21806],{},"≈",[142,21808,21809],{},"0.0833",[166,21811,21812],{"encoding":168},"\\frac{1}{8} \\times 1 \\times \\frac{2}{3} \\approx 0.0833",[123,21814,21816,21899,21917,22000],{"className":21815,"ariaHidden":147},[173],[123,21817,21819,21822,21890,21893,21896],{"className":21818},[177],[123,21820],{"className":21821,"style":1751},[181],[123,21823,21825,21828,21887],{"className":21824},[186],[123,21826],{"className":21827},[458,992],[123,21829,21831],{"className":21830},[426],[123,21832,21834,21879],{"className":21833},[472,473],[123,21835,21837,21876],{"className":21836},[477],[123,21838,21840,21854,21862],{"className":21839,"style":1678},[481],[123,21841,21842,21845],{"style":1008},[123,21843],{"className":21844,"style":1012},[489],[123,21846,21848],{"className":21847},[494,495,496,497],[123,21849,21851],{"className":21850},[186,497],[123,21852,2427],{"className":21853},[186,497],[123,21855,21856,21859],{"style":1046},[123,21857],{"className":21858,"style":1012},[489],[123,21860],{"className":21861,"style":1054},[1053],[123,21863,21864,21867],{"style":1057},[123,21865],{"className":21866,"style":1012},[489],[123,21868,21870],{"className":21869},[494,495,496,497],[123,21871,21873],{"className":21872},[186,497],[123,21874,80],{"className":21875},[186,497],[123,21877,525],{"className":21878},[524],[123,21880,21882],{"className":21881},[477],[123,21883,21885],{"className":21884,"style":532},[481],[123,21886],{},[123,21888],{"className":21889},[538,992],[123,21891],{"className":21892,"style":296},[194],[123,21894,1210],{"className":21895},[300],[123,21897],{"className":21898,"style":296},[194],[123,21900,21902,21905,21908,21911,21914],{"className":21901},[177],[123,21903],{"className":21904,"style":289},[181],[123,21906,80],{"className":21907},[186],[123,21909],{"className":21910,"style":296},[194],[123,21912,1210],{"className":21913},[300],[123,21915],{"className":21916,"style":296},[194],[123,21918,21920,21923,21991,21994,21997],{"className":21919},[177],[123,21921],{"className":21922,"style":1751},[181],[123,21924,21926,21929,21988],{"className":21925},[186],[123,21927],{"className":21928},[458,992],[123,21930,21932],{"className":21931},[426],[123,21933,21935,21980],{"className":21934},[472,473],[123,21936,21938,21977],{"className":21937},[477],[123,21939,21941,21955,21963],{"className":21940,"style":1678},[481],[123,21942,21943,21946],{"style":1008},[123,21944],{"className":21945,"style":1012},[489],[123,21947,21949],{"className":21948},[494,495,496,497],[123,21950,21952],{"className":21951},[186,497],[123,21953,271],{"className":21954},[186,497],[123,21956,21957,21960],{"style":1046},[123,21958],{"className":21959,"style":1012},[489],[123,21961],{"className":21962,"style":1054},[1053],[123,21964,21965,21968],{"style":1057},[123,21966],{"className":21967,"style":1012},[489],[123,21969,21971],{"className":21970},[494,495,496,497],[123,21972,21974],{"className":21973},[186,497],[123,21975,690],{"className":21976},[186,497],[123,21978,525],{"className":21979},[524],[123,21981,21983],{"className":21982},[477],[123,21984,21986],{"className":21985,"style":532},[481],[123,21987],{},[123,21989],{"className":21990},[538,992],[123,21992],{"className":21993,"style":781},[194],[123,21995,21806],{"className":21996},[785],[123,21998],{"className":21999,"style":781},[194],[123,22001,22003,22006],{"className":22002},[177],[123,22004],{"className":22005,"style":252},[181],[123,22007,21809],{"className":22008},[186],[2740,22010,22011],{"align":2898},[123,22012,22014,22028],{"className":22013},[126],[123,22015,22017],{"className":22016},[130],[132,22018,22019],{"xmlns":134},[136,22020,22021,22026],{},[139,22022,22023],{},[142,22024,22025],{},"0.5833",[166,22027,22025],{"encoding":168},[123,22029,22031],{"className":22030,"ariaHidden":147},[173],[123,22032,22034,22037],{"className":22033},[177],[123,22035],{"className":22036,"style":252},[181],[123,22038,22025],{"className":22039},[186],[2638,22041,22042,22044,22280],{},[2740,22043,690],{"align":2643},[2740,22045,22046],{"align":2898},[123,22047,22049,22084],{"className":22048},[126],[123,22050,22052],{"className":22051},[130],[132,22053,22054],{"xmlns":134},[136,22055,22056,22081],{},[139,22057,22058,22064,22066,22068,22070,22076,22078],{},[426,22059,22060,22062],{},[142,22061,80],{},[142,22063,2763],{},[145,22065,1210],{},[142,22067,690],{},[145,22069,1210],{},[426,22071,22072,22074],{},[142,22073,271],{},[142,22075,2760],{},[145,22077,695],{},[142,22079,22080],{},"0.0375",[166,22082,22083],{"encoding":168},"\\frac{1}{32} \\times 2 \\times \\frac{3}{5} = 0.0375",[123,22085,22087,22170,22188,22271],{"className":22086,"ariaHidden":147},[173],[123,22088,22090,22093,22161,22164,22167],{"className":22089},[177],[123,22091],{"className":22092,"style":1751},[181],[123,22094,22096,22099,22158],{"className":22095},[186],[123,22097],{"className":22098},[458,992],[123,22100,22102],{"className":22101},[426],[123,22103,22105,22150],{"className":22104},[472,473],[123,22106,22108,22147],{"className":22107},[477],[123,22109,22111,22125,22133],{"className":22110,"style":1678},[481],[123,22112,22113,22116],{"style":1008},[123,22114],{"className":22115,"style":1012},[489],[123,22117,22119],{"className":22118},[494,495,496,497],[123,22120,22122],{"className":22121},[186,497],[123,22123,2763],{"className":22124},[186,497],[123,22126,22127,22130],{"style":1046},[123,22128],{"className":22129,"style":1012},[489],[123,22131],{"className":22132,"style":1054},[1053],[123,22134,22135,22138],{"style":1057},[123,22136],{"className":22137,"style":1012},[489],[123,22139,22141],{"className":22140},[494,495,496,497],[123,22142,22144],{"className":22143},[186,497],[123,22145,80],{"className":22146},[186,497],[123,22148,525],{"className":22149},[524],[123,22151,22153],{"className":22152},[477],[123,22154,22156],{"className":22155,"style":532},[481],[123,22157],{},[123,22159],{"className":22160},[538,992],[123,22162],{"className":22163,"style":296},[194],[123,22165,1210],{"className":22166},[300],[123,22168],{"className":22169,"style":296},[194],[123,22171,22173,22176,22179,22182,22185],{"className":22172},[177],[123,22174],{"className":22175,"style":289},[181],[123,22177,690],{"className":22178},[186],[123,22180],{"className":22181,"style":296},[194],[123,22183,1210],{"className":22184},[300],[123,22186],{"className":22187,"style":296},[194],[123,22189,22191,22194,22262,22265,22268],{"className":22190},[177],[123,22192],{"className":22193,"style":1751},[181],[123,22195,22197,22200,22259],{"className":22196},[186],[123,22198],{"className":22199},[458,992],[123,22201,22203],{"className":22202},[426],[123,22204,22206,22251],{"className":22205},[472,473],[123,22207,22209,22248],{"className":22208},[477],[123,22210,22212,22226,22234],{"className":22211,"style":1678},[481],[123,22213,22214,22217],{"style":1008},[123,22215],{"className":22216,"style":1012},[489],[123,22218,22220],{"className":22219},[494,495,496,497],[123,22221,22223],{"className":22222},[186,497],[123,22224,2760],{"className":22225},[186,497],[123,22227,22228,22231],{"style":1046},[123,22229],{"className":22230,"style":1012},[489],[123,22232],{"className":22233,"style":1054},[1053],[123,22235,22236,22239],{"style":1057},[123,22237],{"className":22238,"style":1012},[489],[123,22240,22242],{"className":22241},[494,495,496,497],[123,22243,22245],{"className":22244},[186,497],[123,22246,271],{"className":22247},[186,497],[123,22249,525],{"className":22250},[524],[123,22252,22254],{"className":22253},[477],[123,22255,22257],{"className":22256,"style":532},[481],[123,22258],{},[123,22260],{"className":22261},[538,992],[123,22263],{"className":22264,"style":781},[194],[123,22266,695],{"className":22267},[785],[123,22269],{"className":22270,"style":781},[194],[123,22272,22274,22277],{"className":22273},[177],[123,22275],{"className":22276,"style":252},[181],[123,22278,22080],{"className":22279},[186],[2740,22281,22282],{"align":2898},[123,22283,22285,22299],{"className":22284},[126],[123,22286,22288],{"className":22287},[130],[132,22289,22290],{"xmlns":134},[136,22291,22292,22297],{},[139,22293,22294],{},[142,22295,22296],{},"0.6208",[166,22298,22296],{"encoding":168},[123,22300,22302],{"className":22301,"ariaHidden":147},[173],[123,22303,22305,22308],{"className":22304},[177],[123,22306],{"className":22307,"style":252},[181],[123,22309,22296],{"className":22310},[186],[2638,22312,22313,22315,22551],{},[2740,22314,271],{"align":2643},[2740,22316,22317],{"align":2898},[123,22318,22320,22355],{"className":22319},[126],[123,22321,22323],{"className":22322},[130],[132,22324,22325],{"xmlns":134},[136,22326,22327,22352],{},[139,22328,22329,22335,22337,22339,22341,22347,22349],{},[426,22330,22331,22333],{},[142,22332,80],{},[142,22334,17088],{},[145,22336,1210],{},[142,22338,2760],{},[145,22340,1210],{},[426,22342,22343,22345],{},[142,22344,151],{},[142,22346,16985],{},[145,22348,21806],{},[142,22350,22351],{},"0.0223",[166,22353,22354],{"encoding":168},"\\frac{1}{128} \\times 5 \\times \\frac{4}{7} \\approx 