[{"data":1,"prerenderedAt":12946},["ShallowReactive",2],{"blog-post-en-flipping-a-coin-until-you-get-pi":3,"locale-switcher-blog-en-flipping-a-coin-until-you-get-pi":12943,"blog-siblings-en-flipping-a-coin-until-you-get-pi":12945},{"id":4,"title":5,"_locale":6,"body":7,"date":12788,"description":12789,"extension":12790,"meta":12791,"navigation":12938,"path":12939,"published":12938,"seo":12940,"slug":13,"stem":12941,"__hash__":12942},"blog\u002Fblog\u002Fpi-coin-flips-catalan-numbers.en.md","Flipping a Coin Until You Get Pi","en",{"type":8,"value":9,"toc":12772},"minimal",[10,14,175,214,223,226,248,251,543,548,564,569,572,613,833,837,844,889,1247,1251,1258,1265,1268,1278,1358,1488,1491,1497,1616,1653,1791,2140,2144,2150,2154,3532,3535,3958,4056,4059,4063,4197,4226,4229,4261,4534,4844,4847,5356,5863,6127,6134,6138,6340,6344,6416,7165,7224,7666,7669,9000,9154,9158,9161,9421,9684,10205,10208,10981,10988,11490,11579,11634,12002,12115,12255,12258,12566,12611,12614,12650,12656,12660,12663,12694,12726,12729,12761,12764],[11,12,5],"h1",{"id":13},"flipping-a-coin-until-you-get-pi",[15,16,17,18,25,26,31,32,174],"p",{},"Matt Parker, from the ",[19,20,24],"a",{"href":21,"rel":22},"https:\u002F\u002Fwww.youtube.com\u002F@standupmaths",[23],"nofollow","Stand-up Maths"," channel, recently published ",[19,27,30],{"href":28,"rel":29},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=kahGSss6SsU",[23],"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 ",[33,34,37,67],"span",{"className":35},[36],"katex",[33,38,41],{"className":39},[40],"katex-mathml",[42,43,45],"math",{"xmlns":44},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[46,47,48,62],"semantics",{},[49,50,51],"mrow",{},[52,53,54,58],"mfrac",{},[55,56,57],"mi",{},"π",[59,60,61],"mn",{},"4",[63,64,66],"annotation",{"encoding":65},"application\u002Fx-tex","\\frac{\\pi}{4}",[33,68,72],{"className":69,"ariaHidden":71},[70],"katex-html","true",[33,73,76,81],{"className":74},[75],"base",[33,77],{"className":78,"style":80},[79],"strut","height:1.0404em;vertical-align:-0.345em;",[33,82,85,90,170],{"className":83},[84],"mord",[33,86],{"className":87},[88,89],"mopen","nulldelimiter",[33,91,93],{"className":92},[52],[33,94,98,161],{"className":95},[96,97],"vlist-t","vlist-t2",[33,99,102,156],{"className":100},[101],"vlist-r",[33,103,107,128,139],{"className":104,"style":106},[105],"vlist","height:0.6954em;",[33,108,110,115],{"style":109},"top:-2.655em;",[33,111],{"className":112,"style":114},[113],"pstrut","height:3em;",[33,116,122],{"className":117},[118,119,120,121],"sizing","reset-size6","size3","mtight",[33,123,125],{"className":124},[84,121],[33,126,61],{"className":127},[84,121],[33,129,131,134],{"style":130},"top:-3.23em;",[33,132],{"className":133,"style":114},[113],[33,135],{"className":136,"style":138},[137],"frac-line","border-bottom-width:0.04em;",[33,140,142,145],{"style":141},"top:-3.394em;",[33,143],{"className":144,"style":114},[113],[33,146,148],{"className":147},[118,119,120,121],[33,149,151],{"className":150},[84,121],[33,152,57],{"className":153,"style":155},[84,154,121],"mathnormal","margin-right:0.0359em;",[33,157,160],{"className":158},[159],"vlist-s","​",[33,162,164],{"className":163},[101],[33,165,168],{"className":166,"style":167},[105],"height:0.345em;",[33,169],{},[33,171],{"className":172},[173,89],"mclose",".",[15,176,177,178,208,209,213],{},"Yes, ",[33,179,181,195],{"className":180},[36],[33,182,184],{"className":183},[40],[42,185,186],{"xmlns":44},[46,187,188,192],{},[49,189,190],{},[55,191,57],{},[63,193,194],{"encoding":65},"\\pi",[33,196,198],{"className":197,"ariaHidden":71},[70],[33,199,201,205],{"className":200},[75],[33,202],{"className":203,"style":204},[79],"height:0.4306em;",[33,206,57],{"className":207,"style":155},[84,154],". ",[210,211,212],"em",{},"The number that belongs to circles",". Hiding inside coin flips.",[15,215,216,217,222],{},"In our ",[19,218,221],{"href":219,"rel":220},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fpascals-triangle-combinatorics-and-coin-flips\u002F",[23],"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.",[15,224,225],{},"The rules of the experiment are straightforward:",[227,228,229,233,241],"ol",{},[230,231,232],"li",{},"Flip a fair coin repeatedly.",[230,234,235,236,240],{},"Stop ",[237,238,239],"strong",{},"the moment"," the number of heads exceeds the number of tails.",[230,242,243,244,247],{},"Record the ",[237,245,246],{},"ratio of heads"," to total flips for that sequence.",[15,249,250],{},"The question is: if we repeat this process infinitely many times and compute the average of all those ratios... what do we get?",[15,252,253,254,383,384,483,484,513,514,542],{},"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 ",[33,255,257,282],{"className":256},[36],[33,258,260],{"className":259},[40],[42,261,262],{"xmlns":44},[46,263,264,279],{},[49,265,266,273,277],{},[52,267,268,271],{},[59,269,270],{},"1",[59,272,270],{},[274,275,276],"mo",{},"=",[59,278,270],{},[63,280,281],{"encoding":65},"\\frac{1}{1} = 1",[33,283,285,373],{"className":284,"ariaHidden":71},[70],[33,286,288,292,361,366,370],{"className":287},[75],[33,289],{"className":290,"style":291},[79],"height:1.1901em;vertical-align:-0.345em;",[33,293,295,298,358],{"className":294},[84],[33,296],{"className":297},[88,89],[33,299,301],{"className":300},[52],[33,302,304,350],{"className":303},[96,97],[33,305,307,347],{"className":306},[101],[33,308,311,325,333],{"className":309,"style":310},[105],"height:0.8451em;",[33,312,313,316],{"style":109},[33,314],{"className":315,"style":114},[113],[33,317,319],{"className":318},[118,119,120,121],[33,320,322],{"className":321},[84,121],[33,323,270],{"className":324},[84,121],[33,326,327,330],{"style":130},[33,328],{"className":329,"style":114},[113],[33,331],{"className":332,"style":138},[137],[33,334,335,338],{"style":141},[33,336],{"className":337,"style":114},[113],[33,339,341],{"className":340},[118,119,120,121],[33,342,344],{"className":343},[84,121],[33,345,270],{"className":346},[84,121],[33,348,160],{"className":349},[159],[33,351,353],{"className":352},[101],[33,354,356],{"className":355,"style":167},[105],[33,357],{},[33,359],{"className":360},[173,89],[33,362],{"className":363,"style":365},[364],"mspace","margin-right:0.2778em;",[33,367,276],{"className":368},[369],"mrel",[33,371],{"className":372,"style":365},[364],[33,374,376,380],{"className":375},[75],[33,377],{"className":378,"style":379},[79],"height:0.6444em;",[33,381,270],{"className":382},[84]," (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 ",[33,385,387,406],{"className":386},[36],[33,388,390],{"className":389},[40],[42,391,392],{"xmlns":44},[46,393,394,403],{},[49,395,396],{},[52,397,398,400],{},[59,399,270],{},[59,401,402],{},"2",[63,404,405],{"encoding":65},"\\frac{1}{2}",[33,407,409],{"className":408,"ariaHidden":71},[70],[33,410,412,415],{"className":411},[75],[33,413],{"className":414,"style":291},[79],[33,416,418,421,480],{"className":417},[84],[33,419],{"className":420},[88,89],[33,422,424],{"className":423},[52],[33,425,427,472],{"className":426},[96,97],[33,428,430,469],{"className":429},[101],[33,431,433,447,455],{"className":432,"style":310},[105],[33,434,435,438],{"style":109},[33,436],{"className":437,"style":114},[113],[33,439,441],{"className":440},[118,119,120,121],[33,442,444],{"className":443},[84,121],[33,445,402],{"className":446},[84,121],[33,448,449,452],{"style":130},[33,450],{"className":451,"style":114},[113],[33,453],{"className":454,"style":138},[137],[33,456,457,460],{"style":141},[33,458],{"className":459,"style":114},[113],[33,461,463],{"className":462},[118,119,120,121],[33,464,466],{"className":465},[84,121],[33,467,270],{"className":468},[84,121],[33,470,160],{"className":471},[159],[33,473,475],{"className":474},[101],[33,476,478],{"className":477,"style":167},[105],[33,479],{},[33,481],{"className":482},[173,89],". So the average must land somewhere between ",[33,485,487,501],{"className":486},[36],[33,488,490],{"className":489},[40],[42,491,492],{"xmlns":44},[46,493,494,499],{},[49,495,496],{},[59,497,498],{},"0.5",[63,500,498],{"encoding":65},[33,502,504],{"className":503,"ariaHidden":71},[70],[33,505,507,510],{"className":506},[75],[33,508],{"className":509,"style":379},[79],[33,511,498],{"className":512},[84]," and ",[33,515,517,530],{"className":516},[36],[33,518,520],{"className":519},[40],[42,521,522],{"xmlns":44},[46,523,524,528],{},[49,525,526],{},[59,527,270],{},[63,529,270],{"encoding":65},[33,531,533],{"className":532,"ariaHidden":71},[70],[33,534,536,539],{"className":535},[75],[33,537],{"className":538,"style":379},[79],[33,540,270],{"className":541},[84],". Let's find out exactly where.",[544,545,547],"h2",{"id":546},"building-the-sequences-step-by-step","Building the sequences, step by step",[15,549,550,551,554,555,558,559,174],{},"Let's construct the possible sequences starting from the shortest ones. We will use ",[237,552,553],{},"O"," for heads and ",[237,556,557],{},"X"," for tails — the same notation we use in ",[19,560,563],{"href":561,"rel":562},"https:\u002F\u002Fflipthecoin.app\u002Fplay\u002F",[23],"our coin flip tool",[565,566,568],"h3",{"id":567},"length-1","Length 1",[15,570,571],{},"The shortest possible sequence: flip once, get heads.",[15,573,576],{"className":574},[575],"math-block",[33,577,579,595],{"className":578},[36],[33,580,582],{"className":581},[40],[42,583,584],{"xmlns":44},[46,585,586,592],{},[49,587,588],{},[589,590,553],"mtext",{"mathvariant":591},"bold",[63,593,594],{"encoding":65},"\\textbf{O}",[33,596,598],{"className":597,"ariaHidden":71},[70],[33,599,601,605],{"className":600},[75],[33,602],{"className":603,"style":604},[79],"height:0.6861em;",[33,606,609],{"className":607},[84,608],"text",[33,610,553],{"className":611},[84,612],"textbf",[15,614,615,616,735,736,174],{},"We stop. We have 1 heads and 0 tails. The ratio is ",[33,617,619,640],{"className":618},[36],[33,620,622],{"className":621},[40],[42,623,624],{"xmlns":44},[46,625,626,638],{},[49,627,628,634,636],{},[52,629,630,632],{},[59,631,270],{},[59,633,270],{},[274,635,276],{},[59,637,270],{},[63,639,281],{"encoding":65},[33,641,643,726],{"className":642,"ariaHidden":71},[70],[33,644,646,649,717,720,723],{"className":645},[75],[33,647],{"className":648,"style":291},[79],[33,650,652,655,714],{"className":651},[84],[33,653],{"className":654},[88,89],[33,656,658],{"className":657},[52],[33,659,661,706],{"className":660},[96,97],[33,662,664,703],{"className":663},[101],[33,665,667,681,689],{"className":666,"style":310},[105],[33,668,669,672],{"style":109},[33,670],{"className":671,"style":114},[113],[33,673,675],{"className":674},[118,119,120,121],[33,676,678],{"className":677},[84,121],[33,679,270],{"className":680},[84,121],[33,682,683,686],{"style":130},[33,684],{"className":685,"style":114},[113],[33,687],{"className":688,"style":138},[137],[33,690,691,694],{"style":141},[33,692],{"className":693,"style":114},[113],[33,695,697],{"className":696},[118,119,120,121],[33,698,700],{"className":699},[84,121],[33,701,270],{"className":702},[84,121],[33,704,160],{"className":705},[159],[33,707,709],{"className":708},[101],[33,710,712],{"className":711,"style":167},[105],[33,713],{},[33,715],{"className":716},[173,89],[33,718],{"className":719,"style":365},[364],[33,721,276],{"className":722},[369],[33,724],{"className":725,"style":365},[364],[33,727,729,732],{"className":728},[75],[33,730],{"className":731,"style":379},[79],[33,733,270],{"className":734},[84]," and the probability of this happening is ",[33,737,739,756],{"className":738},[36],[33,740,742],{"className":741},[40],[42,743,744],{"xmlns":44},[46,745,746,754],{},[49,747,748],{},[52,749,750,752],{},[59,751,270],{},[59,753,402],{},[63,755,405],{"encoding":65},[33,757,759],{"className":758,"ariaHidden":71},[70],[33,760,762,765],{"className":761},[75],[33,763],{"className":764,"style":291},[79],[33,766,768,771,830],{"className":767},[84],[33,769],{"className":770},[88,89],[33,772,774],{"className":773},[52],[33,775,777,822],{"className":776},[96,97],[33,778,780,819],{"className":779},[101],[33,781,783,797,805],{"className":782,"style":310},[105],[33,784,785,788],{"style":109},[33,786],{"className":787,"style":114},[113],[33,789,791],{"className":790},[118,119,120,121],[33,792,794],{"className":793},[84,121],[33,795,402],{"className":796},[84,121],[33,798,799,802],{"style":130},[33,800],{"className":801,"style":114},[113],[33,803],{"className":804,"style":138},[137],[33,806,807,810],{"style":141},[33,808],{"className":809,"style":114},[113],[33,811,813],{"className":812},[118,119,120,121],[33,814,816],{"className":815},[84,121],[33,817,270],{"className":818},[84,121],[33,820,160],{"className":821},[159],[33,823,825],{"className":824},[101],[33,826,828],{"className":827,"style":167},[105],[33,829],{},[33,831],{"className":832},[173,89],[565,834,836],{"id":835},"length-3","Length 3",[15,838,839,840,843],{},"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 ",[210,841,842],{},"exceed"," tails. So the next possible sequence has length 3:",[15,845,847],{"className":846},[575],[33,848,850,873],{"className":849},[36],[33,851,853],{"className":852},[40],[42,854,855],{"xmlns":44},[46,856,857,870],{},[49,858,859,861,864,866,868],{},[589,860,557],{"mathvariant":591},[589,862,863],{}," ",[589,865,553],{"mathvariant":591},[589,867,863],{},[589,869,553],{"mathvariant":591},[63,871,872],{"encoding":65},"\\textbf{X O O}",[33,874,876],{"className":875,"ariaHidden":71},[70],[33,877,879,882],{"className":878},[75],[33,880],{"className":881,"style":604},[79],[33,883,885],{"className":884},[84,608],[33,886,888],{"className":887},[84,612],"X O O",[15,890,891,892,991,992,1246],{},"Tails, heads, heads. The third flip gives us 2 heads versus 1 tails. Ratio: ",[33,893,895,914],{"className":894},[36],[33,896,898],{"className":897},[40],[42,899,900],{"xmlns":44},[46,901,902,911],{},[49,903,904],{},[52,905,906,908],{},[59,907,402],{},[59,909,910],{},"3",[63,912,913],{"encoding":65},"\\frac{2}{3}",[33,915,917],{"className":916,"ariaHidden":71},[70],[33,918,920,923],{"className":919},[75],[33,921],{"className":922,"style":291},[79],[33,924,926,929,988],{"className":925},[84],[33,927],{"className":928},[88,89],[33,930,932],{"className":931},[52],[33,933,935,980],{"className":934},[96,97],[33,936,938,977],{"className":937},[101],[33,939,941,955,963],{"className":940,"style":310},[105],[33,942,943,946],{"style":109},[33,944],{"className":945,"style":114},[113],[33,947,949],{"className":948},[118,119,120,121],[33,950,952],{"className":951},[84,121],[33,953,910],{"className":954},[84,121],[33,956,957,960],{"style":130},[33,958],{"className":959,"style":114},[113],[33,961],{"className":962,"style":138},[137],[33,964,965,968],{"style":141},[33,966],{"className":967,"style":114},[113],[33,969,971],{"className":970},[118,119,120,121],[33,972,974],{"className":973},[84,121],[33,975,402],{"className":976},[84,121],[33,978,160],{"className":979},[159],[33,981,983],{"className":982},[101],[33,984,986],{"className":985,"style":167},[105],[33,987],{},[33,989],{"className":990},[173,89]," (2 heads out of 3 flips). Probability: ",[33,993,995,1035],{"className":994},[36],[33,996,998],{"className":997},[40],[42,999,1000],{"xmlns":44},[46,1001,1002,1032],{},[49,1003,1004,1023,1025],{},[1005,1006,1007,1021],"msup",{},[49,1008,1009,1012,1018],{},[274,1010,1011],{"fence":71},"(",[52,1013,1014,1016],{},[59,1015,270],{},[59,1017,402],{},[274,1019,1020],{"fence":71},")",[59,1022,910],{},[274,1024,276],{},[52,1026,1027,1029],{},[59,1028,270],{},[59,1030,1031],{},"8",[63,1033,1034],{"encoding":65},"\\left(\\frac{1}{2}\\right)^3 = \\frac{1}{8}",[33,1036,1038,1172],{"className":1037,"ariaHidden":71},[70],[33,1039,1041,1045,1163,1166,1169],{"className":1040},[75],[33,1042],{"className":1043,"style":1044},[79],"height:1.404em;vertical-align:-0.35em;",[33,1046,1049,1136],{"className":1047},[1048],"minner",[33,1050,1052,1062,1130],{"className":1051},[1048],[33,1053,1057],{"className":1054,"style":1056},[88,1055],"delimcenter","top:0em;",[33,1058,1011],{"className":1059},[1060,1061],"delimsizing","size1",[33,1063,1065,1068,1127],{"className":1064},[84],[33,1066],{"className":1067},[88,89],[33,1069,1071],{"className":1070},[52],[33,1072,1074,1119],{"className":1073},[96,97],[33,1075,1077,1116],{"className":1076},[101],[33,1078,1080,1094,1102],{"className":1079,"style":310},[105],[33,1081,1082,1085],{"style":109},[33,1083],{"className":1084,"style":114},[113],[33,1086,1088],{"className":1087},[118,119,120,121],[33,1089,1091],{"className":1090},[84,121],[33,1092,402],{"className":1093},[84,121],[33,1095,1096,1099],{"style":130},[33,1097],{"className":1098,"style":114},[113],[33,1100],{"className":1101,"style":138},[137],[33,1103,1104,1107],{"style":141},[33,1105],{"className":1106,"style":114},[113],[33,1108,1110],{"className":1109},[118,119,120,121],[33,1111,1113],{"className":1112},[84,121],[33,1114,270],{"className":1115},[84,121],[33,1117,160],{"className":1118},[159],[33,1120,1122],{"className":1121},[101],[33,1123,1125],{"className":1124,"style":167},[105],[33,1126],{},[33,1128],{"className":1129},[173,89],[33,1131,1133],{"className":1132,"style":1056},[173,1055],[33,1134,1020],{"className":1135},[1060,1061],[33,1137,1140],{"className":1138},[1139],"msupsub",[33,1141,1143],{"className":1142},[96],[33,1144,1146],{"className":1145},[101],[33,1147,1150],{"className":1148,"style":1149},[105],"height:1.054em;",[33,1151,1153,1157],{"style":1152},"top:-3.3029em;margin-right:0.05em;",[33,1154],{"className":1155,"style":1156},[113],"height:2.7em;",[33,1158,1160],{"className":1159},[118,119,120,121],[33,1161,910],{"className":1162},[84,121],[33,1164],{"className":1165,"style":365},[364],[33,1167,276],{"className":1168},[369],[33,1170],{"className":1171,"style":365},[364],[33,1173,1175,1178],{"className":1174},[75],[33,1176],{"className":1177,"style":291},[79],[33,1179,1181,1184,1243],{"className":1180},[84],[33,1182],{"className":1183},[88,89],[33,1185,1187],{"className":1186},[52],[33,1188,1190,1235],{"className":1189},[96,97],[33,1191,1193,1232],{"className":1192},[101],[33,1194,1196,1210,1218],{"className":1195,"style":310},[105],[33,1197,1198,1201],{"style":109},[33,1199],{"className":1200,"style":114},[113],[33,1202,1204],{"className":1203},[118,119,120,121],[33,1205,1207],{"className":1206},[84,121],[33,1208,1031],{"className":1209},[84,121],[33,1211,1212,1215],{"style":130},[33,1213],{"className":1214,"style":114},[113],[33,1216],{"className":1217,"style":138},[137],[33,1219,1220,1223],{"style":141},[33,1221],{"className":1222,"style":114},[113],[33,1224,1226],{"className":1225},[118,119,120,121],[33,1227,1229],{"className":1228},[84,121],[33,1230,270],{"className":1231},[84,121],[33,1233,160],{"className":1234},[159],[33,1236,1238],{"className":1237},[101],[33,1239,1241],{"className":1240,"style":167},[105],[33,1242],{},[33,1244],{"className":1245},[173,89],". And