0.0223",[123,22356,22358,22441,22459,22542],{"className":22357,"ariaHidden":147},[173],[123,22359,22361,22364,22432,22435,22438],{"className":22360},[177],[123,22362],{"className":22363,"style":1751},[181],[123,22365,22367,22370,22429],{"className":22366},[186],[123,22368],{"className":22369},[458,992],[123,22371,22373],{"className":22372},[426],[123,22374,22376,22421],{"className":22375},[472,473],[123,22377,22379,22418],{"className":22378},[477],[123,22380,22382,22396,22404],{"className":22381,"style":1678},[481],[123,22383,22384,22387],{"style":1008},[123,22385],{"className":22386,"style":1012},[489],[123,22388,22390],{"className":22389},[494,495,496,497],[123,22391,22393],{"className":22392},[186,497],[123,22394,17088],{"className":22395},[186,497],[123,22397,22398,22401],{"style":1046},[123,22399],{"className":22400,"style":1012},[489],[123,22402],{"className":22403,"style":1054},[1053],[123,22405,22406,22409],{"style":1057},[123,22407],{"className":22408,"style":1012},[489],[123,22410,22412],{"className":22411},[494,495,496,497],[123,22413,22415],{"className":22414},[186,497],[123,22416,80],{"className":22417},[186,497],[123,22419,525],{"className":22420},[524],[123,22422,22424],{"className":22423},[477],[123,22425,22427],{"className":22426,"style":532},[481],[123,22428],{},[123,22430],{"className":22431},[538,992],[123,22433],{"className":22434,"style":296},[194],[123,22436,1210],{"className":22437},[300],[123,22439],{"className":22440,"style":296},[194],[123,22442,22444,22447,22450,22453,22456],{"className":22443},[177],[123,22445],{"className":22446,"style":289},[181],[123,22448,2760],{"className":22449},[186],[123,22451],{"className":22452,"style":296},[194],[123,22454,1210],{"className":22455},[300],[123,22457],{"className":22458,"style":296},[194],[123,22460,22462,22465,22533,22536,22539],{"className":22461},[177],[123,22463],{"className":22464,"style":1751},[181],[123,22466,22468,22471,22530],{"className":22467},[186],[123,22469],{"className":22470},[458,992],[123,22472,22474],{"className":22473},[426],[123,22475,22477,22522],{"className":22476},[472,473],[123,22478,22480,22519],{"className":22479},[477],[123,22481,22483,22497,22505],{"className":22482,"style":1678},[481],[123,22484,22485,22488],{"style":1008},[123,22486],{"className":22487,"style":1012},[489],[123,22489,22491],{"className":22490},[494,495,496,497],[123,22492,22494],{"className":22493},[186,497],[123,22495,16985],{"className":22496},[186,497],[123,22498,22499,22502],{"style":1046},[123,22500],{"className":22501,"style":1012},[489],[123,22503],{"className":22504,"style":1054},[1053],[123,22506,22507,22510],{"style":1057},[123,22508],{"className":22509,"style":1012},[489],[123,22511,22513],{"className":22512},[494,495,496,497],[123,22514,22516],{"className":22515},[186,497],[123,22517,151],{"className":22518},[186,497],[123,22520,525],{"className":22521},[524],[123,22523,22525],{"className":22524},[477],[123,22526,22528],{"className":22527,"style":532},[481],[123,22529],{},[123,22531],{"className":22532},[538,992],[123,22534],{"className":22535,"style":781},[194],[123,22537,21806],{"className":22538},[785],[123,22540],{"className":22541,"style":781},[194],[123,22543,22545,22548],{"className":22544},[177],[123,22546],{"className":22547,"style":252},[181],[123,22549,22351],{"className":22550},[186],[2740,22552,22553],{"align":2898},[123,22554,22556,22570],{"className":22555},[126],[123,22557,22559],{"className":22558},[130],[132,22560,22561],{"xmlns":134},[136,22562,22563,22568],{},[139,22564,22565],{},[142,22566,22567],{},"0.6431",[166,22569,22567],{"encoding":168},[123,22571,22573],{"className":22572,"ariaHidden":147},[173],[123,22574,22576,22579],{"className":22575},[177],[123,22577],{"className":22578,"style":252},[181],[123,22580,22567],{"className":22581},[186],[2638,22583,22584,22586,22822],{},[2740,22585,151],{"align":2643},[2740,22587,22588],{"align":2898},[123,22589,22591,22626],{"className":22590},[126],[123,22592,22594],{"className":22593},[130],[132,22595,22596],{"xmlns":134},[136,22597,22598,22623],{},[139,22599,22600,22606,22608,22610,22612,22618,22620],{},[426,22601,22602,22604],{},[142,22603,80],{},[142,22605,17349],{},[145,22607,1210],{},[142,22609,17328],{},[145,22611,1210],{},[426,22613,22614,22616],{},[142,22615,2760],{},[142,22617,17245],{},[145,22619,21806],{},[142,22621,22622],{},"0.0152",[166,22624,22625],{"encoding":168},"\\frac{1}{512} \\times 14 \\times \\frac{5}{9} \\approx 0.0152",[123,22627,22629,22712,22730,22813],{"className":22628,"ariaHidden":147},[173],[123,22630,22632,22635,22703,22706,22709],{"className":22631},[177],[123,22633],{"className":22634,"style":1751},[181],[123,22636,22638,22641,22700],{"className":22637},[186],[123,22639],{"className":22640},[458,992],[123,22642,22644],{"className":22643},[426],[123,22645,22647,22692],{"className":22646},[472,473],[123,22648,22650,22689],{"className":22649},[477],[123,22651,22653,22667,22675],{"className":22652,"style":1678},[481],[123,22654,22655,22658],{"style":1008},[123,22656],{"className":22657,"style":1012},[489],[123,22659,22661],{"className":22660},[494,495,496,497],[123,22662,22664],{"className":22663},[186,497],[123,22665,17349],{"className":22666},[186,497],[123,22668,22669,22672],{"style":1046},[123,22670],{"className":22671,"style":1012},[489],[123,22673],{"className":22674,"style":1054},[1053],[123,22676,22677,22680],{"style":1057},[123,22678],{"className":22679,"style":1012},[489],[123,22681,22683],{"className":22682},[494,495,496,497],[123,22684,22686],{"className":22685},[186,497],[123,22687,80],{"className":22688},[186,497],[123,22690,525],{"className":22691},[524],[123,22693,22695],{"className":22694},[477],[123,22696,22698],{"className":22697,"style":532},[481],[123,22699],{},[123,22701],{"className":22702},[538,992],[123,22704],{"className":22705,"style":296},[194],[123,22707,1210],{"className":22708},[300],[123,22710],{"className":22711,"style":296},[194],[123,22713,22715,22718,22721,22724,22727],{"className":22714},[177],[123,22716],{"className":22717,"style":289},[181],[123,22719,17328],{"className":22720},[186],[123,22722],{"className":22723,"style":296},[194],[123,22725,1210],{"className":22726},[300],[123,22728],{"className":22729,"style":296},[194],[123,22731,22733,22736,22804,22807,22810],{"className":22732},[177],[123,22734],{"className":22735,"style":1751},[181],[123,22737,22739,22742,22801],{"className":22738},[186],[123,22740],{"className":22741},[458,992],[123,22743,22745],{"className":22744},[426],[123,22746,22748,22793],{"className":22747},[472,473],[123,22749,22751,22790],{"className":22750},[477],[123,22752,22754,22768,22776],{"className":22753,"style":1678},[481],[123,22755,22756,22759],{"style":1008},[123,22757],{"className":22758,"style":1012},[489],[123,22760,22762],{"className":22761},[494,495,496,497],[123,22763,22765],{"className":22764},[186,497],[123,22766,17245],{"className":22767},[186,497],[123,22769,22770,22773],{"style":1046},[123,22771],{"className":22772,"style":1012},[489],[123,22774],{"className":22775,"style":1054},[1053],[123,22777,22778,22781],{"style":1057},[123,22779],{"className":22780,"style":1012},[489],[123,22782,22784],{"className":22783},[494,495,496,497],[123,22785,22787],{"className":22786},[186,497],[123,22788,2760],{"className":22789},[186,497],[123,22791,525],{"className":22792},[524],[123,22794,22796],{"className":22795},[477],[123,22797,22799],{"className":22798,"style":532},[481],[123,22800],{},[123,22802],{"className":22803},[538,992],[123,22805],{"className":22806,"style":781},[194],[123,22808,21806],{"className":22809},[785],[123,22811],{"className":22812,"style":781},[194],[123,22814,22816,22819],{"className":22815},[177],[123,22817],{"className":22818,"style":252},[181],[123,22820,22622],{"className":22821},[186],[2740,22823,22824],{"align":2898},[123,22825,22827,22841],{"className":22826},[126],[123,22828,22830],{"className":22829},[130],[132,22831,22832],{"xmlns":134},[136,22833,22834,22839],{},[139,22835,22836],{},[142,22837,22838],{},"0.6583",[166,22840,22838],{"encoding":168},[123,22842,22844],{"className":22843,"ariaHidden":147},[173],[123,22845,22847,22850],{"className":22846},[177],[123,22848],{"className":22849,"style":252},[181],[123,22851,22838],{"className":22852},[186],[30,22854,22855,22856,22884,22885,23006],{},"With just five terms we are at ",[123,22857,22859,22872],{"className":22858},[126],[123,22860,22862],{"className":22861},[130],[132,22863,22864],{"xmlns":134},[136,22865,22866,22870],{},[139,22867,22868],{},[142,22869,22838],{},[166,22871,22838],{"encoding":168},[123,22873,22875],{"className":22874,"ariaHidden":147},[173],[123,22876,22878,22881],{"className":22877},[177],[123,22879],{"className":22880,"style":252},[181],[123,22882,22838],{"className":22883},[186],", slowly approaching ",[123,22886,22888,22911],{"className":22887},[126],[123,22889,22891],{"className":22890},[130],[132,22892,22893],{"xmlns":134},[136,22894,22895,22908],{},[139,22896,22897,22903,22905],{},[426,22898,22899,22901],{},[430,22900,11453],{},[142,22902,151],{},[145,22904,21806],{},[142,22906,22907],{},"0.7854",[166,22909,22910],{"encoding":168},"\\frac{\\pi}{4} \\approx 0.7854",[123,22912,22914,22997],{"className":22913,"ariaHidden":147},[173],[123,22915,22917,22920,22988,22991,22994],{"className":22916},[177],[123,22918],{"className":22919,"style":14066},[181],[123,22921,22923,22926,22985],{"className":22922},[186],[123,22924],{"className":22925},[458,992],[123,22927,22929],{"className":22928},[426],[123,22930,22932,22977],{"className":22931},[472,473],[123,22933,22935,22974],{"className":22934},[477],[123,22936,22938,22952,22960],{"className":22937,"style":14085},[481],[123,22939,22940,22943],{"style":1008},[123,22941],{"className":22942,"style":1012},[489],[123,22944,22946],{"className":22945},[494,495,496,497],[123,22947,22949],{"className":22948},[186,497],[123,22950,151],{"className":22951},[186,497],[123,22953,22954,22957],{"style":1046},[123,22955],{"className":22956,"style":1012},[489],[123,22958],{"className":22959,"style":1054},[1053],[123,22961,22962,22965],{"style":1057},[123,22963],{"className":22964,"style":1012},[489],[123,22966,22968],{"className":22967},[494,495,496,497],[123,22969,22971],{"className":22970},[186,497],[123,22972,11453],{"className":22973,"style":11545},[186,504,497],[123,22975,525],{"className":22976},[524],[123,22978,22980],{"className":22979},[477],[123,22981,22983],{"className":22982,"style":532},[481],[123,22984],{},[123,22986],{"className":22987},[538,992],[123,22989],{"className":22990,"style":781},[194],[123,22992,21806],{"className":22993},[785],[123,22995],{"className":22996,"style":781},[194],[123,22998,23000,23003],{"className":22999},[177],[123,23001],{"className":23002,"style":252},[181],[123,23004,22907],{"className":23005},[186],". The convergence is slow, but the direction is unmistakable when the number of flips is sufficiently large.",[34,23008,23010],{"id":23009},"the-appearance-of-arcsine","The appearance of arcsine",[30,23012,23013],{},"This is where the magic happens. Let's substitute the Catalan number formula into our series:",[30,23015,23017],{"className":23016},[409],[123,23018,23020,23067],{"className":23019},[126],[123,23021,23023],{"className":23022},[130],[132,23024,23025],{"xmlns":134},[136,23026,23027,23065],{},[139,23028,23029,23035,23037,23049],{},[11964,23030,23031,23033],{},[430,23032,18173],{},[430,23034,432],{},[145,23036,695],{},[426,23038,23039,23041],{},[142,23040,80],{},[139,23042,23043,23045,23047],{},[430,23044,432],{},[145,23046,274],{},[142,23048,80],{},[139,23050,23051,23053,23063],{},[145,23052,424],{"fence":147},[426,23054,23055,23061],{"linethickness":428},[139,23056,23057,23059],{},[142,23058,690],{},[430,23060,432],{},[430,23062,432],{},[145,23064,438],{"fence":147},[166,23066,18208],{"encoding":168},[123,23068,23070,23125],{"className":23069,"ariaHidden":147},[173],[123,23071,23073,23076,23116,23119,23122],{"className":23072},[177],[123,23074],{"className":23075,"style":18218},[181],[123,23077,23079,23082],{"className":23078},[186],[123,23080,18173],{"className":23081,"style":18225},[186,504],[123,23083,23085],{"className":23084},[2548],[123,23086,23088,23108],{"className":23087},[472,473],[123,23089,23091,23105],{"className":23090},[477],[123,23092,23094],{"className":23093,"style":18238},[481],[123,23095,23096,23099],{"style":18241},[123,23097],{"className":23098,"style":490},[489],[123,23100,23102],{"className":23101},[494,495,496,497],[123,23103,432],{"className":23104},[186,504,497],[123,23106,525],{"className":23107},[524],[123,23109,23111],{"className":23110},[477],[123,23112,23114],{"className":23113,"style":18260},[481],[123,23115],{},[123,23117],{"className":23118,"style":781},[194],[123,23120,695],{"className":23121},[785],[123,23123],{"className":23124,"style":781},[194],[123,23126,23128,23131,23205],{"className":23127},[177],[123,23129],{"className":23130,"style":18278},[181],[123,23132,23134,23137,23202],{"className":23133},[186],[123,23135],{"className":23136},[458,992],[123,23138,23140],{"className":23139},[426],[123,23141,23143,23194],{"className":23142},[472,473],[123,23144,23146,23191],{"className":23145},[477],[123,23147,23149,23169,23177],{"className":23148,"style":1678},[481],[123,23150,23151,23154],{"style":1008},[123,23152],{"className":23153,"style":1012},[489],[123,23155,23157],{"className":23156},[494,495,496,497],[123,23158,23160,23163,23166],{"className":23159},[186,497],[123,23161,432],{"className":23162},[186,504,497],[123,23164,274],{"className":23165},[300,497],[123,23167,80],{"className":23168},[186,497],[123,23170,23171,23174],{"style":1046},[123,23172],{"className":23173,"style":1012},[489],[123,23175],{"className":23176,"style":1054},[1053],[123,23178,23179,23182],{"style":1057},[123,23180],{"className":23181,"style":1012},[489],[123,23183,23185],{"className":23184},[494,495,496,497],[123,23186,23188],{"className":23187},[186,497],[123,23189,80],{"className":23190},[186,497],[123,23192,525],{"className":23193},[524],[123,23195,23197],{"className":23196},[477],[123,23198,23200],{"className":23199,"style":1730},[481],[123,23201],{},[123,23203],{"className":23204},[538,992],[123,23206,23208,23214,23268],{"className":23207},[186],[123,23209,23211],{"className":23210,"style":460},[458,459],[123,23212,424],{"className":23213},[464,465],[123,23215,23217],{"className":23216},[426],[123,23218,23220,23260],{"className":23219},[472,473],[123,23221,23223,23257],{"className":23222},[477],[123,23224,23226,23240],{"className":23225,"style":732},[481],[123,23227,23228,23231],{"style":485},[123,23229],{"className":23230,"style":490},[489],[123,23232,23234],{"className":23233},[494,495,496,497],[123,23235,23237],{"className":23236},[186,497],[123,23238,432],{"className":23239},[186,504,497],[123,23241,23242,23245],{"style":508},[123,23243],{"className":23244,"style":490},[489],[123,23246,23248],{"className":23247},[494,495,496,497],[123,23249,23251,23254],{"className":23250},[186,497],[123,23252,690],{"className":23253},[186,497],[123,23255,432],{"className":23256},[186,504,497],[123,23258,525],{"className":23259},[524],[123,23261,23263],{"className":23262},[477],[123,23264,23266],{"className":23265,"style":532},[481],[123,23267],{},[123,23269,23271],{"className":23270,"style":460},[538,459],[123,23272,438],{"className":23273},[464,465],[30,23275,23276,23277,23347,23348,23474,23475,23536],{},"When we multiply ",[123,23278,23280,23298],{"className":23279},[126],[123,23281,23283],{"className":23282},[130],[132,23284,23285],{"xmlns":134},[136,23286,23287,23295],{},[139,23288,23289],{},[11964,23290,23291,23293],{},[430,23292,18173],{},[430,23294,432],{},[166,23296,23297],{"encoding":168},"C_n",[123,23299,23301],{"className":23300,"ariaHidden":147},[173],[123,23302,23304,23307],{"className":23303},[177],[123,23305],{"className":23306,"style":18218},[181],[123,23308,23310,23313],{"className":23309},[186],[123,23311,18173],{"className":23312,"style":18225},[186,504],[123,23314,23316],{"className":23315},[2548],[123,23317,23319,23339],{"className":23318},[472,473],[123,23320,23322,23336],{"className":23321},[477],[123,23323,23325],{"className":23324,"style":18238},[481],[123,23326,23327,23330],{"style":18241},[123,23328],{"className":23329,"style":490},[489],[123,23331,23333],{"className":23332},[494,495,496,497],[123,23334,432],{"className":23335},[186,504,497],[123,23337,525],{"className":23338},[524],[123,23340,23342],{"className":23341},[477],[123,23343,23345],{"className":23344,"style":18260},[481],[123,23346],{}," by ",[123,23349,23351,23382],{"className":23350},[126],[123,23352,23354],{"className":23353},[130],[132,23355,23356],{"xmlns":134},[136,23357,23358,23380],{},[139,23359,23360],{},[426,23361,23362,23370],{},[139,23363,23364,23366,23368],{},[430,23365,432],{},[145,23367,274],{},[142,23369,80],{},[139,23371,23372,23374,23376,23378],{},[142,23373,690],{},[430,23375,432],{},[145,23377,274],{},[142,23379,80],{},[166,23381,17531],{"encoding":168},[123,23383,23385],{"className":23384,"ariaHidden":147},[173],[123,23386,23388,23391],{"className":23387},[177],[123,23389],{"className":23390,"style":1659},[181],[123,23392,23394,23397,23471],{"className":23393},[186],[123,23395],{"className":23396},[458,992],[123,23398,23400],{"className":23399},[426],[123,23401,23403,23463],{"className":23402},[472,473],[123,23404,23406,23460],{"className":23405},[477],[123,23407,23409,23432,23440],{"className":23408,"style":1678},[481],[123,23410,23411,23414],{"style":1008},[123,23412],{"className":23413,"style":1012},[489],[123,23415,23417],{"className":23416},[494,495,496,497],[123,23418,23420,23423,23426,23429],{"className":23419},[186,497],[123,23421,690],{"className":23422},[186,497],[123,23424,432],{"className":23425},[186,504,497],[123,23427,274],{"className":23428},[300,497],[123,23430,80],{"className":23431},[186,497],[123,23433,23434,23437],{"style":1046},[123,23435],{"className":23436,"style":1012},[489],[123,23438],{"className":23439,"style":1054},[1053],[123,23441,23442,23445],{"style":1057},[123,23443],{"className":23444,"style":1012},[489],[123,23446,23448],{"className":23447},[494,495,496,497],[123,23449,23451,23454,23457],{"className":23450},[186,497],[123,23452,432],{"className":23453},[186,504,497],[123,23455,274],{"className":23456},[300,497],[123,23458,80],{"className":23459},[186,497],[123,23461,525],{"className":23462},[524],[123,23464,23466],{"className":23465},[477],[123,23467,23469],{"className":23468,"style":1730},[481],[123,23470],{},[123,23472],{"className":23473},[538,992],", the factor ",[123,23476,23478,23500],{"className":23477},[126],[123,23479,23481],{"className":23480},[130],[132,23482,23483],{"xmlns":134},[136,23484,23485,23497],{},[139,23486,23487,23489,23491,23493,23495],{},[145,23488,424],{"stretchy":868},[430,23490,432],{},[145,23492,274],{},[142,23494,80],{},[145,23496,438],{"stretchy":868},[166,23498,23499],{"encoding":168},"(n+1)",[123,23501,23503,23524],{"className":23502,"ariaHidden":147},[173],[123,23504,23506,23509,23512,23515,23518,23521],{"className":23505},[177],[123,23507],{"className":23508,"style":3490},[181],[123,23510,424],{"className":23511},[458],[123,23513,432],{"className":23514},[186,504],[123,23516],{"className":23517,"style":296},[194],[123,23519,274],{"className":23520},[300],[123,23522],{"className":23523,"style":296},[194],[123,23525,23527,23530,23533],{"className":23526},[177],[123,23528],{"className":23529,"style":3490},[181],[123,23531,80],{"className":23532},[186],[123,23534,438],{"className":23535},[538]," cancels cleanly:",[30,23538,23540],{"className":23539},[409],[123,23541,23543,23635],{"className":23542},[126],[123,23544,23546],{"className":23545},[130],[132,23547,23548],{"xmlns":134},[136,23549,23550,23632],{},[139,23551,23552,23564,23580,23582,23602,23604],{},[426,23553,23554,23556],{},[142,23555,80],{},[139,23557,23558,23560,23562],{},[430,23559,432],{},[145,23561,274],{},[142,23563,80],{},[139,23565,23566,23568,23578],{},[145,23567,424],{"fence":147},[426,23569,23570,23576],{"linethickness":428},[139,23571,23572,23574],{},[142,23573,690],{},[430,23575,432],{},[430,23577,432],{},[145,23579,438],{"fence":147},[145,23581,878],{},[426,23583,23584,23592],{},[139,23585,23586,23588,23590],{},[430,23587,432],{},[145,23589,274],{},[142,23591,80],{},[139,23593,23594,23596,23598,23600],{},[142,23595,690],{},[430,23597,432],{},[145,23599,274],{},[142,23601,80],{},[145,23603,695],{},[426,23605,23606,23622],{},[139,23607,23608,23610,23620],{},[145,23609,424],{"fence":147},[426,23611,23612,23618],{"linethickness":428},[139,23613,23614,23616],{},[142,23615,690],{},[430,23617,432],{},[430,23619,432],{},[145,23621,438],{"fence":147},[139,23623,23624,23626,23628,23630],{},[142,23625,690],{},[430,23627,432],{},[145,23629,274],{},[142,23631,80],{},[166,23633,23634],{"encoding":168},"\\frac{1}{n+1}\\binom{2n}{n} \\cdot \\frac{n+1}{2n+1} = \\frac{\\binom{2n}{n}}{2n+1}",[123,23636,23638,23796,23894],{"className":23637,"ariaHidden":147},[173],[123,23639,23641,23644,23718,23787,23790,23793],{"className":23640},[177],[123,23642],{"className":23643,"style":18278},[181],[123,23645,23647,23650,23715],{"className":23646},[186],[123,23648],{"className":23649},[458,992],[123,23651,23653],{"className":23652},[426],[123,23654,23656,23707],{"className":23655},[472,473],[123,23657,23659,23704],{"className":23658},[477],[123,23660,23662,23682,23690],{"className":23661,"style":1678},[481],[123,23663,23664,23667],{"style":1008},[123,23665],{"className":23666,"style":1012},[489],[123,23668,23670],{"className":23669},[494,495,496,497],[123,23671,23673,23676,23679],{"className":23672},[186,497],[123,23674,432],{"className":23675},[186,504,497],[123,23677,274],{"className":23678},[300,497],[123,23680,80],{"className":23681},[186,497],[123,23683,23684,23687],{"style":1046},[123,23685],{"className":23686,"style":1012},[489],[123,23688],{"className":23689,"style":1054},[1053],[123,23691,23692,23695],{"style":1057},[123,23693],{"className":23694,"style":1012},[489],[123,23696,23698],{"className":23697},[494,495,496,497],[123,23699,23701],{"className":23700},[186,497],[123,23702,80],{"className":23703},[186,497],[123,23705,525],{"className":23706},[524],[123,23708,23710],{"className":23709},[477],[123,23711,23713],{"className":23712,"style":1730},[481],[123,23714],{},[123,23716],{"className":23717},[538,992],[123,23719,23721,23727,23781],{"className":23720},[186],[123,23722,23724],{"className":23723,"style":460},[458,459],[123,23725,424],{"className":23726},[464,465],[123,23728,23730],{"className":23729},[426],[123,23731,23733,23773],{"className":23732},[472,473],[123,23734,23736,23770],{"className":23735},[477],[123,23737,23739,23753],{"className":23738,"style":732},[481],[123,23740,23741,23744],{"style":485},[123,23742],{"className":23743,"style":490},[489],[123,23745,23747],{"className":23746},[494,495,496,497],[123,23748,23750],{"className":23749},[186,497],[123,23751,432],{"className":23752},[186,504,497],[123,23754,23755,23758],{"style":508},[123,23756],{"className":23757,"style":490},[489],[123,23759,23761],{"className":23760},[494,495,496,497],[123,23762,23764,23767],{"className":23763},[186,497],[123,23765,690],{"className":23766},[186,497],[123,23768,432],{"className":23769},[186,504,497],[123,23771,525],{"className":23772},[524],[123,23774,23776],{"className":23775},[477],[123,23777,23779],{"className":23778,"style":532},[481],[123,23780],{},[123,23782,23784],{"className":23783,"style":460},[538,459],[123,23785,438],{"className":23786},[464,465],[123,23788],{"className":23789,"style":296},[194],[123,23791,878],{"className":23792},[300],[123,23794],{"className":23795,"style":296},[194],[123,23797,23799,23802,23885,23888,23891],{"className":23798},[177],[123,23800],{"className":23801,"style":1659},[181],[123,23803,23805,23808,23882],{"className":23804},[186],[123,23806],{"className":23807},[458,992],[123,23809,23811],{"className":23810},[426],[123,23812,23814,23874],{"className":23813},[472,473],[123,23815,23817,23871],{"className":23816},[477],[123,23818,23820,23843,23851],{"className":23819,"style":1678},[481],[123,23821,23822,23825],{"style":1008},[123,23823],{"className":23824,"style":1012},[489],[123,23826,23828],{"className":23827},[494,495,496,497],[123,23829,23831,23834,23837,23840],{"className":23830},[186,497],[123,23832,690],{"className":23833},[186,497],[123,23835,432],{"className":23836},[186,504,497],[123,23838,274],{"className":23839},[300,497],[123,23841,80],{"className":23842},[186,497],[123,23844,23845,23848],{"style":1046},[123,23846],{"className":23847,"style":1012},[489],[123,23849],{"className":23850,"style":1054},[1053],[123,23852,23853,23856],{"style":1057},[123,23854],{"className":23855,"style":1012},[489],[123,23857,23859],{"className":23858},[494,495,496,497],[123,23860,23862,23865,23868],{"className":23861},[186,497],[123,23863,432],{"className":23864},[186,504,497],[123,23866,274],{"className":23867},[300,497],[123,23869,80],{"className":23870},[186,497],[123,23872,525],{"className":23873},[524],[123,23875,23877],{"className":23876},[477],[123,23878,23880],{"className":23879,"style":1730},[481],[123,23881],{},[123,23883],{"className":23884},[538,992],[123,23886],{"className":23887,"style":781},[194],[123,23889,695],{"className":23890},[785],[123,23892],{"className":23893,"style":781},[194],[123,23895,23897,23901],{"className":23896},[177],[123,23898],{"className":23899,"style":23900},[181],"height:1.9133em;vertical-align:-0.4033em;",[123,23902,23904,23907,24047],{"className":23903},[186],[123,23905],{"className":23906},[458,992],[123,23908,23910],{"className":23909},[426],[123,23911,23913,24039],{"className":23912},[472,473],[123,23914,23916,24036],{"className":23915},[477],[123,23917,23919,23942,23950],{"className":23918,"style":3543},[481],[123,23920,23921,23924],{"style":1008},[123,23922],{"className":23923,"style":1012},[489],[123,23925,23927],{"className":23926},[494,495,496,497],[123,23928,23930,23933,23936,23939],{"className":23929},[186,497],[123,23931,690],{"className":23932},[186,497],[123,23934,432],{"className":23935},[186,504,497],[123,23937,274],{"className":23938},[300,497],[123,23940,80],{"className":23941},[186,497],[123,23943,23944,23947],{"style":1046},[123,23945],{"className":23946,"style":1012},[489],[123,23948],{"className":23949,"style":1054},[1053],[123,23951,23952,23955],{"style":3598},[123,23953],{"className":23954,"style":1012},[489],[123,23956,23958],{"className":23957},[494,495,496,497],[123,23959,23961],{"className":23960},[186,497],[123,23962,23964,23973,24027],{"className":23963},[186,497],[123,23965,23967],{"className":23966,"style":3615},[458,494,3584,3614,497,459],[123,23968,23970],{"className":23969},[464,465,497],[123,23971,424],{"className":23972},[497],[123,23974,23976],{"className":23975},[426],[123,23977,23979,24019],{"className":23978},[472,473],[123,23980,23982,24016],{"className":23981},[477],[123,23983,23985,23999],{"className":23984,"style":3634},[481],[123,23986,23987,23990],{"style":3637},[123,23988],{"className":23989,"style":3580},[489],[123,23991,23993],{"className":23992},[494,3584,465,497],[123,23994,23996],{"className":23995},[186,497],[123,23997,432],{"className":23998},[186,504,497],[123,24000,24001,24004],{"style":3652},[123,24002],{"className":24003,"style":3580},[489],[123,24005,24007],{"className":24006},[494,3584,465,497],[123,24008,24010,24013],{"className":24009},[186,497],[123,24011,690],{"className":24012},[186,497],[123,24014,432],{"className":24015},[186,504,497],[123,24017,525],{"className":24018},[524],[123,24020,24022],{"className":24021},[477],[123,24023,24025],{"className":24024,"style":3674},[481],[123,24026],{},[123,24028,24030],{"className":24029,"style":3615},[538,494,3584,3614,497,459],[123,24031,24033],{"className":24032},[464,465,497],[123,24034,438],{"className":24035},[497],[123,24037,525],{"className":24038},[524],[123,24040,24042],{"className":24041},[477],[123,24043,24045],{"className":24044,"style":1730},[481],[123,24046],{},[123,24048],{"className":24049},[538,992],[30,24051,24052],{},"And