it is the only possible sequence of this length.",[565,1248,1250],{"id":1249},"length-5","Length 5",[15,1252,1253,1254,1257],{},"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 — ",[237,1255,1256],{},"at no point before the end can we have had more heads than tails",", because if that had happened, we would have stopped earlier.",[15,1259,1260,1261,1264],{},"It is not enough for the sequence to end with more heads than tails; it must be ",[237,1262,1263],{},"the first time"," it happens.",[15,1266,1267],{},"How many sequences satisfy these conditions? Only two:",[1269,1270,1275],"pre",{"className":1271,"code":1273,"language":608,"meta":1274},[1272],"language-text","X X O O O\nX O X O O\n","",[1276,1277,1273],"code",{"__ignoreMap":1274},[15,1279,1280,1281,1284,1285,1321,1322,1357],{},"Let's check the first: ",[237,1282,1283],{},"XXOOO",". Counting O as ",[33,1286,1288,1305],{"className":1287},[36],[33,1289,1291],{"className":1290},[40],[42,1292,1293],{"xmlns":44},[46,1294,1295,1302],{},[49,1296,1297,1300],{},[274,1298,1299],{},"+",[59,1301,270],{},[63,1303,1304],{"encoding":65},"+1",[33,1306,1308],{"className":1307,"ariaHidden":71},[70],[33,1309,1311,1315,1318],{"className":1310},[75],[33,1312],{"className":1313,"style":1314},[79],"height:0.7278em;vertical-align:-0.0833em;",[33,1316,1299],{"className":1317},[84],[33,1319,270],{"className":1320},[84]," and X as ",[33,1323,1325,1342],{"className":1324},[36],[33,1326,1328],{"className":1327},[40],[42,1329,1330],{"xmlns":44},[46,1331,1332,1339],{},[49,1333,1334,1337],{},[274,1335,1336],{},"−",[59,1338,270],{},[63,1340,1341],{"encoding":65},"-1",[33,1343,1345],{"className":1344,"ariaHidden":71},[70],[33,1346,1348,1351,1354],{"className":1347},[75],[33,1349],{"className":1350,"style":1314},[79],[33,1352,1336],{"className":1353},[84],[33,1355,270],{"className":1356},[84],", the running totals are:",[15,1359,1361],{"className":1360},[575],[33,1362,1364,1413],{"className":1363},[36],[33,1365,1367],{"className":1366},[40],[42,1368,1369],{"xmlns":44},[46,1370,1371,1410],{},[49,1372,1373,1375,1377,1380,1383,1385,1387,1389,1391,1393,1395,1397,1399,1402,1404,1406,1408],{},[274,1374,1336],{},[59,1376,270],{},[274,1378,1379],{"separator":71},",",[589,1381,1382],{},"  ",[274,1384,1336],{},[59,1386,402],{},[274,1388,1379],{"separator":71},[589,1390,1382],{},[274,1392,1336],{},[59,1394,270],{},[274,1396,1379],{"separator":71},[589,1398,1382],{},[59,1400,1401],{},"0",[274,1403,1379],{"separator":71},[589,1405,1382],{},[274,1407,1299],{},[59,1409,270],{},[63,1411,1412],{"encoding":65},"-1,\\; -2,\\; -1,\\; 0,\\; +1",[33,1414,1416],{"className":1415,"ariaHidden":71},[70],[33,1417,1419,1423,1426,1429,1433,1436,1440,1443,1446,1449,1452,1455,1458,1461,1464,1467,1470,1473,1476,1479,1482,1485],{"className":1418},[75],[33,1420],{"className":1421,"style":1422},[79],"height:0.8389em;vertical-align:-0.1944em;",[33,1424,1336],{"className":1425},[84],[33,1427,270],{"className":1428},[84],[33,1430,1379],{"className":1431},[1432],"mpunct",[33,1434],{"className":1435,"style":365},[364],[33,1437],{"className":1438,"style":1439},[364],"margin-right:0.1667em;",[33,1441,1336],{"className":1442},[84],[33,1444,402],{"className":1445},[84],[33,1447,1379],{"className":1448},[1432],[33,1450],{"className":1451,"style":365},[364],[33,1453],{"className":1454,"style":1439},[364],[33,1456,1336],{"className":1457},[84],[33,1459,270],{"className":1460},[84],[33,1462,1379],{"className":1463},[1432],[33,1465],{"className":1466,"style":365},[364],[33,1468],{"className":1469,"style":1439},[364],[33,1471,1401],{"className":1472},[84],[33,1474,1379],{"className":1475},[1432],[33,1477],{"className":1478,"style":365},[364],[33,1480],{"className":1481,"style":1439},[364],[33,1483,1299],{"className":1484},[84],[33,1486,270],{"className":1487},[84],[15,1489,1490],{},"Never exceeds zero until the fifth flip. ✅",[15,1492,1493,1494,174],{},"Now the second: ",[237,1495,1496],{},"XOXOO",[15,1498,1500],{"className":1499},[575],[33,1501,1503,1547],{"className":1502},[36],[33,1504,1506],{"className":1505},[40],[42,1507,1508],{"xmlns":44},[46,1509,1510,1544],{},[49,1511,1512,1514,1516,1518,1520,1522,1524,1526,1528,1530,1532,1534,1536,1538,1540,1542],{},[274,1513,1336],{},[59,1515,270],{},[274,1517,1379],{"separator":71},[589,1519,1382],{},[59,1521,1401],{},[274,1523,1379],{"separator":71},[589,1525,1382],{},[274,1527,1336],{},[59,1529,270],{},[274,1531,1379],{"separator":71},[589,1533,1382],{},[59,1535,1401],{},[274,1537,1379],{"separator":71},[589,1539,1382],{},[274,1541,1299],{},[59,1543,270],{},[63,1545,1546],{"encoding":65},"-1,\\; 0,\\; -1,\\; 0,\\; +1",[33,1548,1550],{"className":1549,"ariaHidden":71},[70],[33,1551,1553,1556,1559,1562,1565,1568,1571,1574,1577,1580,1583,1586,1589,1592,1595,1598,1601,1604,1607,1610,1613],{"className":1552},[75],[33,1554],{"className":1555,"style":1422},[79],[33,1557,1336],{"className":1558},[84],[33,1560,270],{"className":1561},[84],[33,1563,1379],{"className":1564},[1432],[33,1566],{"className":1567,"style":365},[364],[33,1569],{"className":1570,"style":1439},[364],[33,1572,1401],{"className":1573},[84],[33,1575,1379],{"className":1576},[1432],[33,1578],{"className":1579,"style":365},[364],[33,1581],{"className":1582,"style":1439},[364],[33,1584,1336],{"className":1585},[84],[33,1587,270],{"className":1588},[84],[33,1590,1379],{"className":1591},[1432],[33,1593],{"className":1594,"style":365},[364],[33,1596],{"className":1597,"style":1439},[364],[33,1599,1401],{"className":1600},[84],[33,1602,1379],{"className":1603},[1432],[33,1605],{"className":1606,"style":365},[364],[33,1608],{"className":1609,"style":1439},[364],[33,1611,1299],{"className":1612},[84],[33,1614,270],{"className":1615},[84],[15,1617,1618,1619,1652],{},"It returns to zero at the second and fourth flip, but never exceeds it. This is the first time it reaches ",[33,1620,1622,1637],{"className":1621},[36],[33,1623,1625],{"className":1624},[40],[42,1626,1627],{"xmlns":44},[46,1628,1629,1635],{},[49,1630,1631,1633],{},[274,1632,1299],{},[59,1634,270],{},[63,1636,1304],{"encoding":65},[33,1638,1640],{"className":1639,"ariaHidden":71},[70],[33,1641,1643,1646,1649],{"className":1642},[75],[33,1644],{"className":1645,"style":1314},[79],[33,1647,1299],{"className":1648},[84],[33,1650,270],{"className":1651},[84],". ✅",[15,1654,1655,1656,1659,1660,1756,1757,1790],{},"What about ",[237,1657,1658],{},"OXXOO","? Let's see: ",[33,1661,1663,1699],{"className":1662},[36],[33,1664,1666],{"className":1665},[40],[42,1667,1668],{"xmlns":44},[46,1669,1670,1696],{},[49,1671,1672,1674,1676,1678,1680,1682,1684,1686,1688,1690,1692,1694],{},[274,1673,1299],{},[59,1675,270],{},[274,1677,1379],{"separator":71},[59,1679,1401],{},[274,1681,1379],{"separator":71},[274,1683,1336],{},[59,1685,270],{},[274,1687,1379],{"separator":71},[59,1689,1401],{},[274,1691,1379],{"separator":71},[274,1693,1299],{},[59,1695,270],{},[63,1697,1698],{"encoding":65},"+1, 0, -1, 0, +1",[33,1700,1702],{"className":1701,"ariaHidden":71},[70],[33,1703,1705,1708,1711,1714,1717,1720,1723,1726,1729,1732,1735,1738,1741,1744,1747,1750,1753],{"className":1704},[75],[33,1706],{"className":1707,"style":1422},[79],[33,1709,1299],{"className":1710},[84],[33,1712,270],{"className":1713},[84],[33,1715,1379],{"className":1716},[1432],[33,1718],{"className":1719,"style":1439},[364],[33,1721,1401],{"className":1722},[84],[33,1724,1379],{"className":1725},[1432],[33,1727],{"className":1728,"style":1439},[364],[33,1730,1336],{"className":1731},[84],[33,1733,270],{"className":1734},[84],[33,1736,1379],{"className":1737},[1432],[33,1739],{"className":1740,"style":1439},[364],[33,1742,1401],{"className":1743},[84],[33,1745,1379],{"className":1746},[1432],[33,1748],{"className":1749,"style":1439},[364],[33,1751,1299],{"className":1752},[84],[33,1754,270],{"className":1755},[84],". The first flip already puts us at ",[33,1758,1760,1775],{"className":1759},[36],[33,1761,1763],{"className":1762},[40],[42,1764,1765],{"xmlns":44},[46,1766,1767,1773],{},[49,1768,1769,1771],{},[274,1770,1299],{},[59,1772,270],{},[63,1774,1304],{"encoding":65},[33,1776,1778],{"className":1777,"ariaHidden":71},[70],[33,1779,1781,1784,1787],{"className":1780},[75],[33,1782],{"className":1783,"style":1314},[79],[33,1785,1299],{"className":1786},[84],[33,1788,270],{"className":1789},[84],": we would have stopped right there. Not valid. ❌",[15,1792,1793,1794,1797,1798,2040,2041,2139],{},"So there are exactly ",[237,1795,1796],{},"2 sequences"," of length 5, each with probability ",[33,1799,1801,1839],{"className":1800},[36],[33,1802,1804],{"className":1803},[40],[42,1805,1806],{"xmlns":44},[46,1807,1808,1836],{},[49,1809,1810,1827,1829],{},[1005,1811,1812,1824],{},[49,1813,1814,1816,1822],{},[274,1815,1011],{"fence":71},[52,1817,1818,1820],{},[59,1819,270],{},[59,1821,402],{},[274,1823,1020],{"fence":71},[59,1825,1826],{},"5",[274,1828,276],{},[52,1830,1831,1833],{},[59,1832,270],{},[59,1834,1835],{},"32",[63,1837,1838],{"encoding":65},"\\left(\\frac{1}{2}\\right)^5 = \\frac{1}{32}",[33,1840,1842,1966],{"className":1841,"ariaHidden":71},[70],[33,1843,1845,1848,1957,1960,1963],{"className":1844},[75],[33,1846],{"className":1847,"style":1044},[79],[33,1849,1851,1934],{"className":1850},[1048],[33,1852,1854,1860,1928],{"className":1853},[1048],[33,1855,1857],{"className":1856,"style":1056},[88,1055],[33,1858,1011],{"className":1859},[1060,1061],[33,1861,1863,1866,1925],{"className":1862},[84],[33,1864],{"className":1865},[88,89],[33,1867,1869],{"className":1868},[52],[33,1870,1872,1917],{"className":1871},[96,97],[33,1873,1875,1914],{"className":1874},[101],[33,1876,1878,1892,1900],{"className":1877,"style":310},[105],[33,1879,1880,1883],{"style":109},[33,1881],{"className":1882,"style":114},[113],[33,1884,1886],{"className":1885},[118,119,120,121],[33,1887,1889],{"className":1888},[84,121],[33,1890,402],{"className":1891},[84,121],[33,1893,1894,1897],{"style":130},[33,1895],{"className":1896,"style":114},[113],[33,1898],{"className":1899,"style":138},[137],[33,1901,1902,1905],{"style":141},[33,1903],{"className":1904,"style":114},[113],[33,1906,1908],{"className":1907},[118,119,120,121],[33,1909,1911],{"className":1910},[84,121],[33,1912,270],{"className":1913},[84,121],[33,1915,160],{"className":1916},[159],[33,1918,1920],{"className":1919},[101],[33,1921,1923],{"className":1922,"style":167},[105],[33,1924],{},[33,1926],{"className":1927},[173,89],[33,1929,1931],{"className":1930,"style":1056},[173,1055],[33,1932,1020],{"className":1933},[1060,1061],[33,1935,1937],{"className":1936},[1139],[33,1938,1940],{"className":1939},[96],[33,1941,1943],{"className":1942},[101],[33,1944,1946],{"className":1945,"style":1149},[105],[33,1947,1948,1951],{"style":1152},[33,1949],{"className":1950,"style":1156},[113],[33,1952,1954],{"className":1953},[118,119,120,121],[33,1955,1826],{"className":1956},[84,121],[33,1958],{"className":1959,"style":365},[364],[33,1961,276],{"className":1962},[369],[33,1964],{"className":1965,"style":365},[364],[33,1967,1969,1972],{"className":1968},[75],[33,1970],{"className":1971,"style":291},[79],[33,1973,1975,1978,2037],{"className":1974},[84],[33,1976],{"className":1977},[88,89],[33,1979,1981],{"className":1980},[52],[33,1982,1984,2029],{"className":1983},[96,97],[33,1985,1987,2026],{"className":1986},[101],[33,1988,1990,2004,2012],{"className":1989,"style":310},[105],[33,1991,1992,1995],{"style":109},[33,1993],{"className":1994,"style":114},[113],[33,1996,1998],{"className":1997},[118,119,120,121],[33,1999,2001],{"className":2000},[84,121],[33,2002,1835],{"className":2003},[84,121],[33,2005,2006,2009],{"style":130},[33,2007],{"className":2008,"style":114},[113],[33,2010],{"className":2011,"style":138},[137],[33,2013,2014,2017],{"style":141},[33,2015],{"className":2016,"style":114},[113],[33,2018,2020],{"className":2019},[118,119,120,121],[33,2021,2023],{"className":2022},[84,121],[33,2024,270],{"className":2025},[84,121],[33,2027,160],{"className":2028},[159],[33,2030,2032],{"className":2031},[101],[33,2033,2035],{"className":2034,"style":167},[105],[33,2036],{},[33,2038],{"className":2039},[173,89]," and ratio ",[33,2042,2044,2062],{"className":2043},[36],[33,2045,2047],{"className":2046},[40],[42,2048,2049],{"xmlns":44},[46,2050,2051,2059],{},[49,2052,2053],{},[52,2054,2055,2057],{},[59,2056,910],{},[59,2058,1826],{},[63,2060,2061],{"encoding":65},"\\frac{3}{5}",[33,2063,2065],{"className":2064,"ariaHidden":71},[70],[33,2066,2068,2071],{"className":2067},[75],[33,2069],{"className":2070,"style":291},[79],[33,2072,2074,2077,2136],{"className":2073},[84],[33,2075],{"className":2076},[88,89],[33,2078,2080],{"className":2079},[52],[33,2081,2083,2128],{"className":2082},[96,97],[33,2084,2086,2125],{"className":2085},[101],[33,2087,2089,2103,2111],{"className":2088,"style":310},[105],[33,2090,2091,2094],{"style":109},[33,2092],{"className":2093,"style":114},[113],[33,2095,2097],{"className":2096},[118,119,120,121],[33,2098,2100],{"className":2099},[84,121],[33,2101,1826],{"className":2102},[84,121],[33,2104,2105,2108],{"style":130},[33,2106],{"className":2107,"style":114},[113],[33,2109],{"className":2110,"style":138},[137],[33,2112,2113,2116],{"style":141},[33,2114],{"className":2115,"style":114},[113],[33,2117,2119],{"className":2118},[118,119,120,121],[33,2120,2122],{"className":2121},[84,121],[33,2123,910],{"className":2124},[84,121],[33,2126,160],{"className":2127},[159],[33,2129,2131],{"className":2130},[101],[33,2132,2134],{"className":2133,"style":167},[105],[33,2135],{},[33,2137],{"className":2138},[173,89]," (3 heads out of 5 flips).",[565,2141,2143],{"id":2142},"length-7","Length 7",[15,2145,2146,2147,174],{},"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 ",[237,2148,2149],{},"5 valid sequences",[565,2151,2153],{"id":2152},"the-summary-table","The summary table",[2155,2156,2157,2236],"table",{},[2158,2159,2160],"thead",{},[2161,2162,2163,2227,2230,2233],"tr",{},[2164,2165,2167,2168,1020],"th",{"align":2166},"center","Length (",[33,2169,2171,2192],{"className":2170},[36],[33,2172,2174],{"className":2173},[40],[42,2175,2176],{"xmlns":44},[46,2177,2178,2189],{},[49,2179,2180,2182,2185,2187],{},[59,2181,402],{},[55,2183,2184],{},"n",[274,2186,1299],{},[59,2188,270],{},[63,2190,2191],{"encoding":65},"2n+1",[33,2193,2195,2218],{"className":2194,"ariaHidden":71},[70],[33,2196,2198,2201,2204,2207,2211,2215],{"className":2197},[75],[33,2199],{"className":2200,"style":1314},[79],[33,2202,402],{"className":2203},[84],[33,2205,2184],{"className":2206},[84,154],[33,2208],{"className":2209,"style":2210},[364],"margin-right:0.2222em;",[33,2212,1299],{"className":2213},[2214],"mbin",[33,2216],{"className":2217,"style":2210},[364],[33,2219,2221,2224],{"className":2220},[75],[33,2222],{"className":2223,"style":379},[79],[33,2225,270],{"className":2226},[84],[2164,2228,2229],{"align":2166},"Heads ratio",[2164,2231,2232],{"align":2166},"Valid sequences",[2164,2234,2235],{"align":2166},"Probability of each",[2237,2238,2239,2497,2754,3011,3271],"tbody",{},[2161,2240,2241,2296,2396,2398],{},[2242,2243,2244,2245,1020],"td",{"align":2166},"1 (",[33,2246,2248,2266],{"className":2247},[36],[33,2249,2251],{"className":2250},[40],[42,2252,2253],{"xmlns":44},[46,2254,2255,2263],{},[49,2256,2257,2259,2261],{},[55,2258,2184],{},[274,2260,276],{},[59,2262,1401],{},[63,2264,2265],{"encoding":65},"n=0",[33,2267,2269,2287],{"className":2268,"ariaHidden":71},[70],[33,2270,2272,2275,2278,2281,2284],{"className":2271},[75],[33,2273],{"className":2274,"style":204},[79],[33,2276,2184],{"className":2277},[84,154],[33,2279],{"className":2280,"style":365},[364],[33,2282,276],{"className":2283},[369],[33,2285],{"className":2286,"style":365},[364],[33,2288,2290,2293],{"className":2289},[75],[33,2291],{"className":2292,"style":379},[79],[33,2294,1401],{"className":2295},[84],[2242,2297,2298],{"align":2166},[33,2299,2301,2319],{"className":2300},[36],[33,2302,2304],{"className":2303},[40],[42,2305,2306],{"xmlns":44},[46,2307,2308,2316],{},[49,2309,2310],{},[52,2311,2312,2314],{},[59,2313,270],{},[59,2315,270],{},[63,2317,2318],{"encoding":65},"\\frac{1}{1}",[33,2320,2322],{"className":2321,"ariaHidden":71},[70],[33,2323,2325,2328],{"className":2324},[75],[33,2326],{"className":2327,"style":291},[79],[33,2329,2331,2334,2393],{"className":2330},[84],[33,2332],{"className":2333},[88,89],[33,2335,2337],{"className":2336},[52],[33,2338,2340,2385],{"className":2339},[96,97],[33,2341,2343,2382],{"className":2342},[101],[33,2344,2346,2360,2368],{"className":2345,"style":310},[105],[33,2347,2348,2351],{"style":109},[33,2349],{"className":2350,"style":114},[113],[33,2352,2354],{"className":2353},[118,119,120,121],[33,2355,2357],{"className":2356},[84,121],[33,2358,270],{"className":2359},[84,121],[33,2361,2362,2365],{"style":130},[33,2363],{"className":2364,"style":114},[113],[33,2366],{"className":2367,"style":138},[137],[33,2369,2370,2373],{"style":141},[33,2371],{"className":2372,"style":114},[113],[33,2374,2376],{"className":2375},[118,119,120,121],[33,2377,2379],{"className":2378},[84,121],[33,2380,270],{"className":2381},[84,121],[33,2383,160],{"className":2384},[159],[33,2386,2388],{"className":2387},[101],[33,2389,2391],{"className":2390,"style":167},[105],[33,2392],{},[33,2394],{"className":2395},[173,89],[2242,2397,270],{"align":2166},[2242,2399,2400],{"align":2166},[33,2401,2403,2420],{"className":2402},[36],[33,2404,2406],{"className":2405},[40],[42,2407,2408],{"xmlns":44},[46,2409,2410,2418],{},[49,2411,2412],{},[52,2413,2414,2416],{},[59,2415,270],{},[59,2417,402],{},[63,2419,405],{"encoding":65},[33,2421,2423],{"className":2422,"ariaHidden":71},[70],[33,2424,2426,2429],{"className":2425},[75],[33,2427],{"className":2428,"style":291},[79],[33,2430,2432,2435,2494],{"className":2431},[84],[33,2433],{"className":2434},[88,89],[33,2436,2438],{"className":2437},[52],[33,2439,2441,2486],{"className":2440},[96,97],[33,2442,2444,2483],{"className":2443},[101],[33,2445,2447,2461,2469],{"className":2446,"style":310},[105],[33,2448,2449,2452],{"style":109},[33,2450],{"className":2451,"style":114},[113],[33,2453,2455],{"className":2454},[118,119,120,121],[33,2456,2458],{"className":2457},[84,121],[33,2459,402],{"className":2460},[84,121],[33,2462,2463,2466],{"style":130},[33,2464],{"className":2465,"style":114},[113],[33,2467],{"className":2468,"style":138},[137],[33,2470,2471,2474],{"style":141},[33,2472],{"className":2473,"style":114},[113],[33,2475,2477],{"className":2476},[118,119,120,121],[33,2478,2480],{"className":2479},[84,121],[33,2481,270],{"className":2482},[84,121],[33,2484,160],{"className":2485},[159],[33,2487,2489],{"className":2488},[101],[33,2490,2492],{"className":2491,"style":167},[105],[33,2493],{},[33,2495],{"className":2496},[173,89],[2161,2498,2499,2553,2652,2654],{},[2242,2500,2501,2502,1020],{"align":2166},"3 