the series becomes:",[30,24054,24056],{"className":24055},[409],[123,24057,24059,24195],{"className":24058},[126],[123,24060,24062],{"className":24061},[130],[132,24063,24064],{"xmlns":134},[136,24065,24066,24192],{},[139,24067,24068,24070,24072,24086,24132,24134,24140,24154],{},[1901,24069,21110],{},[145,24071,695],{},[7906,24073,24074,24076,24084],{},[145,24075,7910],{},[139,24077,24078,24080,24082],{},[430,24079,432],{},[145,24081,695],{},[142,24083,1349],{},[430,24085,21128],{"mathvariant":21127},[426,24087,24088,24104],{},[139,24089,24090,24092,24102],{},[145,24091,424],{"fence":147},[426,24093,24094,24100],{"linethickness":428},[139,24095,24096,24098],{},[142,24097,690],{},[430,24099,432],{},[430,24101,432],{},[145,24103,438],{"fence":147},[139,24105,24106,24120,24122,24124,24126,24128,24130],{},[2520,24107,24108,24110],{},[142,24109,690],{},[139,24111,24112,24114,24116,24118],{},[142,24113,690],{},[430,24115,432],{},[145,24117,274],{},[142,24119,80],{},[145,24121,424],{"stretchy":868},[142,24123,690],{},[430,24125,432],{},[145,24127,274],{},[142,24129,80],{},[145,24131,438],{"stretchy":868},[145,24133,695],{},[426,24135,24136,24138],{},[142,24137,80],{},[142,24139,690],{},[7906,24141,24142,24144,24152],{},[145,24143,7910],{},[139,24145,24146,24148,24150],{},[430,24147,432],{},[145,24149,695],{},[142,24151,1349],{},[430,24153,21128],{"mathvariant":21127},[426,24155,24156,24172],{},[139,24157,24158,24160,24170],{},[145,24159,424],{"fence":147},[426,24161,24162,24168],{"linethickness":428},[139,24163,24164,24166],{},[142,24165,690],{},[430,24167,432],{},[430,24169,432],{},[145,24171,438],{"fence":147},[139,24173,24174,24180,24182,24184,24186,24188,24190],{},[2520,24175,24176,24178],{},[142,24177,151],{},[430,24179,432],{},[145,24181,424],{"stretchy":868},[142,24183,690],{},[430,24185,432],{},[145,24187,274],{},[142,24189,80],{},[145,24191,438],{"stretchy":868},[166,24193,24194],{"encoding":168},"\\text{Average} = \\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)}",[123,24196,24198,24219,24497],{"className":24197,"ariaHidden":147},[173],[123,24199,24201,24204,24210,24213,24216],{"className":24200},[177],[123,24202],{"className":24203,"style":21185},[181],[123,24205,24207],{"className":24206},[186,104],[123,24208,21110],{"className":24209},[186],[123,24211],{"className":24212,"style":781},[194],[123,24214,695],{"className":24215},[785],[123,24217],{"className":24218,"style":781},[194],[123,24220,24222,24226,24289,24292,24488,24491,24494],{"className":24221},[177],[123,24223],{"className":24224,"style":24225},[181],"height:2.03em;vertical-align:-0.52em;",[123,24227,24229,24232],{"className":24228},[8039],[123,24230,7910],{"className":24231,"style":8045},[8039,8043,8044],[123,24233,24235],{"className":24234},[2548],[123,24236,24238,24281],{"className":24237},[472,473],[123,24239,24241,24278],{"className":24240},[477],[123,24242,24244,24264],{"className":24243,"style":8058},[481],[123,24245,24246,24249],{"style":8061},[123,24247],{"className":24248,"style":490},[489],[123,24250,24252],{"className":24251},[494,495,496,497],[123,24253,24255,24258,24261],{"className":24254},[186,497],[123,24256,432],{"className":24257},[186,504,497],[123,24259,695],{"className":24260},[785,497],[123,24262,1349],{"className":24263},[186,497],[123,24265,24266,24269],{"style":8082},[123,24267],{"className":24268,"style":490},[489],[123,24270,24272],{"className":24271},[494,495,496,497],[123,24273,24275],{"className":24274},[186,497],[123,24276,21128],{"className":24277},[186,497],[123,24279,525],{"className":24280},[524],[123,24282,24284],{"className":24283},[477],[123,24285,24287],{"className":24286,"style":8104},[481],[123,24288],{},[123,24290],{"className":24291,"style":195},[194],[123,24293,24295,24298,24485],{"className":24294},[186],[123,24296],{"className":24297},[458,992],[123,24299,24301],{"className":24300},[426],[123,24302,24304,24477],{"className":24303},[472,473],[123,24305,24307,24474],{"className":24306},[477],[123,24308,24310,24380,24388],{"className":24309,"style":3543},[481],[123,24311,24312,24315],{"style":1008},[123,24313],{"className":24314,"style":1012},[489],[123,24316,24318],{"className":24317},[494,495,496,497],[123,24319,24321,24362,24365,24368,24371,24374,24377],{"className":24320},[186,497],[123,24322,24324,24327],{"className":24323},[186,497],[123,24325,690],{"className":24326},[186,497],[123,24328,24330],{"className":24329},[2548],[123,24331,24333],{"className":24332},[472],[123,24334,24336],{"className":24335},[477],[123,24337,24339],{"className":24338,"style":3573},[481],[123,24340,24341,24344],{"style":3576},[123,24342],{"className":24343,"style":3580},[489],[123,24345,24347],{"className":24346},[494,3584,465,497],[123,24348,24350,24353,24356,24359],{"className":24349},[186,497],[123,24351,690],{"className":24352},[186,497],[123,24354,432],{"className":24355},[186,504,497],[123,24357,274],{"className":24358},[300,497],[123,24360,80],{"className":24361},[186,497],[123,24363,424],{"className":24364},[458,497],[123,24366,690],{"className":24367},[186,497],[123,24369,432],{"className":24370},[186,504,497],[123,24372,274],{"className":24373},[300,497],[123,24375,80],{"className":24376},[186,497],[123,24378,438],{"className":24379},[538,497],[123,24381,24382,24385],{"style":1046},[123,24383],{"className":24384,"style":1012},[489],[123,24386],{"className":24387,"style":1054},[1053],[123,24389,24390,24393],{"style":3598},[123,24391],{"className":24392,"style":1012},[489],[123,24394,24396],{"className":24395},[494,495,496,497],[123,24397,24399],{"className":24398},[186,497],[123,24400,24402,24411,24465],{"className":24401},[186,497],[123,24403,24405],{"className":24404,"style":3615},[458,494,3584,3614,497,459],[123,24406,24408],{"className":24407},[464,465,497],[123,24409,424],{"className":24410},[497],[123,24412,24414],{"className":24413},[426],[123,24415,24417,24457],{"className":24416},[472,473],[123,24418,24420,24454],{"className":24419},[477],[123,24421,24423,24437],{"className":24422,"style":3634},[481],[123,24424,24425,24428],{"style":3637},[123,24426],{"className":24427,"style":3580},[489],[123,24429,24431],{"className":24430},[494,3584,465,497],[123,24432,24434],{"className":24433},[186,497],[123,24435,432],{"className":24436},[186,504,497],[123,24438,24439,24442],{"style":3652},[123,24440],{"className":24441,"style":3580},[489],[123,24443,24445],{"className":24444},[494,3584,465,497],[123,24446,24448,24451],{"className":24447},[186,497],[123,24449,690],{"className":24450},[186,497],[123,24452,432],{"className":24453},[186,504,497],[123,24455,525],{"className":24456},[524],[123,24458,24460],{"className":24459},[477],[123,24461,24463],{"className":24462,"style":3674},[481],[123,24464],{},[123,24466,24468],{"className":24467,"style":3615},[538,494,3584,3614,497,459],[123,24469,24471],{"className":24470},[464,465,497],[123,24472,438],{"className":24473},[497],[123,24475,525],{"className":24476},[524],[123,24478,24480],{"className"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comes the moment that makes Matt Parker \"angry out of sheer disbelief.\" It turns out there is a classical function in mathematical analysis called the ",[42,24828,24829],{},"arcsine"," — the inverse function of sine — whose power series expansion is:",[30,24832,24834],{"className":24833},[409],[123,24835,24837,24927],{"className":24836},[126],[123,24838,24840],{"className":24839},[130],[132,24841,24842],{"xmlns":134},[136,24843,24844,24924],{},[139,24845,24846,24848,24850,24852,24854,24856,24858,24872,24896,24898],{},[430,24847,13591],{},[145,24849,13594],{},[145,24851,424],{"stretchy":868},[430,24853,6365],{},[145,24855,438],{"stretchy":868},[145,24857,695],{},[7906,24859,24860,24862,24870],{},[145,24861,7910],{},[139,24863,24864,24866,24868],{},[430,24865,432],{},[145,24867,695],{},[142,24869,1349],{},[430,24871,21128],{"mathvariant":21127},[426,24873,24874,24890],{},[139,24875,24876,24878,24888],{},[145,24877,424],{"fence":147},[426,24879,24880,24886],{"linethickness":428},[139,24881,24882,24884],{},[142,24883,690],{},[430,24885,432],{},[430,24887,432],{},[145,24889,438],{"fence":147},[2520,24891,24892,24894],{},[142,24893,151],{},[430,24895,432],{},[145,24897,878],{},[426,24899,24900,24914],{},[2520,24901,24902,24904],{},[430,24903,6365],{},[139,24905,24906,24908,24910,24912],{},[142,24907,690],{},[430,24909,432],{},[145,24911,274],{},[142,24913,80],{},[139,24915,24916,24918,24920,24922],{},[142,24917,690],{},[430,24919,432],{},[145,24921,274],{},[142,24923,80],{},[166,24925,24926],{"encoding":168},"\\arcsin(x) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n} \\cdot 