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(",[33,2760,2762,2780],{"className":2761},[36],[33,2763,2765],{"className":2764},[40],[42,2766,2767],{"xmlns":44},[46,2768,2769,2777],{},[49,2770,2771,2773,2775],{},[55,2772,2184],{},[274,2774,276],{},[59,2776,402],{},[63,2778,2779],{"encoding":65},"n=2",[33,2781,2783,2801],{"className":2782,"ariaHidden":71},[70],[33,2784,2786,2789,2792,2795,2798],{"className":2785},[75],[33,2787],{"className":2788,"style":204},[79],[33,2790,2184],{"className":2791},[84,154],[33,2793],{"className":2794,"style":365},[364],[33,2796,276],{"className":2797},[369],[33,2799],{"className":2800,"style":365},[364],[33,2802,2804,2807],{"className":2803},[75],[33,2805],{"className":2806,"style":379},[79],[33,2808,402],{"className":2809},[84],[2242,2811,2812],{"align":2166},[33,2813,2815,2832],{"className":2814},[36],[33,2816,2818],{"className":2817},[40],[42,2819,2820],{"xmlns":44},[46,2821,2822,2830],{},[49,2823,2824],{},[52,2825,2826,2828],{},[59,2827,910],{},[59,2829,1826],{},[63,2831,2061],{"encoding":65},[33,2833,2835],{"className":2834,"ariaHidden":71},[70],[33,2836,2838,2841],{"className":2837},[75],[33,2839],{"className":2840,"style":291},[79],[33,2842,2844,2847,2906],{"className":2843},[84],[33,2845],{"className":2846},[88,89],[33,2848,2850],{"className":2849},[52],[33,2851,2853,2898],{"className":2852},[96,97],[33,2854,2856,2895],{"className":2855},[101],[33,2857,2859,2873,2881],{"className":2858,"style":310},[105],[33,2860,2861,2864],{"style":109},[33,2862],{"className":2863,"style":114},[113],[33,2865,2867],{"className":2866},[118,119,120,121],[33,2868,2870],{"className":2869},[84,121],[33,2871,1826],{"className":2872},[84,121],[33,2874,2875,2878],{"style":130},[33,2876],{"className":2877,"style":114},[113],[33,2879],{"className":2880,"style":138},[137],[33,2882,2883,2886],{"style":141},[33,2884],{"className":2885,"style":114},[113],[33,2887,2889],{"className":2888},[118,119,120,121],[33,2890,2892],{"className":2891},[84,121],[33,2893,910],{"className":2894},[84,121],[33,2896,160],{"className":2897},[159],[33,2899,2901],{"className":2900},[101],[33,2902,2904],{"className":2903,"style":167},[105],[33,2905],{},[33,2907],{"className":2908},[173,89],[2242,2910,402],{"align":2166},[2242,2912,2913],{"align":2166},[33,2914,2916,2934],{"className":2915},[36],[33,2917,2919],{"className":2918},[40],[42,2920,2921],{"xmlns":44},[46,2922,2923,2931],{},[49,2924,2925],{},[52,2926,2927,2929],{},[59,2928,270],{},[59,2930,1835],{},[63,2932,2933],{"encoding":65},"\\frac{1}{32}",[33,2935,2937],{"className":2936,"ariaHidden":71},[70],[33,2938,2940,2943],{"className":2939},[75],[33,2941],{"className":2942,"style":291},[79],[33,2944,2946,2949,3008],{"className":2945},[84],[33,2947],{"className":2948},[88,89],[33,2950,2952],{"className":2951},[52],[33,2953,2955,3000],{"className":2954},[96,97],[33,2956,2958,2997],{"className":2957},[101],[33,2959,2961,2975,2983],{"className":2960,"style":310},[105],[33,2962,2963,2966],{"style":109},[33,2964],{"className":2965,"style":114},[113],[33,2967,2969],{"className":2968},[118,119,120,121],[33,2970,2972],{"className":2971},[84,121],[33,2973,1835],{"className":2974},[84,121],[33,2976,2977,2980],{"style":130},[33,2978],{"className":2979,"style":114},[113],[33,2981],{"className":2982,"style":138},[137],[33,2984,2985,2988],{"style":141},[33,2986],{"className":2987,"style":114},[113],[33,2989,2991],{"className":2990},[118,119,120,121],[33,2992,2994],{"className":2993},[84,121],[33,2995,270],{"className":2996},[84,121],[33,2998,160],{"className":2999},[159],[33,3001,3003],{"className":3002},[101],[33,3004,3006],{"className":3005,"style":167},[105],[33,3007],{},[33,3009],{"className":3010},[173,89],[2161,3012,3013,3067,3168,3170],{},[2242,3014,3015,3016,1020],{"align":2166},"7 (",[33,3017,3019,3037],{"className":3018},[36],[33,3020,3022],{"className":3021},[40],[42,3023,3024],{"xmlns":44},[46,3025,3026,3034],{},[49,3027,3028,3030,3032],{},[55,3029,2184],{},[274,3031,276],{},[59,3033,910],{},[63,3035,3036],{"encoding":65},"n=3",[33,3038,3040,3058],{"className":3039,"ariaHidden":71},[70],[33,3041,3043,3046,3049,3052,3055],{"className":3042},[75],[33,3044],{"className":3045,"style":204},[79],[33,3047,2184],{"className":3048},[84,154],[33,3050],{"className":3051,"style":365},[364],[33,3053,276],{"className":3054},[369],[33,3056],{"className":3057,"style":365},[364],[33,3059,3061,3064],{"className":3060},[75],[33,3062],{"className":3063,"style":379},[79],[33,3065,910],{"className":3066},[84],[2242,3068,3069],{"align":2166},[33,3070,3072,3091],{"className":3071},[36],[33,3073,3075],{"className":3074},[40],[42,3076,3077],{"xmlns":44},[46,3078,3079,3088],{},[49,3080,3081],{},[52,3082,3083,3085],{},[59,3084,61],{},[59,3086,3087],{},"7",[63,3089,3090],{"encoding":65},"\\frac{4}{7}",[33,3092,3094],{"className":3093,"ariaHidden":71},[70],[33,3095,3097,3100],{"className":3096},[75],[33,3098],{"className":3099,"style":291},[79],[33,3101,3103,3106,3165],{"className":3102},[84],[33,3104],{"className":3105},[88,89],[33,3107,3109],{"className":3108},[52],[33,3110,3112,3157],{"className":3111},[96,97],[33,3113,3115,3154],{"className":3114},[101],[33,3116,3118,3132,3140],{"className":3117,"style":310},[105],[33,3119,3120,3123],{"style":109},[33,3121],{"className":3122,"style":114},[113],[33,3124,3126],{"className":3125},[118,119,120,121],[33,3127,3129],{"className":3128},[84,121],[33,3130,3087],{"className":3131},[84,121],[33,3133,3134,3137],{"style":130},[33,3135],{"className":3136,"style":114},[113],[33,3138],{"className":3139,"style":138},[137],[33,3141,3142,3145],{"style":141},[33,3143],{"className":3144,"style":114},[113],[33,3146,3148],{"className":3147},[118,119,120,121],[33,3149,3151],{"className":3150},[84,121],[33,3152,61],{"className":3153},[84,121],[33,3155,160],{"className":3156},[159],[33,3158,3160],{"className":3159},[101],[33,3161,3163],{"className":3162,"style":167},[105],[33,3164],{},[33,3166],{"className":3167},[173,89],[2242,3169,1826],{"align":2166},[2242,3171,3172],{"align":2166},[33,3173,3175,3194],{"className":3174},[36],[33,3176,3178],{"className":3177},[40],[42,3179,3180],{"xmlns":44},[46,3181,3182,3191],{},[49,3183,3184],{},[52,3185,3186,3188],{},[59,3187,270],{},[59,3189,3190],{},"128",[63,3192,3193],{"encoding":65},"\\frac{1}{128}",[33,3195,3197],{"className":3196,"ariaHidden":71},[70],[33,3198,3200,3203],{"className":3199},[75],[33,3201],{"className":3202,"style":291},[79],[33,3204,3206,3209,3268],{"className":3205},[84],[33,3207],{"className":3208},[88,89],[33,3210,3212],{"className":3211},[52],[33,3213,3215,3260],{"className":3214},[96,97],[33,3216,3218,3257],{"className":3217},[101],[33,3219,3221,3235,3243],{"className":3220,"style":310},[105],[33,3222,3223,3226],{"style":109},[33,3224],{"className":3225,"style":114},[113],[33,3227,3229],{"className":3228},[118,119,120,121],[33,3230,3232],{"className":3231},[84,121],[33,3233,3190],{"className":3234},[84,121],[33,3236,3237,3240],{"style":130},[33,3238],{"className":3239,"style":114},[113],[33,3241],{"className":3242,"style":138},[137],[33,3244,3245,3248],{"style":141},[33,3246],{"className":3247,"style":114},[113],[33,3249,3251],{"className":3250},[118,119,120,121],[33,3252,3254],{"className":3253},[84,121],[33,3255,270],{"className":3256},[84,121],[33,3258,160],{"className":3259},[159],[33,3261,3263],{"className":3262},[101],[33,3264,3266],{"className":3265,"style":167},[105],[33,3267],{},[33,3269],{"className":3270},[173,89],[2161,3272,3273,3327,3428,3431],{},[2242,3274,3275,3276,1020],{"align":2166},"9 (",[33,3277,3279,3297],{"className":3278},[36],[33,3280,3282],{"className":3281},[40],[42,3283,3284],{"xmlns":44},[46,3285,3286,3294],{},[49,3287,3288,3290,3292],{},[55,3289,2184],{},[274,3291,276],{},[59,3293,61],{},[63,3295,3296],{"encoding":65},"n=4",[33,3298,3300,3318],{"className":3299,"ariaHidden":71},[70],[33,3301,3303,3306,3309,3312,3315],{"className":3302},[75],[33,3304],{"className":3305,"style":204},[79],[33,3307,2184],{"className":3308},[84,154],[33,3310],{"className":3311,"style":365},[364],[33,3313,276],{"className":3314},[369],[33,3316],{"className":3317,"style":365},[364],[33,3319,3321,3324],{"className":3320},[75],[33,3322],{"className":3323,"style":379},[79],[33,3325,61],{"className":3326},[84],[2242,3328,3329],{"align":2166},[33,3330,3332,3351],{"className":3331},[36],[33,3333,3335],{"className":3334},[40],[42,3336,3337],{"xmlns":44},[46,3338,3339,3348],{},[49,3340,3341],{},[52,3342,3343,3345],{},[59,3344,1826],{},[59,3346,3347],{},"9",[63,3349,3350],{"encoding":65},"\\frac{5}{9}",[33,3352,3354],{"className":3353,"ariaHidden":71},[70],[33,3355,3357,3360],{"className":3356},[75],[33,3358],{"className":3359,"style":291},[79],[33,3361,3363,3366,3425],{"className":3362},[84],[33,3364],{"className":3365},[88,89],[33,3367,3369],{"className":3368},[52],[33,3370,3372,3417],{"className":3371},[96,97],[33,3373,3375,3414],{"className":3374},[101],[33,3376,3378,3392,3400],{"className":3377,"style":310},[105],[33,3379,3380,3383],{"style":109},[33,3381],{"className":3382,"style":114},[113],[33,3384,3386],{"className":3385},[118,119,120,121],[33,3387,3389],{"className":3388},[84,121],[33,3390,3347],{"className":3391},[84,121],[33,3393,3394,3397],{"style":130},[33,3395],{"className":3396,"style":114},[113],[33,3398],{"className":3399,"style":138},[137],[33,3401,3402,3405],{"style":141},[33,3403],{"className":3404,"style":114},[113],[33,3406,3408],{"className":3407},[118,119,120,121],[33,3409,3411],{"className":3410},[84,121],[33,3412,1826],{"className":3413},[84,121],[33,3415,160],{"className":3416},[159],[33,3418,3420],{"className":3419},[101],[33,3421,3423],{"className":3422,"style":167},[105],[33,3424],{},[33,3426],{"className":3427},[173,89],[2242,3429,3430],{"align":2166},"14",[2242,3432,3433],{"align":2166},[33,3434,3436,3455],{"className":3435},[36],[33,3437,3439],{"className":3438},[40],[42,3440,3441],{"xmlns":44},[46,3442,3443,3452],{},[49,3444,3445],{},[52,3446,3447,3449],{},[59,3448,270],{},[59,3450,3451],{},"512",[63,3453,3454],{"encoding":65},"\\frac{1}{512}",[33,3456,3458],{"className":3457,"ariaHidden":71},[70],[33,3459,3461,3464],{"className":3460},[75],[33,3462],{"className":3463,"style":291},[79],[33,3465,3467,3470,3529],{"className":3466},[84],[33,3468],{"className":3469},[88,89],[33,3471,3473],{"className":3472},[52],[33,3474,3476,3521],{"className":3475},[96,97],[33,3477,3479,3518],{"className":3478},[101],[33,3480,3482,3496,3504],{"className":3481,"style":310},[105],[33,3483,3484,3487],{"style":109},[33,3485],{"className":3486,"style":114},[113],[33,3488,3490],{"className":3489},[118,119,120,121],[33,3491,3493],{"className":3492},[84,121],[33,3494,3451],{"className":3495},[84,121],[33,3497,3498,3501],{"style":130},[33,3499],{"className":3500,"style":114},[113],[33,3502],{"className":3503,"style":138},[137],[33,3505,3506,3509],{"style":141},[33,3507],{"className":3508,"style":114},[113],[33,3510,3512],{"className":3511},[118,119,120,121],[33,3513,3515],{"className":3514},[84,121],[33,3516,270],{"className":3517},[84,121],[33,3519,160],{"className":3520},[159],[33,3522,3524],{"className":3523},[101],[33,3525,3527],{"className":3526,"style":167},[105],[33,3528],{},[33,3530],{"className":3531},[173,89],[15,3533,3534],{},"Notice the patterns:",[3536,3537,3538,3597,3729],"ul",{},[230,3539,3540,3541,174],{},"The length is always odd: ",[33,3542,3544,3564],{"className":3543},[36],[33,3545,3547],{"className":3546},[40],[42,3548,3549],{"xmlns":44},[46,3550,3551,3561],{},[49,3552,3553,3555,3557,3559],{},[59,3554,402],{},[55,3556,2184],{},[274,3558,1299],{},[59,3560,270],{},[63,3562,3563],{"encoding":65},"2n + 1",[33,3565,3567,3588],{"className":3566,"ariaHidden":71},[70],[33,3568,3570,3573,3576,3579,3582,3585],{"className":3569},[75],[33,3571],{"className":3572,"style":1314},[79],[33,3574,402],{"className":3575},[84],[33,3577,2184],{"className":3578},[84,154],[33,3580],{"className":3581,"style":2210},[364],[33,3583,1299],{"className":3584},[2214],[33,3586],{"className":3587,"style":2210},[364],[33,3589,3591,3594],{"className":3590},[75],[33,3592],{"className":3593,"style":379},[79],[33,3595,270],{"className":3596},[84],[230,3598,3599,3600,174],{},"The ratio of heads is always ",[33,3601,3603,3635],{"className":3602},[36],[33,3604,3606],{"className":3605},[40],[42,3607,3608],{"xmlns":44},[46,3609,3610,3632],{},[49,3611,3612],{},[52,3613,3614,3622],{},[49,3615,3616,3618,3620],{},[55,3617,2184],{},[274,3619,1299],{},[59,3621,270],{},[49,3623,3624,3626,3628,3630],{},[59,3625,402],{},[55,3627,2184],{},[274,3629,1299],{},[59,3631,270],{},[63,3633,3634],{"encoding":65},"\\frac{n+1}{2n+1}",[33,3636,3638],{"className":3637,"ariaHidden":71},[70],[33,3639,3641,3645],{"className":3640},[75],[33,3642],{"className":3643,"style":3644},[79],"height:1.2484em;vertical-align:-0.4033em;",[33,3646,3648,3651,3726],{"className":3647},[84],[33,3649],{"className":3650},[88,89],[33,3652,3654],{"className":3653},[52],[33,3655,3657,3717],{"className":3656},[96,97],[33,3658,3660,3714],{"className":3659},[101],[33,3661,3663,3686,3694],{"className":3662,"style":310},[105],[33,3664,3665,3668],{"style":109},[33,3666],{"className":3667,"style":114},[113],[33,3669,3671],{"className":3670},[118,119,120,121],[33,3672,3674,3677,3680,3683],{"className":3673},[84,121],[33,3675,402],{"className":3676},[84,121],[33,3678,2184],{"className":3679},[84,154,121],[33,3681,1299],{"className":3682},[2214,121],[33,3684,270],{"className":3685},[84,121],[33,3687,3688,3691],{"style":130},[33,3689],{"className":3690,"style":114},[113],[33,3692],{"className":3693,"style":138},[137],[33,3695,3696,3699],{"style":141},[33,3697],{"className":3698,"style":114},[113],[33,3700,3702],{"className":3701},[118,119,120,121],[33,3703,3705,3708,3711],{"className":3704},[84,121],[33,3706,2184],{"className":3707},[84,154,121],[33,3709,1299],{"className":3710},[2214,121],[33,3712,270],{"className":3713},[84,121],[33,3715,160],{"className":3716},[159],[33,3718,3720],{"className":3719},[101],[33,3721,3724],{"className":3722,"style":3723},[105],"height:0.4033em;",[33,3725],{},[33,3727],{"className":3728},[173,89],[230,3730,3731,3732,3901,3902,3957],{},"The probability of each individual sequence is ",[33,3733,3735,3771],{"className":3734},[36],[33,3736,3738],{"className":3737},[40],[42,3739,3740],{"xmlns":44},[46,3741,3742,3768],{},[49,3743,3744],{},[1005,3745,3746,3758],{},[49,3747,3748,3750,3756],{},[274,3749,1011],{"fence":71},[52,3751,3752,3754],{},[59,3753,270],{},[59,3755,402],{},[274,3757,1020],{"fence":71},[49,3759,3760,3762,3764,3766],{},[59,3761,402],{},[55,3763,2184],{},[274,3765,1299],{},[59,3767,270],{},[63,3769,3770],{"encoding":65},"\\left(\\frac{1}{2}\\right)^{2n+1}",[33,3772,3774],{"className":3773,"ariaHidden":71},[70],[33,3775,3777,3780],{"className":3776},[75],[33,3778],{"className":3779,"style":1044},[79],[33,3781,3783,3866],{"className":3782},[1048],[33,3784,3786,3792,3860],{"className":3785},[1048],[33,3787,3789],{"className":3788,"style":1056},[88,1055],[33,3790,1011],{"className":3791},[1060,1061],[33,3793,3795,3798,3857],{"className":3794},[84],[33,3796],{"className":3797},[88,89],[33,3799,3801],{"className":3800},[52],[33,3802,3804,3849],{"className":3803},[96,97],[33,3805,3807,3846],{"className":3806},[101],[33,3808,3810,3824,3832],{"className":3809,"style":310},[105],[33,3811,3812,3815],{"style":109},[33,3813],{"className":3814,"style":114},[113],[33,3816,3818],{"className":3817},[118,119,120,121],[33,3819,3821],{"className":3820},[84,121],[33,3822,402],{"className":3823},[84,121],[33,3825,3826,3829],{"style":130},[33,3827],{"className":3828,"style":114},[113],[33,3830],{"className":3831,"style":138},[137],[33,3833,3834,3837],{"style":141},[33,3835],{"className":3836,"style":114},[113],[33,3838,3840],{"className":3839},[118,119,120,121],[33,3841,3843],{"className":3842},[84,121],[33,3844,270],{"className":3845},[84,121],[33,3847,160],{"className":3848},[159],[33,3850,3852],{"className":3851},[101],[33,3853,3855],{"className":3854,"style":167},[105],[33,3856],{},[33,3858],{"className":3859},[173,89],[33,3861,3863],{"className":3862,"style":1056},[173,1055],[33,3864,1020],{"className":3865},[1060,1061],[33,3867,3869],{"className":3868},[1139],[33,3870,3872],{"className":3871},[96],[33,3873,3875],{"className":3874},[101],[33,3876,3878],{"className":3877,"style":1149},[105],[33,3879,3880,3883],{"style":1152},[33,3881],{"className":3882,"style":1156},[113],[33,3884,3886],{"className":3885},[118,119,120,121],[33,3887,3889,3892,3895,3898],{"className":3888},[84,121],[33,3890,402],{"className":3891},[84,121],[33,3893,2184],{"className":3894},[84,154,121],[33,3896,1299],{"className":3897},[2214,121],[33,3899,270],{"className":3900},[84,121]," — each of the ",[33,3903,3905,3924],{"className":3904},[36],[33,3906,3908],{"className":3907},[40],[42,3909,3910],{"xmlns":44},[46,3911,3912,3922],{},[49,3913,3914,3916,3918,3920],{},[59,3915,402],{},[55,3917,2184],{},[274,3919,1299],{},[59,3921,270],{},[63,3923,2191],{"encoding":65},[33,3925,3927,3948],{"className":3926,"ariaHidden":71},[70],[33,3928,3930,3933,3936,3939,3942,3945],{"className":3929},[75],[33,3931],{"className":3932,"style":1314},[79],[33,3934,402],{"className":3935},[84],[33,3937,2184],{"className":3938},[84,154],[33,3940],{"className":3941,"style":2210},[364],[33,3943,1299],{"className":3944},[2214],[33,3946],{"className":3947,"style":2210},[364],[33,3949,3951,3954],{"className":3950},[75],[33,3952],{"className":3953,"style":379},[79],[33,3955,270],{"className":3956},[84]," flips has a one-in-two chance.",[15,3959,3960,3961],{},"But the column that really matters is the number of valid sequences: ",[33,3962,3964,3999],{"className":3963},[36],[33,3965,3967],{"className":3966},[40],[42,3968,3969],{"xmlns":44},[46,3970,3971,3996],{},[49,3972,3973,3975,3977,3979,3981,3983,3985,3987,3989,3991,3993],{},[59,3974,270],{},[274,3976,1379],{"separator":71},[59,3978,270],{},[274,3980,1379],{"separator":71},[59,3982,402],{},[274,3984,1379],{"separator":71},[59,3986,1826],{},[274,3988,1379],{"separator":71},[59,3990,3430],{},[274,3992,1379],{"separator":71},[274,3994,3995],{},"…",[63,3997,3998],{"encoding":65},"1, 1, 2, 5, 14, \\ldots",[33,4000,4002],{"className":4001,"ariaHidden":71},[70],[33,4003,4005,4008,4011,4014,4017,4020,4023,4026,4029,4032,4035,4038,4041,4044,4047,4050,4053],{"className":4004},[75],[33,4006],{"className":4007,"style":1422},[79],[33,4009,270],{"className":4010},[84],[33,4012,1379],{"className":4013},[1432],[33,4015],{"className":4016,"style":1439},[364],[33,4018,270],{"className":4019},[84],[33,4021,1379],{"className":4022},[1432],[33,4024],{"className":4025,"style":1439},[364],[33,4027,402],{"className":4028},[84],[33,4030,1379],{"className":4031},[1432],[33,4033],{"className":4034,"style":1439},[364],[33,4036,1826],{"className":4037},[84],[33,4039,1379],{"className":4040},[1432],[33,4042],{"className":4043,"style":1439},[364],[33,4045,3430],{"className":4046},[84],[33,4048,1379],{"className":4049},[1432],[33,4051],{"className":4052,"style":1439},[364],[33,4054,3995],{"className":4055},[1048],[15,4057,4058],{},"Ring