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],[123,25215],{"className":25216},[458,992],[123,25218,25220],{"className":25219},[426],[123,25221,25223,25318],{"className":25222},[472,473],[123,25224,25226,25315],{"className":25225},[477],[123,25227,25230,25253,25261],{"className":25228,"style":25229},[481],"height:1.0179em;",[123,25231,25232,25235],{"style":1008},[123,25233],{"className":25234,"style":1012},[489],[123,25236,25238],{"className":25237},[494,495,496,497],[123,25239,25241,25244,25247,25250],{"className":25240},[186,497],[123,25242,690],{"className":25243},[186,497],[123,25245,432],{"className":25246},[186,504,497],[123,25248,274],{"className":25249},[300,497],[123,25251,80],{"className":25252},[186,497],[123,25254,25255,25258],{"style":1046},[123,25256],{"className":25257,"style":1012},[489],[123,25259],{"className":25260,"style":1054},[1053],[123,25262,25263,25266],{"style":1057},[123,25264],{"className":25265,"style":1012},[489],[123,25267,25269],{"className":25268},[494,495,496,497],[123,25270,25272],{"className":25271},[186,497],[123,25273,25275,25278],{"className":25274},[186,497],[123,25276,6365],{"className":25277},[186,504,497],[123,25279,25281],{"className":25280},[2548],[123,25282,25284],{"className":25283},[472],[123,25285,25287],{"className":25286},[477],[123,25288,25291],{"className":25289,"style":25290},[481],"height:0.8913em;",[123,25292,25294,25297],{"style":25293},"top:-2.931em;margin-right:0.0714em;",[123,25295],{"className":25296,"style":3580},[489],[123,25298,25300],{"className":25299},[494,3584,465,497],[123,25301,25303,25306,25309,25312],{"className":25302},[186,497],[123,25304,690],{"className":25305},[186,497],[123,25307,432],{"className":25308},[186,504,497],[123,25310,274],{"className":25311},[300,497],[123,25313,80],{"className":25314},[186,497],[123,25316,525],{"className":25317},[524],[123,25319,25321],{"className":25320},[477],[123,25322,25324],{"className":25323,"style":1730},[481],[123,25325],{},[123,25327],{"className":25328},[538,992],[30,25330,25331,25332,25360,25361,25389,25390,827],{},"Look familiar? It is exactly our series, multiplied by ",[123,25333,25335,25348],{"className":25334},[126],[123,25336,25338],{"className":25337},[130],[132,25339,25340],{"xmlns":134},[136,25341,25342,25346],{},[139,25343,25344],{},[142,25345,690],{},[166,25347,690],{"encoding":168},[123,25349,25351],{"className":25350,"ariaHidden":147},[173],[123,25352,25354,25357],{"className":25353},[177],[123,25355],{"className":25356,"style":252},[181],[123,25358,690],{"className":25359},[186],", with ",[123,25362,25364,25377],{"className":25363},[126],[123,25365,25367],{"className":25366},[130],[132,25368,25369],{"xmlns":134},[136,25370,25371,25375],{},[139,25372,25373],{},[430,25374,6365],{},[166,25376,6365],{"encoding":168},[123,25378,25380],{"className":25379,"ariaHidden":147},[173],[123,25381,25383,25386],{"className":25382},[177],[123,25384],{"className":25385,"style":572},[181],[123,25387,6365],{"className":25388},[186,504]," instead of ",[123,25391,25393,25406],{"className":25392},[126],[123,25394,25396],{"className":25395},[130],[132,25397,25398],{"xmlns":134},[136,25399,25400,25404],{},[139,25401,25402],{},[142,25403,80],{},[166,25405,80],{"encoding":168},[123,25407,25409],{"className":25408,"ariaHidden":147},[173],[123,25410,25412,25415],{"className":25411},[177],[123,25413],{"className":25414,"style":252},[181],[123,25416,80],{"className":25417},[186],[30,25419,25420,25421,2503],{},"If we evaluate at ",[123,25422,25424,25442],{"className":25423},[126],[123,25425,25427],{"className":25426},[130],[132,25428,25429],{"xmlns":134},[136,25430,25431,25439],{},[139,25432,25433,25435,25437],{},[430,25434,6365],{},[145,25436,695],{},[142,25438,80],{},[166,25440,25441],{"encoding":168},"x = 1",[123,25443,25445,25463],{"className":25444,"ariaHidden":147},[173],[123,25446,25448,25451,25454,25457,25460],{"className":25447},[177],[123,25449],{"className":25450,"style":572},[181],[123,25452,6365],{"className":25453},[186,504],[123,25455],{"className":25456,"style":781},[194],[123,25458,695],{"className":25459},[785],[123,25461],{"className":25462,"style":781},[194],[123,25464,25466,25469],{"className":25465},[177],[123,25467],{"className":25468,"style":252},[181],[123,25470,80],{"className":25471},[186],[30,25473,25475],{"className":25474},[409],[123,25476,25478,25554],{"className":25477},[126],[123,25479,25481],{"className":25480},[130],[132,25482,25483],{"xmlns":134},[136,25484,25485,25551],{},[139,25486,25487,25489,25491,25493,25495,25497,25499,25513],{},[430,25488,13591],{},[145,25490,13594],{},[145,25492,424],{"stretchy":868},[142,25494,80],{},[145,25496,438],{"stretchy":868},[145,25498,695],{},[7906,25500,25501,25503,25511],{},[145,25502,7910],{},[139,25504,25505,25507,25509],{},[430,25506,432],{},[145,25508,695],{},[142,25510,1349],{},[430,25512,21128],{"mathvariant":21127},[426,25514,25515,25531],{},[139,25516,25517,25519,25529],{},[145,25518,424],{"fence":147},[426,25520,25521,25527],{"linethickness":428},[139,25522,25523,25525],{},[142,25524,690],{},[430,25526,432],{},[430,25528,432],{},[145,25530,438],{"fence":147},[139,25532,25533,25539,25541,25543,25545,25547,25549],{},[2520,25534,25535,25537],{},[142,25536,151],{},[430,25538,432],{},[145,25540,424],{"stretchy":868},[142,25542,690],{},[430,25544,432],{},[145,25546,274],{},[142,25548,80],{},[145,25550,438],{"stretchy":868},[166,25552,25553],{"encoding":168},"\\arcsin(1) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n(2n+1)}",[123,25555,25557,25584],{"className":25556,"ariaHidden":147},[173],[123,25558,25560,25563,25566,25569,25572,25575,25578,25581],{"className":25559},[177],[123,25561],{"className":25562,"style":3490},[181],[123,25564,13591],{"className":25565},[8039],[123,25567,424],{"className":25568},[458],[123,25570,80],{"className":25571},[186],[123,25573,438],{"className":25574},[538],[123,25576],{"className":25577,"style":781},[194],[123,25579,695],{"className":25580},[785],[123,25582],{"className":25583,"style":781},[194],[123,25585,25587,25590,25653,25656],{"className":25586},[177],[123,25588],{"className":25589,"style":24225},[181],[123,25591,25593,25596],{"className":25592},[8039],[123,25594,7910],{"className":25595,"style":8045},[8039,8043,8044],[123,25597,25599],{"className":25598},[2548],[123,25600,25602,25645],{"className":25601},[472,473],[123,25603,25605,25642],{"className":25604},[477],[123,25606,25608,25628],{"className":25607,"style":8058},[481],[123,25609,25610,25613],{"style":8061},[123,25611],{"className":25612,"style":490},[489],[123,25614,25616],{"className":25615},[494,495,496,497],[123,25617,25619,25622,25625],{"className":25618},[186,497],[123,25620,432],{"className":25621},[186,504,497],[123,25623,695],{"className":25624},[785,497],[123,25626,1349],{"className":25627},[186,497],[123,25629,25630,25633],{"style":8082},[123,25631],{"className":25632,"style":490},[489],[123,25634,25636],{"className":25635},[494,495,496,497],[123,25637,25639],{"className":25638},[186,497],[123,25640,21128],{"className":25641},[186,497],[123,25643,525],{"className":25644},[524],[123,25646,25648],{"className":25647},[477],[123,25649,25651],{"className":25650,"style":8104},[481],[123,25652],{},[123,25654],{"className":25655,"style":195},[194],[123,25657,25659,25662,25837],{"className":25658},[186],[123,25660],{"className":25661},[458,992],[123,25663,25665],{"className":25664},[426],[123,25666,25668,25829],{"className":25667},[472,473],[123,25669,25671,25826],{"className":25670},[477],[123,25672,25674,25732,25740],{"className":25673,"style":3543},[481],[123,25675,25676,25679],{"style":1008},[123,25677],{"className":25678,"style":1012},[489],[123,25680,25682],{"className":25681},[494,495,496,497],[123,25683,25685,25714,25717,25720,25723,25726,25729],{"className":25684},[186,497],[123,25686,25688,25691],{"className":25687},[186,497],[123,25689,151],{"className":25690},[186,497],[123,25692,25694],{"className":25693},[2548],[123,25695,25697],{"className":25696},[472],[123,25698,25700],{"className":25699},[477],[123,25701,25703],{"className":25702,"style":4830},[481],[123,25704,25705,25708],{"style":3576},[123,25706],{"className":25707,"style":3580},[489],[123,25709,25711],{"className":25710},[494,3584,465,497],[123,25712,432],{"className":25713},[186,504,497],[123,25715,424],{"className":25716},[458,497],[123,25718,690],{"className":25719},[186,497],[123,25721,432],{"className":25722},[186,504,497],[123,25724,274],{"className":25725},[300,497],[123,25727,80],{"className":25728},[186,497],[123,25730,438],{"className":25731},[538,497],[123,25733,25734,25737],{"style":1046},[123,25735],{"className":25736,"style":1012},[489],[123,25738],{"className":25739,"style":1054},[1053],[123,25741,25742,25745],{"style":3598},[123,25743],{"className":25744,"style":1012},[489],[123,25746,25748],{"className":25747},[494,495,496,497],[123,25749,25751],{"className":25750},[186,497],[123,25752,25754,25763,25817],{"className":25753},[186,497],[123,25755,25757],{"className":25756,"style":3615},[458,494,3584,3614,497,459],[123,25758,25760],{"className":25759},[464,465,497],[123,25761,424],{"className":25762},[497],[123,25764,25766],{"className":25765},[426],[123,25767,25769,25809],{"className":25768},[472,473],[123,25770,25772,25806],{"className":25771},[477],[123,25773,25775,25789],{"className":25774,"style":3634},[481],[123,25776,25777,25780],{"style":3637},[123,25778],{"className":25779,"style":3580},[489],[123,25781,25783],{"className":25782},[494,3584,465,497],[123,25784,25786],{"className":25785},[186,497],[123,25787,432],{"className":25788},[186,504,497],[123,25790,25791,25794],{"style":3652},[123,25792],{"className":25793,"style":3580},[489],[123,25795,25797],{"className":25796},[494,3584,465,497],[123,25798,25800,25803],{"className":25799},[186,497],[123,25801,690],{"className":25802},[186,497],[123,25804,432],{"className":25805},[186,504,497],[123,25807,525],{"className":25808},[524],[123,25810,25812],{"className":25811},[477],[123,25813,25815],{"className":25814,"style":3674},[481],[123,25816],{},[123,25818,25820],{"className":25819,"style":3615},[538,494,3584,3614,497,459],[123,25821,25823],{"className":25822},[464,465,497],[123,25824,438],{"className":25825},[497],[123,25827,525],{"className":25828},[524],[123,25830,25832],{"className":25831},[477],[123,25833,25835],{"className":25834,"style":1082},[481],[123,25836],{},[123,25838],{"className":25839},[538,992],[30,25841,25842,25843,25889,25890,25918,25919,2503],{},"And