a bell?",[544,4060,4062],{"id":4061},"from-manual-counting-to-the-catalan-numbers","From manual counting to the Catalan numbers",[15,4064,4065,4066,4188,4189,4196],{},"Those numbers — ",[33,4067,4069,4113],{"className":4068},[36],[33,4070,4072],{"className":4071},[40],[42,4073,4074],{"xmlns":44},[46,4075,4076,4110],{},[49,4077,4078,4080,4082,4084,4086,4088,4090,4092,4094,4096,4098,4101,4103,4106,4108],{},[59,4079,270],{},[274,4081,1379],{"separator":71},[59,4083,270],{},[274,4085,1379],{"separator":71},[59,4087,402],{},[274,4089,1379],{"separator":71},[59,4091,1826],{},[274,4093,1379],{"separator":71},[59,4095,3430],{},[274,4097,1379],{"separator":71},[59,4099,4100],{},"42",[274,4102,1379],{"separator":71},[59,4104,4105],{},"132",[274,4107,1379],{"separator":71},[274,4109,3995],{},[63,4111,4112],{"encoding":65},"1, 1, 2, 5, 14, 42, 132, \\ldots",[33,4114,4116],{"className":4115,"ariaHidden":71},[70],[33,4117,4119,4122,4125,4128,4131,4134,4137,4140,4143,4146,4149,4152,4155,4158,4161,4164,4167,4170,4173,4176,4179,4182,4185],{"className":4118},[75],[33,4120],{"className":4121,"style":1422},[79],[33,4123,270],{"className":4124},[84],[33,4126,1379],{"className":4127},[1432],[33,4129],{"className":4130,"style":1439},[364],[33,4132,270],{"className":4133},[84],[33,4135,1379],{"className":4136},[1432],[33,4138],{"className":4139,"style":1439},[364],[33,4141,402],{"className":4142},[84],[33,4144,1379],{"className":4145},[1432],[33,4147],{"className":4148,"style":1439},[364],[33,4150,1826],{"className":4151},[84],[33,4153,1379],{"className":4154},[1432],[33,4156],{"className":4157,"style":1439},[364],[33,4159,3430],{"className":4160},[84],[33,4162,1379],{"className":4163},[1432],[33,4165],{"className":4166,"style":1439},[364],[33,4168,4100],{"className":4169},[84],[33,4171,1379],{"className":4172},[1432],[33,4174],{"className":4175,"style":1439},[364],[33,4177,4105],{"className":4178},[84],[33,4180,1379],{"className":4181},[1432],[33,4183],{"className":4184,"style":1439},[364],[33,4186,3995],{"className":4187},[1048]," — have a name. They are the ",[19,4190,4193],{"href":4191,"rel":4192},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCatalan_number",[23],[237,4194,4195],{},"Catalan numbers",", one of the most ubiquitous sequences in all of combinatorics.",[15,4198,4199,4200,4207,4208,4211,4212,4215,4216,4221,4222,4225],{},"Their history is itself a story of unexpected connections. The Mongolian mathematician ",[19,4201,4204],{"href":4202,"rel":4203},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMinggatu",[23],[237,4205,4206],{},"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 ",[210,4209,4210],{},"Ge Yuan Mi Lu Jie Fa",". Two decades later, in 1751, ",[237,4213,4214],{},"Leonhard Euler"," ",[19,4217,4220],{"href":4218,"rel":4219},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FCatalan_number#History",[23],"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 ",[237,4223,4224],{},"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.",[15,4227,4228],{},"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.",[15,4230,4231,4232,4260],{},"The ",[33,4233,4235,4248],{"className":4234},[36],[33,4236,4238],{"className":4237},[40],[42,4239,4240],{"xmlns":44},[46,4241,4242,4246],{},[49,4243,4244],{},[55,4245,2184],{},[63,4247,2184],{"encoding":65},[33,4249,4251],{"className":4250,"ariaHidden":71},[70],[33,4252,4254,4257],{"className":4253},[75],[33,4255],{"className":4256,"style":204},[79],[33,4258,2184],{"className":4259},[84,154],"-th Catalan number is defined as:",[15,4262,4264],{"className":4263},[575],[33,4265,4267,4318],{"className":4266},[36],[33,4268,4270],{"className":4269},[40],[42,4271,4272],{"xmlns":44},[46,4273,4274,4315],{},[49,4275,4276,4284,4286,4298],{},[4277,4278,4279,4282],"msub",{},[55,4280,4281],{},"C",[55,4283,2184],{},[274,4285,276],{},[52,4287,4288,4290],{},[59,4289,270],{},[49,4291,4292,4294,4296],{},[55,4293,2184],{},[274,4295,1299],{},[59,4297,270],{},[49,4299,4300,4302,4313],{},[274,4301,1011],{"fence":71},[52,4303,4305,4311],{"linethickness":4304},"0px",[49,4306,4307,4309],{},[59,4308,402],{},[55,4310,2184],{},[55,4312,2184],{},[274,4314,1020],{"fence":71},[63,4316,4317],{"encoding":65},"C_n = \\frac{1}{n+1}\\binom{2n}{n}",[33,4319,4321,4381],{"className":4320,"ariaHidden":71},[70],[33,4322,4324,4328,4372,4375,4378],{"className":4323},[75],[33,4325],{"className":4326,"style":4327},[79],"height:0.8333em;vertical-align:-0.15em;",[33,4329,4331,4335],{"className":4330},[84],[33,4332,4281],{"className":4333,"style":4334},[84,154],"margin-right:0.0715em;",[33,4336,4338],{"className":4337},[1139],[33,4339,4341,4363],{"className":4340},[96,97],[33,4342,4344,4360],{"className":4343},[101],[33,4345,4348],{"className":4346,"style":4347},[105],"height:0.1514em;",[33,4349,4351,4354],{"style":4350},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[33,4352],{"className":4353,"style":1156},[113],[33,4355,4357],{"className":4356},[118,119,120,121],[33,4358,2184],{"className":4359},[84,154,121],[33,4361,160],{"className":4362},[159],[33,4364,4366],{"className":4365},[101],[33,4367,4370],{"className":4368,"style":4369},[105],"height:0.15em;",[33,4371],{},[33,4373],{"className":4374,"style":365},[364],[33,4376,276],{"className":4377},[369],[33,4379],{"className":4380,"style":365},[364],[33,4382,4384,4388,4462],{"className":4383},[75],[33,4385],{"className":4386,"style":4387},[79],"height:1.2984em;vertical-align:-0.4033em;",[33,4389,4391,4394,4459],{"className":4390},[84],[33,4392],{"className":4393},[88,89],[33,4395,4397],{"className":4396},[52],[33,4398,4400,4451],{"className":4399},[96,97],[33,4401,4403,4448],{"className":4402},[101],[33,4404,4406,4426,4434],{"className":4405,"style":310},[105],[33,4407,4408,4411],{"style":109},[33,4409],{"className":4410,"style":114},[113],[33,4412,4414],{"className":4413},[118,119,120,121],[33,4415,4417,4420,4423],{"className":4416},[84,121],[33,4418,2184],{"className":4419},[84,154,121],[33,4421,1299],{"className":4422},[2214,121],[33,4424,270],{"className":4425},[84,121],[33,4427,4428,4431],{"style":130},[33,4429],{"className":4430,"style":114},[113],[33,4432],{"className":4433,"style":138},[137],[33,4435,4436,4439],{"style":141},[33,4437],{"className":4438,"style":114},[113],[33,4440,4442],{"className":4441},[118,119,120,121],[33,4443,4445],{"className":4444},[84,121],[33,4446,270],{"className":4447},[84,121],[33,4449,160],{"className":4450},[159],[33,4452,4454],{"className":4453},[101],[33,4455,4457],{"className":4456,"style":3723},[105],[33,4458],{},[33,4460],{"className":4461},[173,89],[33,4463,4465,4471,4528],{"className":4464},[84],[33,4466,4468],{"className":4467,"style":1056},[88,1055],[33,4469,1011],{"className":4470},[1060,1061],[33,4472,4474],{"className":4473},[52],[33,4475,4477,4520],{"className":4476},[96,97],[33,4478,4480,4517],{"className":4479},[101],[33,4481,4484,4499],{"className":4482,"style":4483},[105],"height:0.8951em;",[33,4485,4487,4490],{"style":4486},"top:-2.355em;",[33,4488],{"className":4489,"style":1156},[113],[33,4491,4493],{"className":4492},[118,119,120,121],[33,4494,4496],{"className":4495},[84,121],[33,4497,2184],{"className":4498},[84,154,121],[33,4500,4502,4505],{"style":4501},"top:-3.144em;",[33,4503],{"className":4504,"style":1156},[113],[33,4506,4508],{"className":4507},[118,119,120,121],[33,4509,4511,4514],{"className":4510},[84,121],[33,4512,402],{"className":4513},[84,121],[33,4515,2184],{"className":4516},[84,154,121],[33,4518,160],{"className":4519},[159],[33,4521,4523],{"className":4522},[101],[33,4524,4526],{"className":4525,"style":167},[105],[33,4527],{},[33,4529,4531],{"className":4530,"style":1056},[173,1055],[33,4532,1020],{"className":4533},[1060,1061],[15,4535,4536,4537,4645,4646,4680,4681,4709,4710,4713,4714,4742,4743,4776,4777,4805,4806,4839,4840,174],{},"where ",[33,4538,4540,4566],{"className":4539},[36],[33,4541,4543],{"className":4542},[40],[42,4544,4545],{"xmlns":44},[46,4546,4547,4563],{},[49,4548,4549,4551,4561],{},[274,4550,1011],{"fence":71},[52,4552,4553,4559],{"linethickness":4304},[49,4554,4555,4557],{},[59,4556,402],{},[55,4558,2184],{},[55,4560,2184],{},[274,4562,1020],{"fence":71},[63,4564,4565],{"encoding":65},"\\binom{2n}{n}",[33,4567,4569],{"className":4568,"ariaHidden":71},[70],[33,4570,4572,4576],{"className":4571},[75],[33,4573],{"className":4574,"style":4575},[79],"height:1.2451em;vertical-align:-0.35em;",[33,4577,4579,4585,4639],{"className":4578},[84],[33,4580,4582],{"className":4581,"style":1056},[88,1055],[33,4583,1011],{"className":4584},[1060,1061],[33,4586,4588],{"className":4587},[52],[33,4589,4591,4631],{"className":4590},[96,97],[33,4592,4594,4628],{"className":4593},[101],[33,4595,4597,4611],{"className":4596,"style":4483},[105],[33,4598,4599,4602],{"style":4486},[33,4600],{"className":4601,"style":1156},[113],[33,4603,4605],{"className":4604},[118,119,120,121],[33,4606,4608],{"className":4607},[84,121],[33,4609,2184],{"className":4610},[84,154,121],[33,4612,4613,4616],{"style":4501},[33,4614],{"className":4615,"style":1156},[113],[33,4617,4619],{"className":4618},[118,119,120,121],[33,4620,4622,4625],{"className":4621},[84,121],[33,4623,402],{"className":4624},[84,121],[33,4626,2184],{"className":4627},[84,154,121],[33,4629,160],{"className":4630},[159],[33,4632,4634],{"className":4633},[101],[33,4635,4637],{"className":4636,"style":167},[105],[33,4638],{},[33,4640,4642],{"className":4641,"style":1056},[173,1055],[33,4643,1020],{"className":4644},[1060,1061]," — read \"",[33,4647,4649,4665],{"className":4648},[36],[33,4650,4652],{"className":4651},[40],[42,4653,4654],{"xmlns":44},[46,4655,4656,4662],{},[49,4657,4658,4660],{},[59,4659,402],{},[55,4661,2184],{},[63,4663,4664],{"encoding":65},"2n",[33,4666,4668],{"className":4667,"ariaHidden":71},[70],[33,4669,4671,4674,4677],{"className":4670},[75],[33,4672],{"className":4673,"style":379},[79],[33,4675,402],{"className":4676},[84],[33,4678,2184],{"className":4679},[84,154]," choose ",[33,4682,4684,4697],{"className":4683},[36],[33,4685,4687],{"className":4686},[40],[42,4688,4689],{"xmlns":44},[46,4690,4691,4695],{},[49,4692,4693],{},[55,4694,2184],{},[63,4696,2184],{"encoding":65},[33,4698,4700],{"className":4699,"ariaHidden":71},[70],[33,4701,4703,4706],{"className":4702},[75],[33,4704],{"className":4705,"style":204},[79],[33,4707,2184],{"className":4708},[84,154],"\" — is the ",[237,4711,4712],{},"binomial coefficient",": the number of ways to pick ",[33,4715,4717,4730],{"className":4716},[36],[33,4718,4720],{"className":4719},[40],[42,4721,4722],{"xmlns":44},[46,4723,4724,4728],{},[49,4725,4726],{},[55,4727,2184],{},[63,4729,2184],{"encoding":65},[33,4731,4733],{"className":4732,"ariaHidden":71},[70],[33,4734,4736,4739],{"className":4735},[75],[33,4737],{"className":4738,"style":204},[79],[33,4740,2184],{"className":4741},[84,154]," elements from a set of ",[33,4744,4746,4761],{"className":4745},[36],[33,4747,4749],{"className":4748},[40],[42,4750,4751],{"xmlns":44},[46,4752,4753,4759],{},[49,4754,4755,4757],{},[59,4756,402],{},[55,4758,2184],{},[63,4760,4664],{"encoding":65},[33,4762,4764],{"className":4763,"ariaHidden":71},[70],[33,4765,4767,4770,4773],{"className":4766},[75],[33,4768],{"className":4769,"style":379},[79],[33,4771,402],{"className":4772},[84],[33,4774,2184],{"className":4775},[84,154],". In practice, it answers questions like \"how many ways can I arrange ",[33,4778,4780,4793],{"className":4779},[36],[33,4781,4783],{"className":4782},[40],[42,4784,4785],{"xmlns":44},[46,4786,4787,4791],{},[49,4788,4789],{},[55,4790,2184],{},[63,4792,2184],{"encoding":65},[33,4794,4796],{"className":4795,"ariaHidden":71},[70],[33,4797,4799,4802],{"className":4798},[75],[33,4800],{"className":4801,"style":204},[79],[33,4803,2184],{"className":4804},[84,154]," heads among ",[33,4807,4809,4824],{"className":4808},[36],[33,4810,4812],{"className":4811},[40],[42,4813,4814],{"xmlns":44},[46,4815,4816,4822],{},[49,4817,4818,4820],{},[59,4819,402],{},[55,4821,2184],{},[63,4823,4664],{"encoding":65},[33,4825,4827],{"className":4826,"ariaHidden":71},[70],[33,4828,4830,4833,4836],{"className":4829},[75],[33,4831],{"className":4832,"style":379},[79],[33,4834,402],{"className":4835},[84],[33,4837,2184],{"className":4838},[84,154]," flips?\" We explored this notation in detail in the ",[19,4841,4843],{"href":219,"rel":4842},[23],"article on Pascal's triangle",[15,4845,4846],{},"Let's verify against our data:",[15,4848,4850],{"className":4849},[575],[33,4851,4853,4928],{"className":4852},[36],[33,4854,4856],{"className":4855},[40],[42,4857,4858],{"xmlns":44},[46,4859,4860,4925],{},[49,4861,4862,4868,4870,4876,4888,4890,4892,4895,4901,4903,4909,4921,4923],{},[4277,4863,4864,4866],{},[55,4865,4281],{},[59,4867,1401],{},[274,4869,276],{},[52,4871,4872,4874],{},[59,4873,270],{},[59,4875,270],{},[49,4877,4878,4880,4886],{},[274,4879,1011],{"fence":71},[52,4881,4882,4884],{"linethickness":4304},[59,4883,1401],{},[59,4885,1401],{},[274,4887,1020],{"fence":71},[274,4889,276],{},[59,4891,270],{},[364,4893],{"width":4894},"2em",[4277,4896,4897,4899],{},[55,4898,4281],{},[59,4900,270],{},[274,4902,276],{},[52,4904,4905,4907],{},[59,4906,270],{},[59,4908,402],{},[49,4910,4911,4913,4919],{},[274,4912,1011],{"fence":71},[52,4914,4915,4917],{"linethickness":4304},[59,4916,402],{},[59,4918,270],{},[274,4920,1020],{"fence":71},[274,4922,276],{},[59,4924,270],{},[63,4926,4927],{"encoding":65},"C_0 = \\frac{1}{1}\\binom{0}{0} = 1 \\qquad C_1 = \\frac{1}{2}\\binom{2}{1} = 1",[33,4929,4931,4987,5136,5198,5347],{"className":4930,"ariaHidden":71},[70],[33,4932,4934,4937,4978,4981,4984],{"className":4933},[75],[33,4935],{"className":4936,"style":4327},[79],[33,4938,4940,4943],{"className":4939},[84],[33,4941,4281],{"className":4942,"style":4334},[84,154],[33,4944,4946],{"className":4945},[1139],[33,4947,4949,4970],{"className":4948},[96,97],[33,4950,4952,4967],{"className":4951},[101],[33,4953,4956],{"className":4954,"style":4955},[105],"height:0.3011em;",[33,4957,4958,4961],{"style":4350},[33,4959],{"className":4960,"style":1156},[113],[33,4962,4964],{"className":4963},[118,119,120,121],[33,4965,1401],{"className":4966},[84,121],[33,4968,160],{"className":4969},[159],[33,4971,4973],{"className":4972},[101],[33,4974,4976],{"className":4975,"style":4369},[105],[33,4977],{},[33,4979],{"className":4980,"style":365},[364],[33,4982,276],{"className":4983},[369],[33,4985],{"className":4986,"style":365},[364],[33,4988,4990,4993,5061,5127,5130,5133],{"className":4989},[75],[33,4991],{"className":4992,"style":4575},[79],[33,4994,4996,4999,5058],{"className":4995},[84],[33,4997],{"className":4998},[88,89],[33,5000,5002],{"className":5001},[52],[33,5003,5005,5050],{"className":5004},[96,97],[33,5006,5008,5047],{"className":5007},[101],[33,5009,5011,5025,5033],{"className":5010,"style":310},[105],[33,5012,5013,5016],{"style":109},[33,5014],{"className":5015,"style":114},[113],[33,5017,5019],{"className":5018},[118,119,120,121],[33,5020,5022],{"className":5021},[84,121],[33,5023,270],{"className":5024},[84,121],[33,5026,5027,5030],{"style":130},[33,5028],{"className":5029,"style":114},[113],[33,5031],{"className":5032,"style":138},[137],[33,5034,5035,5038],{"style":141},[33,5036],{"className":5037,"style":114},[113],[33,5039,5041],{"className":5040},[118,119,120,121],[33,5042,5044],{"className":5043},[84,121],[33,5045,270],{"className":5046},[84,121],[33,5048,160],{"className":5049},[159],[33,5051,5053],{"className":5052},[101],[33,5054,5056],{"className":5055,"style":167},[105],[33,5057],{},[33,5059],{"className":5060},[173,89],[33,5062,5064,5070,5121],{"className":5063},[84],[33,5065,5067],{"className":5066,"style":1056},[88,1055],[33,5068,1011],{"className":5069},[1060,1061],[33,5071,5073],{"className":5072},[52],[33,5074,5076,5113],{"className":5075},[96,97],[33,5077,5079,5110],{"className":5078},[101],[33,5080,5082,5096],{"className":5081,"style":4483},[105],[33,5083,5084,5087],{"style":4486},[33,5085],{"className":5086,"style":1156},[113],[33,5088,5090],{"className":5089},[118,119,120,121],[33,5091,5093],{"className":5092},[84,121],[33,5094,1401],{"className":5095},[84,121],[33,5097,5098,5101],{"style":4501},[33,5099],{"className":5100,"style":1156},[113],[33,5102,5104],{"className":5103},[118,119,120,121],[33,5105,5107],{"className":5106},[84,121],[33,5108,1401],{"className":5109},[84,121],[33,5111,160],{"className":5112},[159],[33,5114,5116],{"className":5115},[101],[33,5117,5119],{"className":5118,"style":167},[105],[33,5120],{},[33,5122,5124],{"className":5123,"style":1056},[173,1055],[33,5125,1020],{"className":5126},[1060,1061],[33,5128],{"className":5129,"style":365},[364],[33,5131,276],{"className":5132},[369],[33,5134],{"className":5135,"style":365},[364],[33,5137,5139,5142,5145,5149,5189,5192,5195],{"className":5138},[75],[33,5140],{"className":5141,"style":4327},[79],[33,5143,270],{"className":5144},[84],[33,5146],{"className":5147,"style":5148},[364],"margin-right:2em;",[33,5150,5152,5155],{"className":5151},[84],[33,5153,4281],{"className":5154,"style":4334},[84,154],[33,5156,5158],{"className":5157},[1139],[33,5159,5161,5181],{"className":5160},[96,97],[33,5162,5164,5178],{"className":5163},[101],[33,5165,5167],{"className":5166,"style":4955},[105],[33,5168,5169,5172],{"style":4350},[33,5170],{"className":5171,"style":1156},[113],[33,5173,5175],{"className":5174},[118,119,120,121],[33,5176,270],{"className":5177},[84,121],[33,5179,160],{"className":5180},[159],[33,5182,5184],{"className":5183},[101],[33,5185,5187],{"className":5186,"style":4369},[105],[33,5188],{},[33,5190],{"className":5191,"style":365},[364],[33,5193,276],{"className":5194},[369],[33,5196],{"className":5197,"style":365},[364],[33,5199,5201,5204,5272,5338,5341,5344],{"className":5200},[75],[33,5202],{"className":5203,"style":4575},[79],[33,5205,5207,5210,5269],{"className":5206},[84],[33,5208],{"className":5209},[88,89],[33,5211,5213],{"className":5212},[52],[33,5214,5216,5261],{"className":5215},[96,97],[33,5217,5219,5258],{"className":5218},[101],[33,5220,5222,5236,5244],{"className":5221,"style":310},[105],[33,5223,5224,5227],{"style":109},[33,5225],{"className":5226,"style":114},[113],[33,5228,5230],{"className":5229},[118,119,120,121],[33,5231,5233],{"className":5232},[84,121],[33,5234,402],{"className":5235},[84,121],[33,5237,5238,5241],{"style":130},[33,5239],{"className":5240,"style":114},[113],[33,5242],{"className":5243,"style":138},[137],[33,5245,5246,5249],{"style":141},[33,5247],{"className":5248,"style":114},[113],[33,5250,5252],{"className":5251},[118,119,120,121],[33,5253,5255],{"className":5254},[84,121],[33,5256,270],{"className":5257},[84,121],[33,5259,160],{"className":5260},[159],[33,5262,5264],{"className":5263},[101],[33,5265,5267],{"className":5266,"style":167},[105],[33,5268],{},[33,5270],{"className":5271},[173,89],[33,5273,5275,5281,5332],{"className":5274},[84],[33,5276,5278],{"className":5277,"style":1056},[88,1055],[33,5279,1011],{"className":5280},[1060,1061],[33,5282,5284],{"className":5283},[52],[33,5285,5287,5324],{"className":5286},[96,97],[33,5288,5290,5321],{"className":5289},[101],[33,5291,5293,5307],{"className":5292,"style":4483},[105],[33,5294,5295,5298],{"style":4486},[33,5296],{"className":5297,"style":1156},[113],[33,5299,5301],{"className":5300},[118,119,120,121],[33,5302,5304],{"className":5303},[84,121],[33,5305,270],{"className":5306},[84,121],[33,5308,5309,5312],{"style":4501},[33,5310],{"className":5311,"style":1156},[113],[33,5313,5315],{"className":5314},[118,119,120,121],[33,5316,5318],{"className":5317},[84,121],[33,5319,402],{"className":5320},[84,121],[33,5322,160],{"className":5323},[159],[33,5325,5327],{"className":5326},[101],[33,5328,5330],{"className":5329,"style":167},[105],[33,5331],{},[33,5333,5335],{"className":5334,"style":1056},[173,1055],[33,5336,1020],{"className":5337},[1060,1061],[33,5339],{"className":5340,"style":365},[364],[33,5342,276],{"className":5343},[369],[33,5345],{"className":5346,"style":365},[364],[33,5348,5350,5353],{"className":5349},[75],[33,5351],{"className":5352,"style":379},[79],[33,5354,270],{"className":5355},[84],[15,5357,5359],{"className":5358},[575],[33,5360,5362,5437],{"className":5361},[36],[33,5363,5365],{"className":5364},[40],[42,5366,5367],{"xmlns":44},[46,5368,5369,5434],{},[49,5370,5371,5377,5379,5385,5397,5399,5401,5403,5409,5411,5417,5430,5432],{},[4277,5372,5373,5375],{},[55,5374,4281],{},[59,5376,402],{},[274,5378,276],{},[52,5380,5381,5383],{},[59,5382,270],{},[59,5384,910],{},[49,5386,5387,5389,5395],{},[274,5388,1011],{"fence":71},[52,5390,5391,5393],{"linethickness":4304},[59,5392,61],{},[59,5394,402],{},[274,5396,1020],{"fence":71},[274,5398,276],{},[59,5400,402],{},[364,5402],{"width":4894},[4277,5404,5405,5407],{},[55,5406,4281],{},[59,5408,910],{},[274,5410,276],{},[52,5412,5413,5415],{},[59,5414,270],{},[59,5416,61],{},[49,5418,5419,5421,5428],{},[274,5420,1011],{"fence":71},[52,5422,5423,5426],{"linethickness":4304},[59,5424,5425],{},"6",[59,5427,910],{},[274,5429,1020],{"fence":71},[274,5431,276],{},[59,5433,1826],{},[63,5435,5436],{"encoding":65},"C_2 = \\frac{1}{3}\\binom{4}{2} = 2 \\qquad C_3 = \\frac{1}{4}\\binom{6}{3} = 5 ",[33,5438,5440,5495,5644,5705,5854],{"className":5439,"ariaHidden":71},[70],[33,5441,5443,5446,5486,5489,5492],{"className":5442},[75],[33,5444],{"className":5445,"style":4327},[79],[33,5447,5449,5452],{"className":5448},[84],[33,5450,4281],{"className":5451,"style":4334},[84,154],[33,5453,5455],{"className":5454},[1139],[33,5456,5458,5478],{"className":5457},[96,97],[33,5459,5461,5475],{"className":5460},[101],[33,5462,5464],{"className":5463,"style":4955},[105],[33,5465,5466,5469],{"style":4350},[33,5467],{"className":5468,"style":1156},[113],[33,5470,5472],{"className":5471},[118,119,120,121],[33,5473,402],{"className":5474},[84,121],[33,5476,160],{"className":5477},[159],[33,5479,5481],{"className":5480},[101],[33,5482,5484],{"className":5483,"style":4369},[105],[33,5485],{},[33,5487],{"className":5488,"style":365},[364],[33,5490,276],{"className":5491},[369],[33,5493],{"className":5494,"style":365},[364],[33,5496,5498,5501,5569,5635,5638,5641],{"className":5497},[75],[33,5499],{"className":5500,"style":4575},[79],[33,5502,5504,5507,5566],{"className":5503},[84],[33,5505],{"className":5506},[88,89],[33,5508,5510],{"className":5509},[52],[33,5511,5513,5558],{"className":5512},[96,97],[33,5514,5516,5555],{"className":5515},[101],[33,5517,5519,5533,5541],{"className":5518,"style":310},[105],[33,5520,5521,5524],{"style":109},[33,5522],{"className":5523,"style":114},[113],[33,5525,5527],{"className":5526},[118,119,120,121],[33,5528,5530],{"className":5529},[84,121],[33,5531,910],{"className":5532},[84,121],[33,5534,5535,5538],{"style":130},[33,5536],{"className":5537,"style":114},[113],[33,5539],{"className":5540,"style":138},[137],[33,5542,5543,5546],{"style":141},[33,5544],{"className":5545,"style":114},[113],[33,5547,5549],{"className":5548},[118,119,120,121],[33,5550,5552],{"className":5551},[84,121],[33,5553,270],{"className":5554},[84,121],[33,5556,160],{"className":5557},[159],[33,5559,5561],{"className":5560},[101],[33,5562,5564],{"className":5563,"style":167},[105],[33,5565],{},[33,5567],{"className":5568},[173,89],[33,5570,5572,5578,5629],{"className":5571},[84],[33,5573,5575],{"className":5574,"style":1056},[88,1055],[33,5576,1011],{"className":5577},[1060,1061],[33,5579,5581],{"className":5580},[52],[33,5582,5584,5621],{"className":5583},[96,97],[33,5585,5587,5618],{"className":5586},[101],[33,5588,5590,5604],{"className":5589,"style":4483},[105],[33,5591,5592,5595],{"style":4486},[33,5593],{"className":5594,"style":1156},[113],[33,5596,5598],{"className":5597},[118,119,120,121],[33,5599,5601],{"className":5600},[84,121],[33,5602,402],{"className":5603},[84,121],[33,5605,5606,5609],{"style":4501},[33,5607],{"className":5608,"style":1156},[113],[33,5610,5612],{"className":5611},[118,119,120,121],[33,5613,5615],{"className":5614},[84,121],[33,5616,61],{"className":5617},[84,121],[33,5619,160],{"className":5620},[159],[33,5622,5624],{"className":5623},[101],[33,5625,5627],{"className":5626,"style":167},[105],[33,5628],{},[33,5630,5632],{"className":5631,"style":1056},[173,1055],[33,5633,1020],{"className":5634},[1060,1061],[33,5636],{"className":5637,"style":365},[364],[33,5639,276],{"className":5640},[369],[33,5642],{"className":5643,"style":365},[364],[33,5645,5647,5650,5653,5656,5696,5699,5702],{"className":5646},[75],[33,5648],{"className":5649,"style":4327},[79],[33,5651,402],{"className":5652},[84],[33,5654],{"className":5655,"style":5148},[364],[33,5657,5659,5662],{"className":5658},[84],[33,5660,4281],{"className":5661,"style":4334},[84,154],[33,5663,5665],{"className":5664},[1139],[33,5666,5668,5688],{"className":5667},[96,97],[33,5669,5671,5685],{"className":5670},[101],[33,5672,5674],{"className":5673,"style":4955},[105],[33,5675,5676,5679],{"style":4350},[33,5677],{"className":5678,"style":1156},[113],[33,5680,5682],{"className":5681},[118,119,120,121],[33,5683,910],{"className":5684},[84,121],[33,5686,160],{"className":5687},[159],[33,5689,5691],{"className":5690},[101],[33,5692,5694],{"className":5693,"style":4369},[105],[33,5695],{},[33,5697],{"className":5698,"style":365},[364],[33,5700,276],{"className":5701},[369],[33,5703],{"className":5704,"style":365},[364],[33,5706,5708,5711,5779,5845,5848,5851],{"className":5707},[75],[33,5709],{"className":5710,"style":4575},[79],[33,5712,5714,5717,5776],{"className":5713},[84],[33,5715],{"className":5716},[88,89],[33,5718,5720],{"className":5719},[52],[33,5721,5723,5768],{"className":5722},[96,97],[33,5724,5726,5765],{"className":5725},[101],[33,5727,5729,5743,5751],{"className":5728,"style":310},[105],[33,5730,5731,5734],{"style":109},[33,5732],{"className":5733,"style":114},[113],[33,5735,5737],{"className":5736},[118,119,120,121],[33,5738,5740],{"className":5739},[84,121],[33,5741,61],{"className":5742},[84,121],[33,5744,5745,5748],{"style":130},[33,5746],{"className":5747,"style":114},[113],[33,5749],{"className":5750,"style":138},[137],[33,5752,5753,5756],{"style":141},[33,5754],{"className":5755,"style":114},[113],[33,5757,5759],{"className":5758},[118,119,120,121],[33,5760,5762],{"className":5761},[84,121],[33,5763,270],{"className":5764},[84,121],[33,5766,160],{"className":5767},[159],[33,5769,5771],{"className":5770},[101],[33,5772,5774],{"className":5773,"style":167},[105],[33,5775],{},[33,5777],{"className":5778},[173,89],[33,5780,5782,5788,5839],{"className":5781},[84],[33,5783,5785],{"className":5784,"style":1056},[88,1055],[33,5786,1011],{"className":5787},[1060,1061],[33,5789,5791],{"className":5790},[52],[33,5792,5794,5831],{"className":5793},[96,97],[33,5795,5797,5828],{"className":5796},[101],[33,5798,5800,5814],{"className":5799,"style":4483},[105],[33,5801,5802,5805],{"style":4486},[33,5803],{"className":5804,"style":1156},[113],[33,5806,5808],{"className":5807},[118,119,120,121],[33,5809,5811],{"className":5810},[84,121],[33,5812,910],{"className":5813},[84,121],[33,5815,5816,5819],{"style":4501},[33,5817],{"className":5818,"style":1156},[113],[33,5820,5822],{"className":5821},[118,119,120,121],[33,5823,5825],{"className":5824},[84,121],[33,5826,5425],{"className":5827},[84,121],[33,5829,160],{"className":5830},[159],[33,5832,5834],{"className":5833},[101],[33,5835,5837],{"className":5836,"style":167},[105],[33,5838],{},[33,5840,5842],{"className":5841,"style":1056},[173,1055],[33,5843,1020],{"className":5844},[1060,1061],[33,5846],{"className":5847,"style":365},[364],[33,5849,276],{"className":5850},[369],[33,5852],{"className":5853,"style":365},[364],[33,5855,5857,5860],{"className":5856},[75],[33,5858],{"className":5859,"style":379},[79],[33,5861,1826],{"className":5862},[84],[15,5864,5866],{"className":5865},[575],[33,5867,5869,5911],{"className":5868},[36],[33,5870,5872],{"className":5871},[40],[42,5873,5874],{"xmlns":44},[46,5875,5876,5908],{},[49,5877,5878,5884,5886,5892,5904,5906],{},[4277,5879,5880,5882],{},[55,5881,4281],{},[59,5883,61],{},[274,5885,276],{},[52,5887,5888,5890],{},[59,5889,270],{},[59,5891,1826],{},[49,5893,5894,5896,5902],{},[274,5895,1011],{"fence":71},[52,5897,5898,5900],{"linethickness":4304},[59,5899,1031],{},[59,5901,61],{},[274,5903,1020],{"fence":71},[274,5905,276],{},[59,5907,3430],{},[63,5909,5910],{"encoding":65},"C_4 = \\frac{1}{5}\\binom{8}{4} = 14",[33,5912,5914,5969,6118],{"className":5913,"ariaHidden":71},[70],[33,5915,5917,5920,5960,5963,5966],{"className":5916},[75],[33,5918],{"className":5919,"style":4327},[79],[33,5921,5923,5926],{"className":5922},[84],[33,5924,4281],{"className":5925,"style":4334},[84,154],[33,5927,5929],{"className":5928},[1139],[33,5930,5932,5952],{"className":5931},[96,97],[33,5933,5935,5949],{"className":5934},[101],[33,5936,5938],{"className":5937,"style":4955},[105],[33,5939,5940,5943],{"style":4350},[33,5941],{"className":5942,"style":1156},[113],[33,5944,5946],{"className":5945},[118,119,120,121],[33,5947,61],{"className":5948},[84,121],[33,5950,160],{"className":5951},[159],[33,5953,5955],{"className":5954},[101],[33,5956,5958],{"className":5957,"style":4369},[105],[33,5959],{},[33,5961],{"className":5962,"style":365},[364],[33,5964,276],{"className":5965},[369],[33,5967],{"className":5968,"style":365},[364],[33,5970,5972,5975,6043,6109,6112,6115],{"className":5971},[75],[33,5973],{"className":5974,"style":4575},[79],[33,5976,5978,5981,6040],{"className":5977},[84],[33,5979],{"className":5980},[88,89],[33,5982,5984],{"className":5983},[52],[33,5985,5987,6032],{"className":5986},[96,97],[33,5988,5990,6029],{"className":5989},[101],[33,5991,5993,6007,6015],{"className":5992,"style":310},[105],[33,5994,5995,5998],{"style":109},[33,5996],{"className":5997,"style":114},[113],[33,5999,6001],{"className":6000},[118,119,120,121],[33,6002,6004],{"className":6003},[84,121],[33,6005,1826],{"className":6006},[84,121],[33,6008,6009,6012],{"style":130},[33,6010],{"className":6011,"style":114},[113],[33,6013],{"className":6014,"style":138},[137],[33,6016,6017,6020],{"style":141},[33,6018],{"className":6019,"style":114},[113],[33,6021,6023],{"className":6022},[118,119,120,121],[33,6024,6026],{"className":6025},[84,121],[33,6027,270],{"className":6028},[84,121],[33,6030,160],{"className":6031},[159],[33,6033,6035],{"className":6034},[101],[33,6036,6038],{"className":6037,"style":167},[105],[33,6039],{},[33,6041],{"className":6042},[173,89],[33,6044,6046,6052,6103],{"className":6045},[84],[33,6047,6049],{"className":6048,"style":1056},[88,1055],[33,6050,1011],{"className":6051},[1060,1061],[33,6053,6055],{"className":6054},[52],[33,6056,6058,6095],{"className":6057},[96,97],[33,6059,6061,6092],{"className":6060},[101],[33,6062,6064,6078],{"className":6063,"style":4483},[105],[33,6065,6066,6069],{"style":4486},[33,6067],{"className":6068,"style":1156},[113],[33,6070,6072],{"className":6071},[118,119,120,121],[33,6073,6075],{"className":6074},[84,121],[33,6076,61],{"className":6077},[84,121],[33,6079,6080,6083],{"style":4501},[33,6081],{"className":6082,"style":1156},[113],[33,6084,6086],{"className":6085},[118,119,120,121],[33,6087,6089],{"className":6088},[84,121],[33,6090,1031],{"className":6091},[84,121],[33,6093,160],{"className":6094},[159],[33,6096,6098],{"className":6097},[101],[33,6099,6101],{"className":6100,"style":167},[105],[33,6102],{},[33,6104,6106],{"className":6105,"style":1056},[173,1055],[33,6107,1020],{"className":6108},[1060,1061],[33,6110],{"className":6111,"style":365},[364],[33,6113,276],{"className":6114},[369],[33,6116],{"className":6117,"style":365},[364],[33,6119,6121,6124],{"className":6120},[75],[33,6122],{"className":6123,"style":379},[79],[33,6125,3430],{"className":6126},[84],[15,6128,6129,6130,6133],{},"A perfect match. The Catalan numbers count precisely how many flip sequences end ",[237,6131,6132],{},"for the first time"," with more heads than tails at each possible length.",[544,6135,6137],{"id":6136},"the-connection-to-pascals-triangle","The connection to Pascal's triangle",[15,6139,6140,6141,6144,6145,6200,6201,6330,6331,6335,6336,174],{},"There is another way to see why the Catalan numbers appear here. Think of each flip sequence as a ",[237,6142,6143],{},"path"," on a lattice: heads moves one step up, tails moves one step down. The total number of paths of length ",[33,6146,6148,6167],{"className":6147},[36],[33,6149,6151],{"className":6150},[40],[42,6152,6153],{"xmlns":44},[46,6154,6155,6165],{},[49,6156,6157,6159,6161,6163],{},[59,6158,402],{},[55,6160,2184],{},[274,6162,1299],{},[59,6164,270],{},[63,6166,2191],{"encoding":65},[33,6168,6170,6191],{"className":6169,"ariaHidden":71},[70],[33,6171,6173,6176,6179,6182,6185,6188],{"className":6172},[75],[33,6174],{"className":6175,"style":1314},[79],[33,6177,402],{"className":6178},[84],[33,6180,2184],{"className":6181},[84,154],[33,6183],{"className":6184,"style":2210},[364],[33,6186,1299],{"className":6187},[2214],[33,6189],{"className":6190,"style":2210},[364],[33,6192,6194,6197],{"className":6193},[75],[33,6195],{"className":6196,"style":379},[79],[33,6198,270],{"className":6199},[84]," that end one step above zero is the binomial coefficient ",[33,6202,6204,6240],{"className":6203},[36],[33,6205,6207],{"className":6206},[40],[42,6208,6209],{"xmlns":44},[46,6210,6211,6237],{},[49,6212,6213,6215,6235],{},[274,6214,1011],{"fence":71},[52,6216,6217,6227],{"linethickness":4304},[49,6218,6219,6221,6223,6225],{},[59,6220,402],{},[55,6222,2184],{},[274,6224,1299],{},[59,6226,270],{},[49,6228,6229,6231,6233],{},[55,6230,2184],{},[274,6232,1299],{},[59,6234,270],{},[274,6236,1020],{"fence":71},[63,6238,6239],{"encoding":65},"\\binom{2n+1}{n+1}",[33,6241,6243],{"className":6242,"ariaHidden":71},[70],[33,6244,6246,6249],{"className":6245},[75],[33,6247],{"className":6248,"style":4387},[79],[33,6250,6252,6258,6324],{"className":6251},[84],[33,6253,6255],{"className":6254,"style":1056},[88,1055],[33,6256,1011],{"className":6257},[1060,1061],[33,6259,6261],{"className":6260},[52],[33,6262,6264,6316],{"className":6263},[96,97],[33,6265,6267,6313],{"className":6266},[101],[33,6268,6270,6290],{"className":6269,"style":4483},[105],[33,6271,6272,6275],{"style":4486},[33,6273],{"className":6274,"style":1156},[113],[33,6276,6278],{"className":6277},[118,119,120,121],[33,6279,6281,6284,6287],{"className":6280},[84,121],[33,6282,2184],{"className":6283},[84,154,121],[33,6285,1299],{"className":6286},[2214,121],[33,6288,270],{"className":6289},[84,121],[33,6291,6292,6295],{"style":4501},[33,6293],{"className":6294,"style":1156},[113],[33,6296,6298],{"className":6297},[118,119,120,121],[33,6299,6301,6304,6307,6310],{"className":6300},[84,121],[33,6302,402],{"className":6303},[84,121],[33,6305,2184],{"className":6306},[84,154,121],[33,6308,1299],{"className":6309},[2214,121],[33,6311,270],{"className":6312},[84,121],[33,6314,160],{"className":6315},[159],[33,6317,6319],{"className":6318},[101],[33,6320,6322],{"className":6321,"style":3723},[105],[33,6323],{},[33,6325,6327],{"className":6326,"style":1056},[173,1055],[33,6328,1020],{"className":6329},[1060,1061]," — 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 ",[19,6332,6334],{"href":219,"rel":6333},[23],"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 ",[19,6337,6339],{"href":28,"rel":6338},[23],"his video",[544,6341,6343],{"id":6342},"the-infinite-series","The infinite series",[15,6345,6346,6347,6415],{},"Back to our table. The contribution of 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sequences to the overall average 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infinity:",[15,7225,7227],{"className":7226},[575],[33,7228,7230,7310],{"className":7229},[36],[33,7231,7233],{"className":7232},[40],[42,7234,7235],{"xmlns":44},[46,7236,7237,7307],{},[49,7238,7239,7242,7244,7262,7284,7287],{},[589,7240,7241],{},"Average",[274,7243,276],{},[7245,7246,7247,7250,7258],"msubsup",{},[274,7248,7249],{},"∑",[49,7251,7252,7254,7256],{},[55,7253,2184],{},[274,7255,276],{},[59,7257,1401],{},[55,7259,7261],{"mathvariant":7260},"normal","∞",[52,7263,7264,7270],{},[4277,7265,7266,7268],{},[55,7267,4281],{},[55,7269,2184],{},[1005,7271,7272,7274],{},[59,7273,402],{},[49,7275,7276,7278,7280,7282],{},[59,7277,402],{},[55,7279,2184],{},[274,7281,1299],{},[59,7283,270],{},[274,7285,7286],{},"⋅",[52,7288,7289,7297],{},[49,7290,7291,7293,7295],{},[55,7292,2184],{},[274,7294,1299],{},[59,7296,270],{},[49,7298,7299,7301,7303,7305],{},[59,7300,402],{},[55,7302,2184],{},[274,7304,1299],{},[59,7306,270],{},[63,7308,7309],{"encoding":65},"\\text{Average} = \\sum_{n=0}^{\\infty} \\frac{C_n}{2^{2n+1}} \\cdot \\frac{n+1}{2n+1}",[33,7311,7313,7335,7577],{"className":7312,"ariaHidden":71},[70],[33,7314,7316,7320,7326,7329,7332],{"className":7315},[75],[33,7317],{"className":7318,"style":7319},[79],"height:0.8778em;vertical-align:-0.1944em;",[33,7321,7323],{"className":7322},[84,608],[33,7324,7241],{"className":7325},[84],[33,7327],{"className":7328,"style":365},[364],[33,7330,276],{"className":7331},[369],[33,7333],{"className":7334,"style":365},[364],[33,7336,7338,7342,7413,7416,7568,7571,7574],{"className":7337},[75],[33,7339],{"className":7340,"style":7341},[79],"height:1.2334em;vertical-align:-0.345em;",[33,7343,7346,7352],{"className":7344},[7345],"mop",[33,7347,7249],{"className":7348,"style":7351},[7345,7349,7350],"op-symbol","small-op","position:relative;top:0em;",[33,7353,7355],{"className":7354},[1139],[33,7356,7358,7404],{"className":7357},[96,97],[33,7359,7361,7401],{"className":7360},[101],[33,7362,7365,7386],{"className":7363,"style":7364},[105],"height:0.8043em;",[33,7366,7368,7371],{"style":7367},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[33,7369],{"className":7370,"style":1156},[113],[33,7372,7374],{"className":7373},[118,119,120,121],[33,7375,7377,7380,7383],{"className":7376},[84,121],[33,7378,2184],{"className":7379},[84,154,121],[33,7381,276],{"className":7382},[369,121],[33,7384,1401],{"className":7385},[84,121],[33,7387,7389,7392],{"style":7388},"top:-3.2029em;margin-right:0.05em;",[33,7390],{"className":7391,"style":1156},[113],[33,7393,7395],{"className":7394},[118,119,120,121],[33,7396,7398],{"className":7397},[84,121],[33,7399,7261],{"className":7400},[84,121],[33,7402,160],{"className":7403},[159],[33,7405,7407],{"className":7406},[101],[33,7408,7411],{"className":7409,"style":7410},[105],"height:0.2997em;",[33,7412],{},[33,7414],{"className":7415,"style":1439},[364],[33,7417,7419,7422,7565],{"className":7418},[84],[33,7420],{"className":7421},[88,89],[33,7423,7425],{"className":7424},[52],[33,7426,7428,7557],{"className":7427},[96,97],[33,7429,7431,7554],{"className":7430},[101],[33,7432,7435,7491,7499],{"className":7433,"style":7434},[105],"height:0.8884em;",[33,7436,7437,7440],{"style":109},[33,7438],{"className":7439,"style":114},[113],[33,7441,7443],{"className":7442},[118,119,120,121],[33,7444,7446],{"className":7445},[84,121],[33,7447,7449,7452],{"className":7448},[84,121],[33,7450,402],{"className":7451},[84,121],[33,7453,7455],{"className":7454},[1139],[33,7456,7458],{"className":7457},[96],[33,7459,7461],{"className":7460},[101],[33,7462,7465],{"className":7463,"style":7464},[105],"height:0.7463em;",[33,7466,7468,7472],{"style":7467},"top:-2.786em;margin-right:0.0714em;",[33,7469],{"className":7470,"style":7471},[113],"height:2.5em;",[33,7473,7476],{"className":7474},[118,7475,1061,121],"reset-size3",[33,7477,7479,7482,7485,7488],{"className":7478},[84,121],[33,7480,402],{"className":7481},[84,121],[33,7483,2184],{"className":7484},[84,154,121],[33,7486,1299],{"className":7487},[2214,121],[33,7489,270],{"className":7490},[84,121],[33,7492,7493,7496],{"style":130},[33,7494],{"className":7495,"style":114},[113],[33,7497],{"className":7498,"style":138},[137],[33,7500,7502,7505],{"style":7501},"top:-3.4101em;",[33,7503],{"className":7504,"style":114},[113],[33,7506,7508],{"className":7507},[118,119,120,121],[33,7509,7511],{"className":7510},[84,121],[33,7512,7514,7517],{"className":7513},[84,121],[33,7515,4281],{"className":7516,"style":4334},[84,154,121],[33,7518,7520],{"className":7519},[1139],[33,7521,7523,7545],{"className":7522},[96,97],[33,7524,7526,7542],{"className":7525},[101],[33,7527,7530],{"className":7528,"style":7529},[105],"height:0.1645em;",[33,7531,7533,7536],{"style":7532},"top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;",[33,7534],{"className":7535,"style":7471},[113],[33,7537,7539],{"className":7538},[118,7475,1061,121],[33,7540,2184],{"className":7541},[84,154,121],[33,7543,160],{"className":7544},[159],[33,7546,7548],{"className":7547},[101],[33,7549,7552],{"className":7550,"style":7551},[105],"height:0.143em;",[33,7553],{},[33,7555,160],{"className":7556},[159],[33,7558,7560],{"className":7559},[101],[33,7561,7563],{"className":7562,"style":167},[105],[33,7564],{},[33,7566],{"className":7567},[173,89],[33,7569],{"className":7570,"style":2210},[364],[33,7572,7286],{"className":7573},[2214],[33,7575],{"className":7576,"style":2210},[364],[33,7578,7580,7583],{"className":7579},[75],[33,7581],{"className":7582,"style":3644},[79],[33,7584,7586,7589,7663],{"className":7585},[84],[33,7587],{"className":7588},[88,89],[33,7590,7592],{"className":7591},[52],[33,7593,7595,7655],{"className":7594},[96,97],[33,7596,7598,7652],{"className":7597},[101],[33,7599,7601,7624,7632],{"className":7600,"style":310},[105],[33,7602,7603,7606],{"style":109},[33,7604],{"className":7605,"style":114},[113],[33,7607,7609],{"className":7608},[118,119,120,121],[33,7610,7612,7615,7618,7621],{"className":7611},[84,121],[33,7613,402],{"className":7614},[84,121],[33,7616,2184],{"className":7617},[84,154,121],[33,7619,1299],{"className":7620},[2214,121],[33,7622,270],{"className":7623},[84,121],[33,7625,7626,7629],{"style":130},[33,7627],{"className":7628,"style":114},[113],[33,7630],{"className":7631,"style":138},[137],[33,7633,7634,7637],{"style":141},[33,7635],{"className":7636,"style":114},[113],[33,7638,7640],{"className":7639},[118,119,120,121],[33,7641,7643,7646,7649],{"className":7642},[84,121],[33,7644,2184],{"className":7645},[84,154,121],[33,7647,1299],{"className":7648},[2214,121],[33,7650,270],{"className":7651},[84,121],[33,7653,160],{"className":7654},[159],[33,7656,7658],{"className":7657},[101],[33,7659,7661],{"className":7660,"style":3723},[105],[33,7662],{},[33,7664],{"className":7665},[173,89],[15,7667,7668],{},"Let's compute the first few terms:",[2155,7670,7671,7712],{},[2158,7672,7673],{},[2161,7674,7675,7705,7709],{},[2164,7676,7677],{"align":2166},[33,7678,7680,7693],{"className":7679},[36],[33,7681,7683],{"className":7682},[40],[42,7684,7685],{"xmlns":44},[46,7686,7687,7691],{},[49,7688,7689],{},[55,7690,2184],{},[63,7692,2184],{"encoding":65},[33,7694,7696],{"className":7695,"ariaHidden":71},[70],[33,7697,7699,7702],{"className":7698},[75],[33,7700],{"className":7701,"style":204},[79],[33,7703,2184],{"className":7704},[84,154],[2164,7706,7708],{"align":7707},"left","Contribution",[2164,7710,7711],{"align":7707},"Running total",[2237,7713,7714,7915,8187,8458,8729],{},[2161,7715,7716,7718,7884],{},[2242,7717,1401],{"align":2166},[2242,7719,7720],{"align":7707},[33,7721,7723,7753],{"className":7722},[36],[33,7724,7726],{"className":7725},[40],[42,7727,7728],{"xmlns":44},[46,7729,7730,7750],{},[49,7731,7732,7738,7740,7742,7744,7746,7748],{},[52,7733,7734,7736],{},[59,7735,270],{},[59,7737,402],{},[274,7739,6468],{},[59,7741,270],{},[274,7743,6468],{},[59,7745,270],{},[274,7747,276],{},[59,7749,498],{},[63,7751,7752],{"encoding":65},"\\frac{1}{2} \\times 1 \\times 1 = 0.5",[33,7754,7756,7839,7857,7875],{"className":7755,"ariaHidden":71},[70],[33,7757,7759,7762,7830,7833,7836],{"className":7758},[75],[33,7760],{"className":7761,"style":291},[79],[33,7763,7765,7768,7827],{"className":7764},[84],[33,7766],{"className":7767},[88,89],[33,7769,7771],{"className":7770},[52],[33,7772,7774,7819],{"className":7773},[96,97],[33,7775,7777,7816],{"className":7776},[101],[33,7778,7780,7794,7802],{"className":7779,"style":310},[105],[33,7781,7782,7785],{"style":109},[33,7783],{"className":7784,"style":114},[113],[33,7786,7788],{"className":7787},[118,119,120,121],[33,7789,7791],{"className":7790},[84,121],[33,7792,402],{"className":7793},[84,121],[33,7795,7796,7799],{"style":130},[33,7797],{"className":7798,"style":114},[113],[33,7800],{"className":7801,"style":138},[137],[33,7803,7804,7807],{"style":141},[33,7805],{"className":7806,"style":114},[113],[33,7808,7810],{"className":7809},[118,119,120,121],[33,7811,7813],{"className":7812},[84,121],[33,7814,270],{"className":7815},[84,121],[33,7817,160],{"className":7818},[159],[33,7820,7822],{"className":7821},[101],[33,7823,7825],{"className":7824,"style":167},[105],[33,7826],{},[33,7828],{"className":7829},[173,89],[33,7831],{"className":7832,"style":2210},[364],[33,7834,6468],{"className":7835},[2214],[33,7837],{"className":7838,"style":2210},[364],[33,7840,7842,7845,7848,7851,7854],{"className":7841},[75],[33,7843],{"className":7844,"style":1314},[79],[33,7846,270],{"className":7847},[84],[33,7849],{"className":7850,"style":2210},[364],[33,7852,6468],{"className":7853},[2214],[33,7855],{"className":7856,"style":2210},[364],[33,7858,7860,7863,7866,7869,7872],{"className":7859},[75],[33,7861],{"className":7862,"style":379},[79],[33,7864,270],{"className":7865},[84],[33,7867],{"className":7868,"style":365},[364],[33,7870,276],{"className":7871},[369],[33,7873],{"className":7874,"style":365},[364],[33,7876,7878,7881],{"className":7877},[75],[33,7879],{"className":7880,"style":379},[79],[33,7882,498],{"className":7883},[84],[2242,7885,7886],{"align":7707},[33,7887,7889,7903],{"className":7888},[36],[33,7890,7892],{"className":7891},[40],[42,7893,7894],{"xmlns":44},[46,7895,7896,7901],{},[49,7897,7898],{},[59,7899,7900],{},"0.5000",[63,7902,7900],{"encoding":65},[33,7904,7906],{"className":7905,"ariaHidden":71},[70],[33,7907,7909,7912],{"className":7908},[75],[33,7910],{"className":7911,"style":379},[79],[33,7913,7900],{"className":7914},[84],[2161,7916,7917,7919,8156],{},[2242,7918,270],{"align":2166},[2242,7920,7921],{"align":7707},[33,7922,7924,7960],{"className":7923},[36],[33,7925,7927],{"className":7926},[40],[42,7928,7929],{"xmlns":44},[46,7930,7931,7957],{},[49,7932,7933,7939,7941,7943,7945,7951,7954],{},[52,7934,7935,7937],{},[59,7936,270],{},[59,7938,1031],{},[274,7940,6468],{},[59,7942,270],{},[274,7944,6468],{},[52,7946,7947,7949],{},[59,7948,402],{},[59,7950,910],{},[274,7952,7953],{},"≈",[59,7955,7956],{},"0.0833",[63,7958,7959],{"encoding":65},"\\frac{1}{8} \\times 1 \\times \\frac{2}{3} \\approx 0.0833",[33,7961,7963,8046,8064,8147],{"className":7962,"ariaHidden":71},[70],[33,7964,7966,7969,8037,8040,8043],{"className":7965},[75],[33,7967],{"className":7968,"style":291},[79],[33,7970,7972,7975,8034],{"className":7971},[84],[33,7973],{"className":7974},[88,89],[33,7976,7978],{"className":7977},[52],[33,7979,7981,8026],{"className":7980},[96,97],[33,7982,7984,8023],{"className":7983},[101],[33,7985,7987,8001,8009],{"className":7986,"style":310},[105],[33,7988,7989,7992],{"style":109},[33,7990],{"className":7991,"style":114},[113],[33,7993,7995],{"className":7994},[118,119,120,121],[33,7996,7998],{"className":7997},[84,121],[33,7999,1031],{"className":8000},[84,121],[33,8002,8003,8006],{"style":130},[33,8004],{"className":8005,"style":114},[113],[33,8007],{"className":8008,"style":138},[137],[33,8010,8011,8014],{"style":141},[33,8012],{"className":8013,"style":114},[113],[33,8015,8017],{"className":8016},[118,119,120,121],[33,8018,8020],{"className":8019},[84,121],[33,8021,270],{"className":8022},[84,121],[33,8024,160],{"className":8025},[159],[33,8027,8029],{"className":8028},[101],[33,8030,8032],{"className":8031,"style":167},[105],[33,8033],{},[33,8035],{"className":8036},[173,89],[33,8038],{"className":8039,"style":2210},[364],[33,8041,6468],{"className":8042},[2214],[33,8044],{"className":8045,"style":2210},[364],[33,8047,8049,8052,8055,8058,8061],{"className":8048},[75],[33,8050],{"className":8051,"style":1314},[79],[33,8053,270],{"className":8054},[84],[33,8056],{"className":8057,"style":2210},[364],[33,8059,6468],{"className":8060},[2214],[33,8062],{"className":8063,"style":2210},[364],[33,8065,8067,8070,8138,8141,8144],{"className":8066},[75],[33,8068],{"className":8069,"style":291},[79],[33,8071,8073,8076,8135],{"className":8072},[84],[33,8074],{"className":8075},[88,89],[33,8077,8079],{"className":8078},[52],[33,8080,8082,8127],{"className":8081},[96,97],[33,8083,8085,8124],{"className":8084},[101],[33,8086,8088,8102,8110],{"className":8087,"style":310},[105],[33,8089,8090,8093],{"style":109},[33,8091],{"className":8092,"style":114},[113],[33,8094,8096],{"className":8095},[118,119,120,121],[33,8097,8099],{"className":8098},[84,121],[33,8100,910],{"className":8101},[84,121],[33,8103,8104,8107],{"style":130},[33,8105],{"className":8106,"style":114},[113],[33,8108],{"className":8109,"style":138},[137],[33,8111,8112,8115],{"style":141},[33,8113],{"className":8114,"style":114},[113],[33,8116,8118],{"className":8117},[118,119,120,121],[33,8119,8121],{"className":8120},[84,121],[33,8122,402],{"className":8123},[84,121],[33,8125,160],{"className":8126},[159],[33,8128,8130],{"className":8129},[101],[33,8131,8133],{"className":8132,"style":167},[105],[33,8134],{},[33,8136],{"className":8137},[173,89],[33,8139],{"className":8140,"style":365},[364],[33,8142,7953],{"className":8143},[369],[33,8145],{"className":8146,"style":365},[364],[33,8148,8150,8153],{"className":8149},[75],[33,8151],{"className":8152,"style":379},[79],[33,8154,7956],{"className":8155},[84],[2242,8157,8158],{"align":7707},[33,8159,8161,8175],{"className":8160},[36],[33,8162,8164],{"className":8163},[40],[42,8165,8166],{"xmlns":44},[46,8167,8168,8173],{},[49,8169,8170],{},[59,8171,8172],{},"0.5833",[63,8174,8172],{"encoding":65},[33,8176,8178],{"className":8177,"ariaHidden":71},[70],[33,8179,8181,8184],{"className":8180},[75],[33,8182],{"className":8183,"style":379},[79],[33,8185,8172],{"className":8186},[84],[2161,8188,8189,8191,8427],{},[2242,8190,402],{"align":2166},[2242,8192,8193],{"align":7707},[33,8194,8196,8231],{"className":8195},[36],[33,8197,8199],{"className":8198},[40],[42,8200,8201],{"xmlns":44},[46,8202,8203,8228],{},[49,8204,8205,8211,8213,8215,8217,8223,8225],{},[52,8206,8207,8209],{},[59,8208,270],{},[59,8210,1835],{},[274,8212,6468],{},[59,8214,402],{},[274,8216,6468],{},[52,8218,8219,8221],{},[59,8220,910],{},[59,8222,1826],{},[274,8224,276],{},[59,8226,8227],{},"0.0375",[63,8229,8230],{"encoding":65},"\\frac{1}{32} \\times 2 \\times \\frac{3}{5} = 0.0375",[33,8232,8234,8317,8335,8418],{"className":8233,"ariaHidden":71},[70],[33,8235,8237,8240,8308,8311,8314],{"className":8236},[75],[33,8238],{"className":8239,"style":291},[79],[33,8241,8243,8246,8305],{"className":8242},[84],[33,8244],{"className":8245},[88,89],[33,8247,8249],{"className":8248},[52],[33,8250,8252,8297],{"className":8251},[96,97],[33,8253,8255,8294],{"className":8254},[101],[33,8256,8258,8272,8280],{"className":8257,"style":310},[105],[33,8259,8260,8263],{"style":109},[33,8261],{"className":8262,"style":114},[113],[33,8264,8266],{"className":8265},[118,119,120,121],[33,8267,8269],{"className":8268},[84,121],[33,8270,1835],{"className":8271},[84,121],[33,8273,8274,8277],{"style":130},[33,8275],{"className":8276,"style":114},[113],[33,8278],{"className":8279,"style":138},[137],[33,8281,8282,8285],{"style":141},[33,8283],{"className":8284,"style":114},[113],[33,8286,8288],{"className":8287},[118,119,120,121],[33,8289,8291],{"className":8290},[84,121],[33,8292,270],{"className":8293},[84,121],[33,8295,160],{"className":8296},[159],[33,8298,8300],{"className":8299},[101],[33,8301,8303],{"className":8302,"style":167},[105],[33,8304],{},[33,8306],{"className":8307},[173,89],[33,8309],{"className":8310,"style":2210},[364],[33,8312,6468],{"className":8313},[2214],[33,8315],{"className":8316,"style":2210},[364],[33,8318,8320,8323,8326,8329,8332],{"className":8319},[75],[33,8321],{"className":8322,"style":1314},[79],[33,8324,402],{"className":8325},[84],[33,8327],{"className":8328,"style":2210},[364],[33,8330,6468],{"className":8331},[2214],[33,8333],{"className":8334,"style":2210},[364],[33,8336,8338,8341,8409,8412,8415],{"className":8337},[75],[33,8339],{"className":8340,"style":291},[79],[33,8342,8344,8347,8406],{"className":8343},[84],[33,8345],{"className":8346},[88,89],[33,8348,8350],{"className":8349},[52],[33,8351,8353,8398],{"className":8352},[96,97],[33,8354,8356,8395],{"className":8355},[101],[33,8357,8359,8373,8381],{"className":8358,"style":310},[105],[33,8360,8361,8364],{"style":109},[33,8362],{"className":8363,"style":114},[113],[33,8365,8367],{"className":8366},[118,119,120,121],[33,8368,8370],{"className":8369},[84,121],[33,8371,1826],{"className":8372},[84,121],[33,8374,8375,8378],{"style":130},[33,8376],{"className":8377,"style":114},[113],[33,8379],{"className":8380,"style":138},[137],[33,8382,8383,8386],{"style":141},[33,8384],{"className":8385,"style":114},[113],[33,8387,8389],{"className":8388},[118,119,120,121],[33,8390,8392],{"className":8391},[84,121],[33,8393,910],{"className":8394},[84,121],[33,8396,160],{"className":8397},[159],[33,8399,8401],{"className":8400},[101],[33,8402,8404],{"className":8403,"style":167},[105],[33,8405],{},[33,8407],{"className":8408},[173,89],[33,8410],{"className":8411,"style":365},[364],[33,8413,276],{"className":8414},[369],[33,8416],{"className":8417,"style":365},[364],[33,8419,8421,8424],{"className":8420},[75],[33,8422],{"className":8423,"style":379},[79],[33,8425,8227],{"className":8426},[84],[2242,8428,8429],{"align":7707},[33,8430,8432,8446],{"className":8431},[36],[33,8433,8435],{"className":8434},[40],[42,8436,8437],{"xmlns":44},[46,8438,8439,8444],{},[49,8440,8441],{},[59,8442,8443],{},"0.6208",[63,8445,8443],{"encoding":65},[33,8447,8449],{"className":8448,"ariaHidden":71},[70],[33,8450,8452,8455],{"className":8451},[75],[33,8453],{"className":8454,"style":379},[79],[33,8456,8443],{"className":8457},[84],[2161,8459,8460,8462,8698],{},[2242,8461,910],{"align":2166},[2242,8463,8464],{"align":7707},[33,8465,8467,8502],{"className":8466},[36],[33,8468,8470],{"className":8469},[40],[42,8471,8472],{"xmlns":44},[46,8473,8474,8499],{},[49,8475,8476,8482,8484,8486,8488,8494,8496],{},[52,8477,8478,8480],{},[59,8479,270],{},[59,8481,3190],{},[274,8483,6468],{},[59,8485,1826],{},[274,8487,6468],{},[52,8489,8490,8492],{},[59,8491,61],{},[59,8493,3087],{},[274,8495,7953],{},[59,8497,8498],{},"0.0223",[63,8500,8501],{"encoding":65},"\\frac{1}{128} \\times 5 \\times \\frac{4}{7} \\approx 