what is ",[123,25844,25846,25868],{"className":25845},[126],[123,25847,25849],{"className":25848},[130],[132,25850,25851],{"xmlns":134},[136,25852,25853,25865],{},[139,25854,25855,25857,25859,25861,25863],{},[430,25856,13591],{},[145,25858,13594],{},[145,25860,424],{"stretchy":868},[142,25862,80],{},[145,25864,438],{"stretchy":868},[166,25866,25867],{"encoding":168},"\\arcsin(1)",[123,25869,25871],{"className":25870,"ariaHidden":147},[173],[123,25872,25874,25877,25880,25883,25886],{"className":25873},[177],[123,25875],{"className":25876,"style":3490},[181],[123,25878,13591],{"className":25879},[8039],[123,25881,424],{"className":25882},[458],[123,25884,80],{"className":25885},[186],[123,25887,438],{"className":25888},[538],"? It is the angle whose sine is ",[123,25891,25893,25906],{"className":25892},[126],[123,25894,25896],{"className":25895},[130],[132,25897,25898],{"xmlns":134},[136,25899,25900,25904],{},[139,25901,25902],{},[142,25903,80],{},[166,25905,80],{"encoding":168},[123,25907,25909],{"className":25908,"ariaHidden":147},[173],[123,25910,25912,25915],{"className":25911},[177],[123,25913],{"className":25914,"style":252},[181],[123,25916,80],{"className":25917},[186],", that is, ",[123,25920,25922,25940],{"className":25921},[126],[123,25923,25925],{"className":25924},[130],[132,25926,25927],{"xmlns":134},[136,25928,25929,25937],{},[139,25930,25931,25934],{},[142,25932,25933],{},"90",[430,25935,25936],{"mathvariant":21127},"°",[166,25938,25939],{"encoding":168},"90°",[123,25941,25943],{"className":25942,"ariaHidden":147},[173],[123,25944,25946,25949],{"className":25945},[177],[123,25947],{"className":25948,"style":602},[181],[123,25950,25939],{"className":25951},[186],[30,25953,25955],{"className":25954},[409],[123,25956,25958,25988],{"className":25957},[126],[123,25959,25961],{"className":25960},[130],[132,25962,25963],{"xmlns":134},[136,25964,25965,25985],{},[139,25966,25967,25969,25971,25973,25975,25977,25979],{},[430,25968,13591],{},[145,25970,13594],{},[145,25972,424],{"stretchy":868},[142,25974,80],{},[145,25976,438],{"stretchy":868},[145,25978,695],{},[426,25980,25981,25983],{},[430,25982,11453],{},[142,25984,690],{},[166,25986,25987],{"encoding":168},"\\arcsin(1) = \\frac{\\pi}{2}",[123,25989,25991,26018],{"className":25990,"ariaHidden":147},[173],[123,25992,25994,25997,26000,26003,26006,26009,26012,26015],{"className":25993},[177],[123,25995],{"className":25996,"style":3490},[181],[123,25998,13591],{"className":25999},[8039],[123,26001,424],{"className":26002},[458],[123,26004,80],{"className":26005},[186],[123,26007,438],{"className":26008},[538],[123,26010],{"className":26011,"style":781},[194],[123,26013,695],{"className":26014},[785],[123,26016],{"className":26017,"style":781},[194],[123,26019,26021,26024],{"className":26020},[177],[123,26022],{"className":26023,"style":14066},[181],[123,26025,26027,26030,26089],{"className":26026},[186],[123,26028],{"className":26029},[458,992],[123,26031,26033],{"className":26032},[426],[123,26034,26036,26081],{"className":26035},[472,473],[123,26037,26039,26078],{"className":26038},[477],[123,26040,26042,26056,26064],{"className":26041,"style":14085},[481],[123,26043,26044,26047],{"style":1008},[123,26045],{"className":26046,"style":1012},[489],[123,26048,26050],{"className":26049},[494,495,496,497],[123,26051,26053],{"className":26052},[186,497],[123,26054,690],{"className":26055},[186,497],[123,26057,26058,26061],{"style":1046},[123,26059],{"className":26060,"style":1012},[489],[123,26062],{"className":26063,"style":1054},[1053],[123,26065,26066,26069],{"style":1057},[123,26067],{"className":26068,"style":1012},[489],[123,26070,26072],{"className":26071},[494,495,496,497],[123,26073,26075],{"className":26074},[186,497],[123,26076,11453],{"className":26077,"style":11545},[186,504,497],[123,26079,525],{"className":26080},[524],[123,26082,26084],{"className":26083},[477],[123,26085,26087],{"className":26086,"style":532},[481],[123,26088],{},[123,26090],{"className":26091},[538,992],[30,26093,26094],{},"Therefore:",[30,26096,26098],{"className":26097},[409],[123,26099,26101,26139],{"className":26100},[126],[123,26102,26104],{"className":26103},[130],[132,26105,26106],{"xmlns":134},[136,26107,26108,26136],{},[139,26109,26110,26112,26114,26120,26122,26128,26130],{},[1901,26111,21110],{},[145,26113,695],{},[426,26115,26116,26118],{},[142,26117,80],{},[142,26119,690],{},[145,26121,878],{},[426,26123,26124,26126],{},[430,26125,11453],{},[142,26127,690],{},[145,26129,695],{},[426,26131,26132,26134],{},[430,26133,11453],{},[142,26135,151],{},[166,26137,26138],{"encoding":168},"\\text{Average} = \\frac{1}{2} \\cdot \\frac{\\pi}{2} = \\frac{\\pi}{4}",[123,26140,26142,26163,26246,26329],{"className":26141,"ariaHidden":147},[173],[123,26143,26145,26148,26154,26157,26160],{"className":26144},[177],[123,26146],{"className":26147,"style":21185},[181],[123,26149,26151],{"className":26150},[186,104],[123,26152,21110],{"className":26153},[186],[123,26155],{"className":26156,"style":781},[194],[123,26158,695],{"className":26159},[785],[123,26161],{"className":26162,"style":781},[194],[123,26164,26166,26169,26237,26240,26243],{"className":26165},[177],[123,26167],{"className":26168,"style":1751},[181],[123,26170,26172,26175,26234],{"className":26171},[186],[123,26173],{"className":26174},[458,992],[123,26176,26178],{"className":26177},[426],[123,26179,26181,26226],{"className":26180},[472,473],[123,26182,26184,26223],{"className":26183},[477],[123,26185,26187,26201,26209],{"className":26186,"style":1678},[481],[123,26188,26189,26192],{"style":1008},[123,26190],{"className":26191,"style":1012},[489],[123,26193,26195],{"className":26194},[494,495,496,497],[123,26196,26198],{"className":26197},[186,497],[123,26199,690],{"className":26200},[186,497],[123,26202,26203,26206],{"style":1046},[123,26204],{"className":26205,"style":1012},[489],[123,26207],{"className":26208,"style":1054},[1053],[123,26210,26211,26214],{"style":1057},[123,26212],{"className":26213,"style":1012},[489],[123,26215,26217],{"className":26216},[494,495,496,497],[123,26218,26220],{"className":26219},[186,497],[123,26221,80],{"className":26222},[186,497],[123,26224,525],{"className":26225},[524],[123,26227,26229],{"className":26228},[477],[123,26230,26232],{"className":26231,"style":532},[481],[123,26233],{},[123,26235],{"className":26236},[538,992],[123,26238],{"className":26239,"style":296},[194],[123,26241,878],{"className":26242},[300],[123,26244],{"className":26245,"style":296},[194],[123,26247,26249,26252,26320,26323,26326],{"className":26248},[177],[123,26250],{"className":26251,"style":14066},[181],[123,26253,26255,26258,26317],{"className":26254},[186],[123,26256],{"className":26257},[458,992],[123,26259,26261],{"className":26260},[426],[123,26262,26264,26309],{"className":26263},[472,473],[123,26265,26267,26306],{"className":26266},[477],[123,26268,26270,26284,26292],{"className":26269,"style":14085},[481],[123,26271,26272,26275],{"style":1008},[123,26273],{"className":26274,"style":1012},[489],[123,26276,26278],{"className":26277},[494,495,496,497],[123,26279,26281],{"className":26280},[186,497],[123,26282,690],{"className":26283},[186,497],[123,26285,26286,26289],{"style":1046},[123,26287],{"className":26288,"style":1012},[489],[123,26290],{"className":26291,"style":1054},[1053],[123,26293,26294,26297],{"style":1057},[123,26295],{"className":26296,"style":1012},[489],[123,26298,26300],{"className":26299},[494,495,496,497],[123,26301,26303],{"className":26302},[186,497],[123,26304,11453],{"className":26305,"style":11545},[186,504,497],[123,26307,525],{"className":26308},[524],[123,26310,26312],{"className":26311},[477],[123,26313,26315],{"className":26314,"style":532},[481],[123,26316],{},[123,26318],{"className":26319},[538,992],[123,26321],{"className":26322,"style":781},[194],[123,26324,695],{"className":26325},[785],[123,26327],{"className":26328,"style":781},[194],[123,26330,26332,26335],{"className":26331},[177],[123,26333],{"className":26334,"style":14066},[181],[123,26336,26338,26341,26400],{"className":26337},[186],[123,26339],{"className":26340},[458,992],[123,26342,26344],{"className":26343},[426],[123,26345,26347,26392],{"className":26346},[472,473],[123,26348,26350,26389],{"className":26349},[477],[123,26351,26353,26367,26375],{"className":26352,"style":14085},[481],[123,26354,26355,26358],{"style":1008},[123,26356],{"className":26357,"style":1012},[489],[123,26359,26361],{"className":26360},[494,495,496,497],[123,26362,26364],{"className":26363},[186,497],[123,26365,151],{"className":26366},[186,497],[123,26368,26369,26372],{"style":1046},[123,26370],{"className":26371,"style":1012},[489],[123,26373],{"className":26374,"style":1054},[1053],[123,26376,26377,26380],{"style":1057},[123,26378],{"className":26379,"style":1012},[489],[123,26381,26383],{"className":26382},[494,495,496,497],[123,26384,26386],{"className":26385},[186,497],[123,26387,11453],{"className":26388,"style":11545},[186,504,497],[123,26390,525],{"className":26391},[524],[123,26393,26395],{"className":26394},[477],[123,26396,26398],{"className":26397,"style":532},[481],[123,26399],{},[123,26401],{"className":26402},[538,992],[30,26404,26405,26406,26414,26415,26443,26444,26447],{},"This