0.0223",[33,8503,8505,8588,8606,8689],{"className":8504,"ariaHidden":71},[70],[33,8506,8508,8511,8579,8582,8585],{"className":8507},[75],[33,8509],{"className":8510,"style":291},[79],[33,8512,8514,8517,8576],{"className":8513},[84],[33,8515],{"className":8516},[88,89],[33,8518,8520],{"className":8519},[52],[33,8521,8523,8568],{"className":8522},[96,97],[33,8524,8526,8565],{"className":8525},[101],[33,8527,8529,8543,8551],{"className":8528,"style":310},[105],[33,8530,8531,8534],{"style":109},[33,8532],{"className":8533,"style":114},[113],[33,8535,8537],{"className":8536},[118,119,120,121],[33,8538,8540],{"className":8539},[84,121],[33,8541,3190],{"className":8542},[84,121],[33,8544,8545,8548],{"style":130},[33,8546],{"className":8547,"style":114},[113],[33,8549],{"className":8550,"style":138},[137],[33,8552,8553,8556],{"style":141},[33,8554],{"className":8555,"style":114},[113],[33,8557,8559],{"className":8558},[118,119,120,121],[33,8560,8562],{"className":8561},[84,121],[33,8563,270],{"className":8564},[84,121],[33,8566,160],{"className":8567},[159],[33,8569,8571],{"className":8570},[101],[33,8572,8574],{"className":8573,"style":167},[105],[33,8575],{},[33,8577],{"className":8578},[173,89],[33,8580],{"className":8581,"style":2210},[364],[33,8583,6468],{"className":8584},[2214],[33,8586],{"className":8587,"style":2210},[364],[33,8589,8591,8594,8597,8600,8603],{"className":8590},[75],[33,8592],{"className":8593,"style":1314},[79],[33,8595,1826],{"className":8596},[84],[33,8598],{"className":8599,"style":2210},[364],[33,8601,6468],{"className":8602},[2214],[33,8604],{"className":8605,"style":2210},[364],[33,8607,8609,8612,8680,8683,8686],{"className":8608},[75],[33,8610],{"className":8611,"style":291},[79],[33,8613,8615,8618,8677],{"className":8614},[84],[33,8616],{"className":8617},[88,89],[33,8619,8621],{"className":8620},[52],[33,8622,8624,8669],{"className":8623},[96,97],[33,8625,8627,8666],{"className":8626},[101],[33,8628,8630,8644,8652],{"className":8629,"style":310},[105],[33,8631,8632,8635],{"style":109},[33,8633],{"className":8634,"style":114},[113],[33,8636,8638],{"className":8637},[118,119,120,121],[33,8639,8641],{"className":8640},[84,121],[33,8642,3087],{"className":8643},[84,121],[33,8645,8646,8649],{"style":130},[33,8647],{"className":8648,"style":114},[113],[33,8650],{"className":8651,"style":138},[137],[33,8653,8654,8657],{"style":141},[33,8655],{"className":8656,"style":114},[113],[33,8658,8660],{"className":8659},[118,119,120,121],[33,8661,8663],{"className":8662},[84,121],[33,8664,61],{"className":8665},[84,121],[33,8667,160],{"className":8668},[159],[33,8670,8672],{"className":8671},[101],[33,8673,8675],{"className":8674,"style":167},[105],[33,8676],{},[33,8678],{"className":8679},[173,89],[33,8681],{"className":8682,"style":365},[364],[33,8684,7953],{"className":8685},[369],[33,8687],{"className":8688,"style":365},[364],[33,8690,8692,8695],{"className":8691},[75],[33,8693],{"className":8694,"style":379},[79],[33,8696,8498],{"className":8697},[84],[2242,8699,8700],{"align":7707},[33,8701,8703,8717],{"className":8702},[36],[33,8704,8706],{"className":8705},[40],[42,8707,8708],{"xmlns":44},[46,8709,8710,8715],{},[49,8711,8712],{},[59,8713,8714],{},"0.6431",[63,8716,8714],{"encoding":65},[33,8718,8720],{"className":8719,"ariaHidden":71},[70],[33,8721,8723,8726],{"className":8722},[75],[33,8724],{"className":8725,"style":379},[79],[33,8727,8714],{"className":8728},[84],[2161,8730,8731,8733,8969],{},[2242,8732,61],{"align":2166},[2242,8734,8735],{"align":7707},[33,8736,8738,8773],{"className":8737},[36],[33,8739,8741],{"className":8740},[40],[42,8742,8743],{"xmlns":44},[46,8744,8745,8770],{},[49,8746,8747,8753,8755,8757,8759,8765,8767],{},[52,8748,8749,8751],{},[59,8750,270],{},[59,8752,3451],{},[274,8754,6468],{},[59,8756,3430],{},[274,8758,6468],{},[52,8760,8761,8763],{},[59,8762,1826],{},[59,8764,3347],{},[274,8766,7953],{},[59,8768,8769],{},"0.0152",[63,8771,8772],{"encoding":65},"\\frac{1}{512} \\times 14 \\times \\frac{5}{9} \\approx 0.0152",[33,8774,8776,8859,8877,8960],{"className":8775,"ariaHidden":71},[70],[33,8777,8779,8782,8850,8853,8856],{"className":8778},[75],[33,8780],{"className":8781,"style":291},[79],[33,8783,8785,8788,8847],{"className":8784},[84],[33,8786],{"className":8787},[88,89],[33,8789,8791],{"className":8790},[52],[33,8792,8794,8839],{"className":8793},[96,97],[33,8795,8797,8836],{"className":8796},[101],[33,8798,8800,8814,8822],{"className":8799,"style":310},[105],[33,8801,8802,8805],{"style":109},[33,8803],{"className":8804,"style":114},[113],[33,8806,8808],{"className":8807},[118,119,120,121],[33,8809,8811],{"className":8810},[84,121],[33,8812,3451],{"className":8813},[84,121],[33,8815,8816,8819],{"style":130},[33,8817],{"className":8818,"style":114},[113],[33,8820],{"className":8821,"style":138},[137],[33,8823,8824,8827],{"style":141},[33,8825],{"className":8826,"style":114},[113],[33,8828,8830],{"className":8829},[118,119,120,121],[33,8831,8833],{"className":8832},[84,121],[33,8834,270],{"className":8835},[84,121],[33,8837,160],{"className":8838},[159],[33,8840,8842],{"className":8841},[101],[33,8843,8845],{"className":8844,"style":167},[105],[33,8846],{},[33,8848],{"className":8849},[173,89],[33,8851],{"className":8852,"style":2210},[364],[33,8854,6468],{"className":8855},[2214],[33,8857],{"className":8858,"style":2210},[364],[33,8860,8862,8865,8868,8871,8874],{"className":8861},[75],[33,8863],{"className":8864,"style":1314},[79],[33,8866,3430],{"className":8867},[84],[33,8869],{"className":8870,"style":2210},[364],[33,8872,6468],{"className":8873},[2214],[33,8875],{"className":8876,"style":2210},[364],[33,8878,8880,8883,8951,8954,8957],{"className":8879},[75],[33,8881],{"className":8882,"style":291},[79],[33,8884,8886,8889,8948],{"className":8885},[84],[33,8887],{"className":8888},[88,89],[33,8890,8892],{"className":8891},[52],[33,8893,8895,8940],{"className":8894},[96,97],[33,8896,8898,8937],{"className":8897},[101],[33,8899,8901,8915,8923],{"className":8900,"style":310},[105],[33,8902,8903,8906],{"style":109},[33,8904],{"className":8905,"style":114},[113],[33,8907,8909],{"className":8908},[118,119,120,121],[33,8910,8912],{"className":8911},[84,121],[33,8913,3347],{"className":8914},[84,121],[33,8916,8917,8920],{"style":130},[33,8918],{"className":8919,"style":114},[113],[33,8921],{"className":8922,"style":138},[137],[33,8924,8925,8928],{"style":141},[33,8926],{"className":8927,"style":114},[113],[33,8929,8931],{"className":8930},[118,119,120,121],[33,8932,8934],{"className":8933},[84,121],[33,8935,1826],{"className":8936},[84,121],[33,8938,160],{"className":8939},[159],[33,8941,8943],{"className":8942},[101],[33,8944,8946],{"className":8945,"style":167},[105],[33,8947],{},[33,8949],{"className":8950},[173,89],[33,8952],{"className":8953,"style":365},[364],[33,8955,7953],{"className":8956},[369],[33,8958],{"className":8959,"style":365},[364],[33,8961,8963,8966],{"className":8962},[75],[33,8964],{"className":8965,"style":379},[79],[33,8967,8769],{"className":8968},[84],[2242,8970,8971],{"align":7707},[33,8972,8974,8988],{"className":8973},[36],[33,8975,8977],{"className":8976},[40],[42,8978,8979],{"xmlns":44},[46,8980,8981,8986],{},[49,8982,8983],{},[59,8984,8985],{},"0.6583",[63,8987,8985],{"encoding":65},[33,8989,8991],{"className":8990,"ariaHidden":71},[70],[33,8992,8994,8997],{"className":8993},[75],[33,8995],{"className":8996,"style":379},[79],[33,8998,8985],{"className":8999},[84],[15,9001,9002,9003,9031,9032,9153],{},"With just five terms we are at ",[33,9004,9006,9019],{"className":9005},[36],[33,9007,9009],{"className":9008},[40],[42,9010,9011],{"xmlns":44},[46,9012,9013,9017],{},[49,9014,9015],{},[59,9016,8985],{},[63,9018,8985],{"encoding":65},[33,9020,9022],{"className":9021,"ariaHidden":71},[70],[33,9023,9025,9028],{"className":9024},[75],[33,9026],{"className":9027,"style":379},[79],[33,9029,8985],{"className":9030},[84],", slowly approaching ",[33,9033,9035,9058],{"className":9034},[36],[33,9036,9038],{"className":9037},[40],[42,9039,9040],{"xmlns":44},[46,9041,9042,9055],{},[49,9043,9044,9050,9052],{},[52,9045,9046,9048],{},[55,9047,57],{},[59,9049,61],{},[274,9051,7953],{},[59,9053,9054],{},"0.7854",[63,9056,9057],{"encoding":65},"\\frac{\\pi}{4} \\approx 0.7854",[33,9059,9061,9144],{"className":9060,"ariaHidden":71},[70],[33,9062,9064,9067,9135,9138,9141],{"className":9063},[75],[33,9065],{"className":9066,"style":80},[79],[33,9068,9070,9073,9132],{"className":9069},[84],[33,9071],{"className":9072},[88,89],[33,9074,9076],{"className":9075},[52],[33,9077,9079,9124],{"className":9078},[96,97],[33,9080,9082,9121],{"className":9081},[101],[33,9083,9085,9099,9107],{"className":9084,"style":106},[105],[33,9086,9087,9090],{"style":109},[33,9088],{"className":9089,"style":114},[113],[33,9091,9093],{"className":9092},[118,119,120,121],[33,9094,9096],{"className":9095},[84,121],[33,9097,61],{"className":9098},[84,121],[33,9100,9101,9104],{"style":130},[33,9102],{"className":9103,"style":114},[113],[33,9105],{"className":9106,"style":138},[137],[33,9108,9109,9112],{"style":141},[33,9110],{"className":9111,"style":114},[113],[33,9113,9115],{"className":9114},[118,119,120,121],[33,9116,9118],{"className":9117},[84,121],[33,9119,57],{"className":9120,"style":155},[84,154,121],[33,9122,160],{"className":9123},[159],[33,9125,9127],{"className":9126},[101],[33,9128,9130],{"className":9129,"style":167},[105],[33,9131],{},[33,9133],{"className":9134},[173,89],[33,9136],{"className":9137,"style":365},[364],[33,9139,7953],{"className":9140},[369],[33,9142],{"className":9143,"style":365},[364],[33,9145,9147,9150],{"className":9146},[75],[33,9148],{"className":9149,"style":379},[79],[33,9151,9054],{"className":9152},[84],". The convergence is slow, but the direction is unmistakable when the number of flips is sufficiently large.",[544,9155,9157],{"id":9156},"the-appearance-of-arcsine","The appearance of arcsine",[15,9159,9160],{},"This is where the magic happens. Let's substitute the Catalan number formula into our series:",[15,9162,9164],{"className":9163},[575],[33,9165,9167,9214],{"className":9166},[36],[33,9168,9170],{"className":9169},[40],[42,9171,9172],{"xmlns":44},[46,9173,9174,9212],{},[49,9175,9176,9182,9184,9196],{},[4277,9177,9178,9180],{},[55,9179,4281],{},[55,9181,2184],{},[274,9183,276],{},[52,9185,9186,9188],{},[59,9187,270],{},[49,9189,9190,9192,9194],{},[55,9191,2184],{},[274,9193,1299],{},[59,9195,270],{},[49,9197,9198,9200,9210],{},[274,9199,1011],{"fence":71},[52,9201,9202,9208],{"linethickness":4304},[49,9203,9204,9206],{},[59,9205,402],{},[55,9207,2184],{},[55,9209,2184],{},[274,9211,1020],{"fence":71},[63,9213,4317],{"encoding":65},[33,9215,9217,9272],{"className":9216,"ariaHidden":71},[70],[33,9218,9220,9223,9263,9266,9269],{"className":9219},[75],[33,9221],{"className":9222,"style":4327},[79],[33,9224,9226,9229],{"className":9225},[84],[33,9227,4281],{"className":9228,"style":4334},[84,154],[33,9230,9232],{"className":9231},[1139],[33,9233,9235,9255],{"className":9234},[96,97],[33,9236,9238,9252],{"className":9237},[101],[33,9239,9241],{"className":9240,"style":4347},[105],[33,9242,9243,9246],{"style":4350},[33,9244],{"className":9245,"style":1156},[113],[33,9247,9249],{"className":9248},[118,119,120,121],[33,9250,2184],{"className":9251},[84,154,121],[33,9253,160],{"className":9254},[159],[33,9256,9258],{"className":9257},[101],[33,9259,9261],{"className":9260,"style":4369},[105],[33,9262],{},[33,9264],{"className":9265,"style":365},[364],[33,9267,276],{"className":9268},[369],[33,9270],{"className":9271,"style":365},[364],[33,9273,9275,9278,9352],{"className":9274},[75],[33,9276],{"className":9277,"style":4387},[79],[33,9279,9281,9284,9349],{"className":9280},[84],[33,9282],{"className":9283},[88,89],[33,9285,9287],{"className":9286},[52],[33,9288,9290,9341],{"className":9289},[96,97],[33,9291,9293,9338],{"className":9292},[101],[33,9294,9296,9316,9324],{"className":9295,"style":310},[105],[33,9297,9298,9301],{"style":109},[33,9299],{"className":9300,"style":114},[113],[33,9302,9304],{"className":9303},[118,119,120,121],[33,9305,9307,9310,9313],{"className":9306},[84,121],[33,9308,2184],{"className":9309},[84,154,121],[33,9311,1299],{"className":9312},[2214,121],[33,9314,270],{"className":9315},[84,121],[33,9317,9318,9321],{"style":130},[33,9319],{"className":9320,"style":114},[113],[33,9322],{"className":9323,"style":138},[137],[33,9325,9326,9329],{"style":141},[33,9327],{"className":9328,"style":114},[113],[33,9330,9332],{"className":9331},[118,119,120,121],[33,9333,9335],{"className":9334},[84,121],[33,9336,270],{"className":9337},[84,121],[33,9339,160],{"className":9340},[159],[33,9342,9344],{"className":9343},[101],[33,9345,9347],{"className":9346,"style":3723},[105],[33,9348],{},[33,9350],{"className":9351},[173,89],[33,9353,9355,9361,9415],{"className":9354},[84],[33,9356,9358],{"className":9357,"style":1056},[88,1055],[33,9359,1011],{"className":9360},[1060,1061],[33,9362,9364],{"className":9363},[52],[33,9365,9367,9407],{"className":9366},[96,97],[33,9368,9370,9404],{"className":9369},[101],[33,9371,9373,9387],{"className":9372,"style":4483},[105],[33,9374,9375,9378],{"style":4486},[33,9376],{"className":9377,"style":1156},[113],[33,9379,9381],{"className":9380},[118,119,120,121],[33,9382,9384],{"className":9383},[84,121],[33,9385,2184],{"className":9386},[84,154,121],[33,9388,9389,9392],{"style":4501},[33,9390],{"className":9391,"style":1156},[113],[33,9393,9395],{"className":9394},[118,119,120,121],[33,9396,9398,9401],{"className":9397},[84,121],[33,9399,402],{"className":9400},[84,121],[33,9402,2184],{"className":9403},[84,154,121],[33,9405,160],{"className":9406},[159],[33,9408,9410],{"className":9409},[101],[33,9411,9413],{"className":9412,"style":167},[105],[33,9414],{},[33,9416,9418],{"className":9417,"style":1056},[173,1055],[33,9419,1020],{"className":9420},[1060,1061],[15,9422,9423,9424,9494,9495,9621,9622,9683],{},"When we multiply 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by 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the factor ",[33,9623,9625,9647],{"className":9624},[36],[33,9626,9628],{"className":9627},[40],[42,9629,9630],{"xmlns":44},[46,9631,9632,9644],{},[49,9633,9634,9636,9638,9640,9642],{},[274,9635,1011],{"stretchy":6361},[55,9637,2184],{},[274,9639,1299],{},[59,9641,270],{},[274,9643,1020],{"stretchy":6361},[63,9645,9646],{"encoding":65},"(n+1)",[33,9648,9650,9671],{"className":9649,"ariaHidden":71},[70],[33,9651,9653,9656,9659,9662,9665,9668],{"className":9652},[75],[33,9654],{"className":9655,"style":6384},[79],[33,9657,1011],{"className":9658},[88],[33,9660,2184],{"className":9661},[84,154],[33,9663],{"className":9664,"style":2210},[364],[33,9666,1299],{"className":9667},[2214],[33,9669],{"className":9670,"style":2210},[364],[33,9672,9674,9677,9680],{"className":9673},[75],[33,9675],{"className":9676,"style":6384},[79],[33,9678,270],{"className":9679},[84],[33,9681,1020],{"className":9682},[173]," cancels cleanly:",[15,9685,9687],{"className":9686},[575],[33,9688,9690,9782],{"className":9689},[36],[33,9691,9693],{"className":9692},[40],[42,9694,9695],{"xmlns":44},[46,9696,9697,9779],{},[49,9698,9699,9711,9727,9729,9749,9751],{},[52,9700,9701,9703],{},[59,9702,270],{},[49,9704,9705,9707,9709],{},[55,9706,2184],{},[274,9708,1299],{},[59,9710,270],{},[49,9712,9713,9715,9725],{},[274,9714,1011],{"fence":71},[52,9716,9717,9723],{"linethickness":4304},[49,9718,9719,9721],{},[59,9720,402],{},[55,9722,2184],{},[55,9724,2184],{},[274,9726,1020],{"fence":71},[274,9728,7286],{},[52,9730,9731,9739],{},[49,9732,9733,9735,9737],{},[55,9734,2184],{},[274,9736,1299],{},[59,9738,270],{},[49,9740,9741,9743,9745,9747],{},[59,9742,402],{},[55,9744,2184],{},[274,9746,1299],{},[59,9748,270],{},[274,9750,276],{},[52,9752,9753,9769],{},[49,9754,9755,9757,9767],{},[274,9756,1011],{"fence":71},[52,9758,9759,9765],{"linethickness":4304},[49,9760,9761,9763],{},[59,9762,402],{},[55,9764,2184],{},[55,9766,2184],{},[274,9768,1020],{"fence":71},[49,9770,9771,9773,9775,9777],{},[59,9772,402],{},[55,9774,2184],{},[274,9776,1299],{},[59,9778,270],{},[63,9780,9781],{"encoding":65},"\\frac{1}{n+1}\\binom{2n}{n} \\cdot \\frac{n+1}{2n+1} = \\frac{\\binom{2n}{n}}{2n+1}",[33,9783,9785,9943,10041],{"className":9784,"ariaHidden":71},[70],[33,9786,9788,9791,9865,9934,9937,9940],{"className":9787},[75],[33,9789],{"className":9790,"style":4387},[79],[33,9792,9794,9797,9862],{"className":9793},[84],[33,9795],{"className":9796},[88,89],[33,9798,9800],{"className":9799},[52],[33,9801,9803,9854],{"className":9802},[96,97],[33,9804,9806,9851],{"className":9805},[101],[33,9807,9809,9829,9837],{"className":9808,"style":310},[105],[33,9810,9811,9814],{"style":109},[33,9812],{"className":9813,"style":114},[113],[33,9815,9817],{"className":9816},[118,119,120,121],[33,9818,9820,9823,9826],{"className":9819},[84,121],[33,9821,2184],{"className":9822},[84,154,121],[33,9824,1299],{"className":9825},[2214,121],[33,9827,270],{"className":9828},[84,121],[33,9830,9831,9834],{"style":130},[33,9832],{"className":9833,"style":114},[113],[33,9835],{"className":9836,"style":138},[137],[33,9838,9839,9842],{"style":141},[33,9840],{"className":9841,"style":114},[113],[33,9843,9845],{"className":9844},[118,119,120,121],[33,9846,9848],{"className":9847},[84,121],[33,9849,270],{"className":9850},[84,121],[33,9852,160],{"className":9853},[159],[33,9855,9857],{"className":9856},[101],[33,9858,9860],{"className":9859,"style":3723},[105],[33,9861],{},[33,9863],{"className":9864},[173,89],[33,9866,9868,9874,9928],{"className":9867},[84],[33,9869,9871],{"className":9870,"style":1056},[88,1055],[33,9872,1011],{"className":9873},[1060,1061],[33,9875,9877],{"className":9876},[52],[33,9878,9880,9920],{"className":9879},[96,97],[33,9881,9883,9917],{"className":9882},[101],[33,9884,9886,9900],{"className":9885,"style":4483},[105],[33,9887,9888,9891],{"style":4486},[33,9889],{"className":9890,"style":1156},[113],[33,9892,9894],{"className":9893},[118,119,120,121],[33,9895,9897],{"className":9896},[84,121],[33,9898,2184],{"className":9899},[84,154,121],[33,9901,9902,9905],{"style":4501},[33,9903],{"className":9904,"style":1156},[113],[33,9906,9908],{"className":9907},[118,119,120,121],[33,9909,9911,9914],{"className":9910},[84,121],[33,9912,402],{"className":9913},[84,121],[33,9915,2184],{"className":9916},[84,154,121],[33,9918,160],{"className":9919},[159],[33,9921,9923],{"className":9922},[101],[33,9924,9926],{"className":9925,"style":167},[105],[33,9927],{},[33,9929,9931],{"className":9930,"style":1056},[173,1055],[33,9932,1020],{"className":9933},[1060,1061],[33,9935],{"className":9936,"style":2210},[364],[33,9938,7286],{"className":9939},[2214],[33,9941],{"className":9942,"style":2210},[364],[33,9944,9946,9949,10032,10035,10038],{"className":9945},[75],[33,9947],{"className":9948,"style":3644},[79],[33,9950,9952,9955,10029],{"className":9951},[84],[33,9953],{"className":9954},[88,89],[33,9956,9958],{"className":9957},[52],[33,9959,9961,10021],{"className":9960},[96,97],[33,9962,9964,10018],{"className":9963},[101],[33,9965,9967,9990,9998],{"className":9966,"style":310},[105],[33,9968,9969,9972],{"style":109},[33,9970],{"className":9971,"style":114},[113],[33,9973,9975],{"className":9974},[118,119,120,121],[33,9976,9978,9981,9984,9987],{"className":9977},[84,121],[33,9979,402],{"className":9980},[84,121],[33,9982,2184],{"className":9983},[84,154,121],[33,9985,1299],{"className":9986},[2214,121],[33,9988,270],{"className":9989},[84,121],[33,9991,9992,9995],{"style":130},[33,9993],{"className":9994,"style":114},[113],[33,9996],{"className":9997,"style":138},[137],[33,9999,10000,10003],{"style":141},[33,10001],{"className":10002,"style":114},[113],[33,10004,10006],{"className":10005},[118,119,120,121],[33,10007,10009,10012,10015],{"className":10008},[84,121],[33,10010,2184],{"className":10011},[84,154,121],[33,10013,1299],{"className":10014},[2214,121],[33,10016,270],{"className":10017},[84,121],[33,10019,160],{"className":10020},[159],[33,10022,10024],{"className":10023},[101],[33,10025,10027],{"className":10026,"style":3723},[105],[33,10028],{},[33,10030],{"className":10031},[173,89],[33,10033],{"className":10034,"style":365},[364],[33,10036,276],{"className":10037},[369],[33,10039],{"className":10040,"style":365},[364],[33,10042,10044,10048],{"className":10043},[75],[33,10045],{"className":10046,"style":10047},[79],"height:1.9133em;vertical-align:-0.4033em;",[33,10049,10051,10054,10202],{"className":10050},[84],[33,10052],{"className":10053},[88,89],[33,10055,10057],{"className":10056},[52],[33,10058,10060,10194],{"className":10059},[96,97],[33,10061,10063,10191],{"className":10062},[101],[33,10064,10067,10090,10098],{"className":10065,"style":10066},[105],"height:1.51em;",[33,10068,10069,10072],{"style":109},[33,10070],{"className":10071,"style":114},[113],[33,10073,10075],{"className":10074},[118,119,120,121],[33,10076,10078,10081,10084,10087],{"className":10077},[84,121],[33,10079,402],{"className":10080},[84,121],[33,10082,2184],{"className":10083},[84,154,121],[33,10085,1299],{"className":10086},[2214,121],[33,10088,270],{"className":10089},[84,121],[33,10091,10092,10095],{"style":1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the