connection was ",[2779,26407,26410,26411],{"href":26408,"rel":26409},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.14487",[2783],"discovered in 2025 by the mathematician ",[42,26412,26413],{},"Jim Propp",". And what makes it extraordinary is not that ",[123,26416,26418,26431],{"className":26417},[126],[123,26419,26421],{"className":26420},[130],[132,26422,26423],{"xmlns":134},[136,26424,26425,26429],{},[139,26426,26427],{},[430,26428,11453],{},[166,26430,14155],{"encoding":168},[123,26432,26434],{"className":26433,"ariaHidden":147},[173],[123,26435,26437,26440],{"className":26436},[177],[123,26438],{"className":26439,"style":572},[181],[123,26441,11453],{"className":26442,"style":11545},[186,504]," appears in an infinite series — that is common enough. What is extraordinary is that a series born from a process as elementary as flipping coins turns out to be ",[47,26445,26446],{},"exactly"," the same series that defines a trigonometric function.",[30,26449,26450],{},"It is not an approximation or a numerical coincidence — it is an exact mathematical identity.",[30,26452,26453,26454,26482,26483,2503],{},"Of course, that does not make it a practical way to compute ",[123,26455,26457,26470],{"className":26456},[126],[123,26458,26460],{"className":26459},[130],[132,26461,26462],{"xmlns":134},[136,26463,26464,26468],{},[139,26465,26466],{},[430,26467,11453],{},[166,26469,14155],{"encoding":168},[123,26471,26473],{"className":26472,"ariaHidden":147},[173],[123,26474,26476,26479],{"className":26475},[177],[123,26477],{"className":26478,"style":572},[181],[123,26480,11453],{"className":26481,"style":11545},[186,504],". As Propp himself ",[2779,26484,26486],{"href":26408,"rel":26485},[2783],"told in the Arxiv paper itself",[10806,26488,26489],{},[30,26490,26491],{},"To get pi to the accuracy of 3.14, it might take up to one trillion coin flips. This is partially because sequences of coin flips can get really long before heads overtake tails, so much so that the expected value of a sequence's length is infinity! On top of that, you can't flip all the coins at once the same way you can drop needles — the order of heads and tails matters.",[34,26493,10581],{"id":10580},[30,26495,26496],{},"In the previous article, we discovered that Pascal's triangle connected combinatorics, algebra, and probability. Three languages for the same story.",[30,26498,26499,26500,827],{},"Today we have added a fourth character to the plot: the number ",[123,26501,26503,26516],{"className":26502},[126],[123,26504,26506],{"className":26505},[130],[132,26507,26508],{"xmlns":134},[136,26509,26510,26514],{},[139,26511,26512],{},[430,26513,11453],{},[166,26515,14155],{"encoding":168},[123,26517,26519],{"className":26518,"ariaHidden":147},[173],[123,26520,26522,26525],{"className":26521},[177],[123,26523],{"className":26524,"style":572},[181],[123,26526,11453],{"className":26527,"style":11545},[186,504],[30,26529,26530,26531,26559],{},"Flipping a coin seems like a purely random process. Pascal's triangle seems like pure arithmetic. The Catalan numbers seem like a curiosity of combinatorics. And ",[123,26532,26534,26547],{"className":26533},[126],[123,26535,26537],{"className":26536},[130],[132,26538,26539],{"xmlns":134},[136,26540,26541,26545],{},[139,26542,26543],{},[430,26544,11453],{},[166,26546,14155],{"encoding":168},[123,26548,26550],{"className":26549,"ariaHidden":147},[173],[123,26551,26553,26556],{"className":26552},[177],[123,26554],{"className":26555,"style":572},[181],[123,26557,11453],{"className":26558,"style":11545},[186,504]," seems to belong exclusively to geometry — to circles, to curves, to the continuous world.",[30,26561,26562],{},"And yet, they all end up connected in a single equation.",[30,26564,26565,26566,26594],{},"There is something deeply satisfying about discovering that a new fact about ",[123,26567,26569,26582],{"className":26568},[126],[123,26570,26572],{"className":26571},[130],[132,26573,26574],{"xmlns":134},[136,26575,26576,26580],{},[139,26577,26578],{},[430,26579,11453],{},[166,26581,14155],{"encoding":168},[123,26583,26585],{"className":26584,"ariaHidden":147},[173],[123,26586,26588,26591],{"className":26587},[177],[123,26589],{"className":26590,"style":572},[181],[123,26592,11453],{"className":26593,"style":11545},[186,504]," was hiding, all along, behind something as mundane as flipping a coin. And that nobody noticed until 2025.",[30,26596,26597],{},"Mathematics still has secrets. And sometimes, they are closer than we think.",[30,26599,26600,26601,26604],{},"Put the theory into practice: ",[2779,26602,10625],{"href":10623,"rel":26603},[2783]," and watch the patterns unfold.",{"title":105,"searchDepth":10628,"depth":10628,"links":26606},[26607,26614,26615,26616,26617,26618],{"id":14490,"depth":10628,"text":14491,"children":26608},[26609,26610,26611,26612,26613],{"id":14509,"depth":10634,"text":14510},{"id":14774,"depth":10634,"text":14775},{"id":15175,"depth":10634,"text":15176},{"id":16051,"depth":10634,"text":16052},{"id":16061,"depth":10634,"text":16062},{"id":17955,"depth":10628,"text":17956},{"id":20016,"depth":10628,"text":20017},{"id":20221,"depth":10628,"text":20222},{"id":23009,"depth":10628,"text":23010},{"id":10580,"depth":10628,"text":10581},"2026-05-27T10:00:00Z","Discover how flipping a coin until heads outnumber tails leads, unexpectedly, to the number π — through the Catalan numbers and the arcsine series.",{"type":23,"value":26622},[26623,26625,26730,26762],[26,26624,14015],{"id":14020},[30,26626,14023,26627,14029,26630,14035,26633,827],{},[2779,26628,14028],{"href":14026,"rel":26629},[2783],[2779,26631,14034],{"href":14032,"rel":26632},[2783],[123,26634,26636,26653],{"className":26635},[126],[123,26637,26639],{"className":26638},[130],[132,26640,26641],{"xmlns":134},[136,26642,26643,26651],{},[139,26644,26645],{},[426,26646,26647,26649],{},[430,26648,11453],{},[142,26650,151],{},[166,26652,14056],{"encoding":168},[123,26654,26656],{"className":26655,"ariaHidden":147},[173],[123,26657,26659,26662],{"className":26658},[177],[123,26660],{"className":26661,"style":14066},[181],[123,26663,26665,26668,26727],{"className":26664},[186],[123,26666],{"className":26667},[458,992],[123,26669,26671],{"className":26670},[426],[123,26672,26674,26719],{"className":26673},[472,473],[123,26675,26677,26716],{"className":26676},[477],[123,26678,26680,26694,26702],{"className":26679,"style":14085},[481],[123,26681,26682,26685],{"style":1008},[123,26683],{"className":26684,"style":1012},[489],[123,26686,26688],{"className":26687},[494,495,496,497],[123,26689,26691],{"className":26690},[186,497],[123,26692,151],{"className":26693},[186,497],[123,26695,26696,26699],{"style":1046},[123,26697],{"className":26698,"style":1012},[489],[123,26700],{"className":26701,"style":1054},[1053],[123,26703,26704,26707],{"style":1057},[123,26705],{"className":26706,"style":1012},[489],[123,26708,26710],{"className":26709},[494,495,496,497],[123,26711,26713],{"className":26712},[186,497],[123,26714,11453],{"className":26715,"style":11545},[186,504,497],[123,26717,525],{"className":26718},[524],[123,26720,26722],{"className":26721},[477],[123,26723,26725],{"className":26724,"style":532},[481],[123,26726],{},[123,26728],{"className":26729},[538,992],[30,26731,14138,26732,14168,26760,14172],{},[123,26733,26735,26748],{"className":26734},[126],[123,26736,26738],{"className":26737},[130],[132,26739,26740],{"xmlns":134},[136,26741,26742,26746],{},[139,26743,26744],{},[430,26745,11453],{},[166,26747,14155],{"encoding":168},[123,26749,26751],{"className":26750,"ariaHidden":147},[173],[123,26752,26754,26757],{"className":26753},[177],[123,26755],{"className":26756,"style":572},[181],[123,26758,11453],{"className":26759,"style":11545},[186,504],[47,26761,14171],{},[30,26763,14175,26764,14181],{},[2779,26765,14180],{"href":14178,"rel":26766},[2783],{},"\u002Fblog\u002Fpi-coin-flips-catalan-numbers.en",{"title":14015,"description":26620},"blog\u002Fpi-coin-flips-catalan-numbers.en",[26772,14009,24829,26773,26774],"pi","catalan numbers","pascal triangle","xasmp5RTEg1qRTAclWJpMNmm1vuTX24ZDcmdIxMVo8U",[26777],{"id":26778,"avatar":26779,"extension":11,"meta":26780,"name":26783,"slug":26784,"stem":26786,"url":26785,"__hash__":26787},"authors\u002Fauthors\u002Fpaul-melero.yml","\u002Fimg\u002Fuploads\u002Fauthors\u002Fpaul-melero.webp",{"path":26781,"body":26782,"title":26783},"\u002Fauthors\u002Fpaul-melero",{"name":26783,"slug":26784,"url":26785,"avatar":26779},"Paul Melero","paul-melero","https:\u002F\u002Fgraficos.net","authors\u002Fpaul-melero","LTEA07PCVRlrCg3-G1fVOu2kWXF5EjQ1BxEF3zgZT6k",1781475125136]