series becomes:",[15,10209,10211],{"className":10210},[575],[33,10212,10214,10350],{"className":10213},[36],[33,10215,10217],{"className":10216},[40],[42,10218,10219],{"xmlns":44},[46,10220,10221,10347],{},[49,10222,10223,10225,10227,10241,10287,10289,10295,10309],{},[589,10224,7241],{},[274,10226,276],{},[7245,10228,10229,10231,10239],{},[274,10230,7249],{},[49,10232,10233,10235,10237],{},[55,10234,2184],{},[274,10236,276],{},[59,10238,1401],{},[55,10240,7261],{"mathvariant":7260},[52,10242,10243,10259],{},[49,10244,10245,10247,10257],{},[274,10246,1011],{"fence":71},[52,10248,10249,10255],{"linethickness":4304},[49,10250,10251,10253],{},[59,10252,402],{},[55,10254,2184],{},[55,10256,2184],{},[274,10258,1020],{"fence":71},[49,10260,10261,10275,10277,10279,10281,10283,10285],{},[1005,10262,10263,10265],{},[59,10264,402],{},[49,10266,10267,10269,10271,10273],{},[59,10268,402],{},[55,10270,2184],{},[274,10272,1299],{},[59,10274,270],{},[274,10276,1011],{"stretchy":6361},[59,10278,402],{},[55,10280,2184],{},[274,10282,1299],{},[59,10284,270],{},[274,10286,1020],{"stretchy":6361},[274,10288,276],{},[52,10290,10291,10293],{},[59,10292,270],{},[59,10294,402],{},[7245,10296,10297,10299,10307],{},[274,10298,7249],{},[49,10300,10301,10303,10305],{},[55,10302,2184],{},[274,10304,276],{},[59,10306,1401],{},[55,10308,7261],{"mathvariant":7260},[52,10310,10311,10327],{},[49,10312,10313,10315,10325],{},[274,10314,1011],{"fence":71},[52,10316,10317,10323],{"linethickness":4304},[49,10318,10319,10321],{},[59,10320,402],{},[55,10322,2184],{},[55,10324,2184],{},[274,10326,1020],{"fence":71},[49,10328,10329,10335,10337,10339,10341,10343,10345],{},[1005,10330,10331,10333],{},[59,10332,61],{},[55,10334,2184],{},[274,10336,1011],{"stretchy":6361},[59,10338,402],{},[55,10340,2184],{},[274,10342,1299],{},[59,10344,270],{},[274,10346,1020],{"stretchy":6361},[63,10348,10349],{"encoding":65},"\\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)}",[33,10351,10353,10374,10653],{"className":10352,"ariaHidden":71},[70],[33,10354,10356,10359,10365,10368,10371],{"className":10355},[75],[33,10357],{"className":10358,"style":7319},[79],[33,10360,10362],{"className":10361},[84,608],[33,10363,7241],{"className":10364},[84],[33,10366],{"className":10367,"style":365},[364],[33,10369,276],{"className":10370},[369],[33,10372],{"className":10373,"style":365},[364],[33,10375,10377,10381,10444,10447,10644,10647,10650],{"className":10376},[75],[33,10378],{"className":10379,"style":10380},[79],"height:2.03em;vertical-align:-0.52em;",[33,10382,10384,10387],{"className":10383},[7345],[33,10385,7249],{"className":10386,"style":7351},[7345,7349,7350],[33,10388,10390],{"className":10389},[1139],[33,10391,10393,10436],{"className":10392},[96,97],[33,10394,10396,10433],{"className":10395},[101],[33,10397,10399,10419],{"className":10398,"style":7364},[105],[33,10400,10401,10404],{"style":7367},[33,10402],{"className":10403,"style":1156},[113],[33,10405,10407],{"className":10406},[118,119,120,121],[33,10408,10410,10413,10416],{"className":10409},[84,121],[33,10411,2184],{"className":10412},[84,154,121],[33,10414,276],{"className":10415},[369,121],[33,10417,1401],{"className":10418},[84,121],[33,10420,10421,10424],{"style":7388},[33,10422],{"className":10423,"style":1156},[113],[33,10425,10427],{"className":10426},[118,119,120,121],[33,10428,10430],{"className":10429},[84,121],[33,10431,7261],{"className":10432},[84,121],[33,10434,160],{"className":10435},[159],[33,10437,10439],{"className":10438},[101],[33,10440,10442],{"className":10441,"style":7410},[105],[33,10443],{},[33,10445],{"className":10446,"style":1439},[364],[33,10448,10450,10453,10641],{"className":10449},[84],[33,10451],{"className":10452},[88,89],[33,10454,10456],{"className":10455},[52],[33,10457,10459,10632],{"className":10458},[96,97],[33,10460,10462,10629],{"className":10461},[101],[33,10463,10465,10535,10543],{"className":10464,"style":10066},[105],[33,10466,10467,10470],{"style":109},[33,10468],{"className":10469,"style":114},[113],[33,10471,10473],{"className":10472},[118,119,120,121],[33,10474,10476,10517,10520,10523,10526,10529,10532],{"className":10475},[84,121],[33,10477,10479,10482],{"className":10478},[84,121],[33,10480,402],{"className":10481},[84,121],[33,10483,10485],{"className":10484},[1139],[33,10486,10488],{"className":10487},[96],[33,10489,10491],{"className":10490},[101],[33,10492,10494],{"className":10493,"style":7464},[105],[33,10495,10496,10499],{"style":7467},[33,10497],{"className":10498,"style":7471},[113],[33,10500,10502],{"className":10501},[118,7475,1061,121],[33,10503,10505,10508,10511,10514],{"className":10504},[84,121],[33,10506,402],{"className":10507},[84,121],[33,10509,2184],{"className":10510},[84,154,121],[33,10512,1299],{"className":10513},[2214,121],[33,10515,270],{"className":10516},[84,121],[33,10518,1011],{"className":10519},[88,121],[33,10521,402],{"className":10522},[84,121],[33,10524,2184],{"className":10525},[84,154,121],[33,10527,1299],{"className":10528},[2214,121],[33,10530,270],{"className":10531},[84,121],[33,10533,1020],{"className":10534},[173,121],[33,10536,10537,10540],{"style":130},[33,10538],{"className":10539,"style":114},[113],[33,10541],{"className":10542,"style":138},[137],[33,10544,10545,10548],{"style":10100},[33,10546],{"className":10547,"style":114},[113],[33,10549,10551],{"className":10550},[118,119,120,121],[33,10552,10554],{"className":10553},[84,121],[33,10555,10557,10566,10620],{"className":10556},[84,121],[33,10558,10560],{"className":10559,"style":10117},[88,118,7475,10116,121,1055],[33,10561,10563],{"className":10562},[1060,1061,121],[33,10564,1011],{"className":10565},[121],[33,10567,10569],{"className":10568},[52],[33,10570,10572,10612],{"className":10571},[96,97],[33,10573,10575,10609],{"className":10574},[101],[33,10576,10578,10592],{"className":10577,"style":10136},[105],[33,10579,10580,10583],{"style":10139},[33,10581],{"className":10582,"style":7471},[113],[33,10584,10586],{"className":10585},[118,7475,1061,121],[33,10587,10589],{"className":10588},[84,121],[33,10590,2184],{"className":10591},[84,154,121],[33,10593,10594,10597],{"style":10154},[33,10595],{"className":10596,"style":7471},[113],[33,10598,10600],{"className":10599},[118,7475,1061,121],[33,10601,10603,10606],{"className":10602},[84,121],[33,10604,402],{"className":10605},[84,121],[33,10607,2184],{"className":10608},[84,154,121],[33,10610,160],{"className":10611},[159],[33,10613,10615],{"className":10614},[101],[33,10616,10618],{"className":10617,"style":10179},[105],[33,10619],{},[33,10621,10623],{"className":10622,"style":10117},[173,118,7475,10116,121,1055],[33,10624,10626],{"className":10625},[1060,1061,121],[33,10627,1020],{"className":10628},[121],[33,10630,160],{"className":10631},[159],[33,10633,10635],{"className":10634},[101],[33,10636,10639],{"className":10637,"style":10638},[105],"height:0.52em;",[33,10640],{},[33,10642],{"className":10643},[173,89],[33,10645],{"className":10646,"style":365},[364],[33,10648,276],{"className":10649},[369],[33,10651],{"className":10652,"style":365},[364],[33,10654,10656,10659,10727,10730,10793,10796],{"className":10655},[75],[33,10657],{"className":10658,"style":10380},[79],[33,10660,10662,10665,10724],{"className":10661},[84],[33,10663],{"className":10664},[88,89],[33,10666,10668],{"className":10667},[52],[33,10669,10671,10716],{"className":10670},[96,97],[33,10672,10674,10713],{"className":10673},[101],[33,10675,10677,10691,10699],{"className":10676,"style":310},[105],[33,10678,10679,10682],{"style":109},[33,10680],{"className":10681,"style":114},[113],[33,10683,10685],{"className":10684},[118,119,120,121],[33,10686,10688],{"className":10687},[84,121],[33,10689,402],{"className":10690},[84,121],[33,10692,10693,10696],{"style":130},[33,10694],{"className":10695,"style":114},[113],[33,10697],{"className":10698,"style":138},[137],[33,10700,10701,10704],{"style":141},[33,10702],{"className":10703,"style":114},[113],[33,10705,10707],{"className":10706},[118,119,120,121],[33,10708,10710],{"className":10709},[84,121],[33,10711,270],{"className":10712},[84,121],[33,10714,160],{"className":10715},[159],[33,10717,10719],{"className":10718},[101],[33,10720,10722],{"className":10721,"style":167},[105],[33,10723],{},[33,10725],{"className":10726},[173,89],[33,10728],{"className":10729,"style":1439},[364],[33,10731,10733,10736],{"className":10732},[7345],[33,10734,7249],{"className":10735,"style":7351},[7345,7349,7350],[33,10737,10739],{"className":10738},[1139],[33,10740,10742,10785],{"className":10741},[96,97],[33,10743,10745,10782],{"className":10744},[101],[33,10746,10748,10768],{"className":10747,"style":7364},[105],[33,10749,10750,10753],{"style":7367},[33,10751],{"className":10752,"style":1156},[113],[33,10754,10756],{"className":10755},[118,119,120,121],[33,10757,10759,10762,10765],{"className":10758},[84,121],[33,10760,2184],{"className":10761},[84,154,121],[33,10763,276],{"className":10764},[369,121],[33,10766,1401],{"className":10767},[84,121],[33,10769,10770,10773],{"style":7388},[33,10771],{"className":10772,"style":1156},[113],[33,10774,10776],{"className":10775},[118,119,120,121],[33,10777,10779],{"className":10778},[84,121],[33,10780,7261],{"className":10781},[84,121],[33,10783,160],{"className":10784},[159],[33,10786,10788],{"className":10787},[101],[33,10789,10791],{"className":10790,"style":7410},[105],[33,10792],{},[33,10794],{"className":10795,"style":1439},[364],[33,10797,10799,10802,10978],{"className":10798},[84],[33,10800],{"className":10801},[88,89],[33,10803,10805],{"className":10804},[52],[33,10806,10808,10970],{"className":10807},[96,97],[33,10809,10811,10967],{"className":10810},[101],[33,10812,10814,10873,10881],{"className":10813,"style":10066},[105],[33,10815,10816,10819],{"style":109},[33,10817],{"className":10818,"style":114},[113],[33,10820,10822],{"className":10821},[118,119,120,121],[33,10823,10825,10855,10858,10861,10864,10867,10870],{"className":10824},[84,121],[33,10826,10828,10831],{"className":10827},[84,121],[33,10829,61],{"className":10830},[84,121],[33,10832,10834],{"className":10833},[1139],[33,10835,10837],{"className":10836},[96],[33,10838,10840],{"className":10839},[101],[33,10841,10844],{"className":10842,"style":10843},[105],"height:0.5935em;",[33,10845,10846,10849],{"style":7467},[33,10847],{"className":10848,"style":7471},[113],[33,10850,10852],{"className":10851},[118,7475,1061,121],[33,10853,2184],{"className":10854},[84,154,121],[33,10856,1011],{"className":10857},[88,121],[33,10859,402],{"className":10860},[84,121],[33,10862,2184],{"className":10863},[84,154,121],[33,10865,1299],{"className":10866},[2214,121],[33,10868,270],{"className":10869},[84,121],[33,10871,1020],{"className":10872},[173,121],[33,10874,10875,10878],{"style":130},[33,10876],{"className":10877,"style":114},[113],[33,10879],{"className":10880,"style":138},[137],[33,10882,10883,10886],{"style":10100},[33,10884],{"className":10885,"style":114},[113],[33,10887,10889],{"className":10888},[118,119,120,121],[33,10890,10892],{"className":10891},[84,121],[33,10893,10895,10904,10958],{"className":10894},[84,121],[33,10896,10898],{"className":10897,"style":10117},[88,118,7475,10116,121,1055],[33,10899,10901],{"className":10900},[1060,1061,121],[33,10902,1011],{"className":10903},[121],[33,10905,10907],{"className":10906},[52],[33,10908,10910,10950],{"className":10909},[96,97],[33,10911,10913,10947],{"className":10912},[101],[33,10914,10916,10930],{"className":10915,"style":10136},[105],[33,10917,10918,10921],{"style":10139},[33,10919],{"className":10920,"style":7471},[113],[33,10922,10924],{"className":10923},[118,7475,1061,121],[33,10925,10927],{"className":10926},[84,121],[33,10928,2184],{"className":10929},[84,154,121],[33,10931,10932,10935],{"style":10154},[33,10933],{"className":10934,"style":7471},[113],[33,10936,10938],{"className":10937},[118,7475,1061,121],[33,10939,10941,10944],{"className":10940},[84,121],[33,10942,402],{"className":10943},[84,121],[33,10945,2184],{"className":10946},[84,154,121],[33,10948,160],{"className":10949},[159],[33,10951,10953],{"className":10952},[101],[33,10954,10956],{"className":10955,"style":10179},[105],[33,10957],{},[33,10959,10961],{"className":10960,"style":10117},[173,118,7475,10116,121,1055],[33,10962,10964],{"className":10963},[1060,1061,121],[33,10965,1020],{"className":10966},[121],[33,10968,160],{"className":10969},[159],[33,10971,10973],{"className":10972},[101],[33,10974,10976],{"className":10975,"style":10638},[105],[33,10977],{},[33,10979],{"className":10980},[173,89],[15,10982,10983,10984,10987],{},"Now 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 ",[237,10985,10986],{},"arcsine"," — the inverse function of sine — whose power series expansion is:",[15,10989,10991],{"className":10990},[575],[33,10992,10994,11087],{"className":10993},[36],[33,10995,10997],{"className":10996},[40],[42,10998,10999],{"xmlns":44},[46,11000,11001,11084],{},[49,11002,11003,11006,11009,11011,11014,11016,11018,11032,11056,11058],{},[55,11004,11005],{},"arcsin",[274,11007,11008],{},"⁡",[274,11010,1011],{"stretchy":6361},[55,11012,11013],{},"x",[274,11015,1020],{"stretchy":6361},[274,11017,276],{},[7245,11019,11020,11022,11030],{},[274,11021,7249],{},[49,11023,11024,11026,11028],{},[55,11025,2184],{},[274,11027,276],{},[59,11029,1401],{},[55,11031,7261],{"mathvariant":7260},[52,11033,11034,11050],{},[49,11035,11036,11038,11048],{},[274,11037,1011],{"fence":71},[52,11039,11040,11046],{"linethickness":4304},[49,11041,11042,11044],{},[59,11043,402],{},[55,11045,2184],{},[55,11047,2184],{},[274,11049,1020],{"fence":71},[1005,11051,11052,11054],{},[59,11053,61],{},[55,11055,2184],{},[274,11057,7286],{},[52,11059,11060,11074],{},[1005,11061,11062,11064],{},[55,11063,11013],{},[49,11065,11066,11068,11070,11072],{},[59,11067,402],{},[55,11069,2184],{},[274,11071,1299],{},[59,11073,270],{},[49,11075,11076,11078,11080,11082],{},[59,11077,402],{},[55,11079,2184],{},[274,11081,1299],{},[59,11083,270],{},[63,11085,11086],{"encoding":65},"\\arcsin(x) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n} \\cdot \\frac{x^{2n+1}}{2n+1}",[33,11088,11090,11117,11365],{"className":11089,"ariaHidden":71},[70],[33,11091,11093,11096,11099,11102,11105,11108,11111,11114],{"className":11092},[75],[33,11094],{"className":11095,"style":6384},[79],[33,11097,11005],{"className":11098},[7345],[33,11100,1011],{"className":11101},[88],[33,11103,11013],{"className":11104},[84,154],[33,11106,1020],{"className":11107},[173],[33,11109],{"className":11110,"style":365},[364],[33,11112,276],{"className":11113},[369],[33,11115],{"className":11116,"style":365},[364],[33,11118,11120,11124,11187,11190,11356,11359,11362],{"className":11119},[75],[33,11121],{"className":11122,"style":11123},[79],"height:1.855em;vertical-align:-0.345em;",[33,11125,11127,11130],{"className":11126},[7345],[33,11128,7249],{"className":11129,"style":7351},[7345,7349,7350],[33,11131,11133],{"className":11132},[1139],[33,11134,11136,11179],{"className":11135},[96,97],[33,11137,11139,11176],{"className":11138},[101],[33,11140,11142,11162],{"className":11141,"style":7364},[105],[33,11143,11144,11147],{"style":7367},[33,11145],{"className":11146,"style":1156},[113],[33,11148,11150],{"className":11149},[118,119,120,121],[33,11151,11153,11156,11159],{"className":11152},[84,121],[33,11154,2184],{"className":11155},[84,154,121],[33,11157,276],{"className":11158},[369,121],[33,11160,1401],{"className":11161},[84,121],[33,11163,11164,11167],{"style":7388},[33,11165],{"className":11166,"style":1156},[113],[33,11168,11170],{"className":11169},[118,119,120,121],[33,11171,11173],{"className":11172},[84,121],[33,11174,7261],{"className":11175},[84,121],[33,11177,160],{"className":11178},[159],[33,11180,11182],{"className":11181},[101],[33,11183,11185],{"className":11184,"style":7410},[105],[33,11186],{},[33,11188],{"className":11189,"style":1439},[364],[33,11191,11193,11196,11353],{"className":11192},[84],[33,11194],{"className":11195},[88,89],[33,11197,11199],{"className":11198},[52],[33,11200,11202,11345],{"className":11201},[96,97],[33,11203,11205,11342],{"className":11204},[101],[33,11206,11208,11248,11256],{"className":11207,"style":10066},[105],[33,11209,11210,11213],{"style":109},[33,11211],{"className":11212,"style":114},[113],[33,11214,11216],{"className":11215},[118,119,120,121],[33,11217,11219],{"className":11218},[84,121],[33,11220,11222,11225],{"className":11221},[84,121],[33,11223,61],{"className":11224},[84,121],[33,11226,11228],{"className":11227},[1139],[33,11229,11231],{"className":11230},[96],[33,11232,11234],{"className":11233},[101],[33,11235,11237],{"className":11236,"style":10843},[105],[33,11238,11239,11242],{"style":7467},[33,11240],{"className":11241,"style":7471},[113],[33,11243,11245],{"className":11244},[118,7475,1061,121],[33,11246,2184],{"className":11247},[84,154,121],[33,11249,11250,11253],{"style":130},[33,11251],{"className":11252,"style":114},[113],[33,11254],{"className":11255,"style":138},[137],[33,11257,11258,11261],{"style":10100},[33,11259],{"className":11260,"style":114},[113],[33,11262,11264],{"className":11263},[118,119,120,121],[33,11265,11267],{"className":11266},[84,121],[33,11268,11270,11279,11333],{"className":11269},[84,121],[33,11271,11273],{"className":11272,"style":10117},[88,118,7475,10116,121,1055],[33,11274,11276],{"className":11275},[1060,1061,121],[33,11277,1011],{"className":11278},[121],[33,11280,11282],{"className":11281},[52],[33,11283,11285,11325],{"className":11284},[96,97],[33,11286,11288,11322],{"className":11287},[101],[33,11289,11291,11305],{"className":11290,"style":10136},[105],[33,11292,11293,11296],{"style":10139},[33,11294],{"className":11295,"style":7471},[113],[33,11297,11299],{"className":11298},[118,7475,1061,121],[33,11300,11302],{"className":11301},[84,121],[33,11303,2184],{"className":11304},[84,154,121],[33,11306,11307,11310],{"style":10154},[33,11308],{"className":11309,"style":7471},[113],[33,11311,11313],{"className":11312},[118,7475,1061,121],[33,11314,11316,11319],{"className":11315},[84,121],[33,11317,402],{"className":11318},[84,121],[33,11320,2184],{"className":11321},[84,154,121],[33,11323,160],{"className":11324},[159],[33,11326,11328],{"className":11327},[101],[33,11329,11331],{"className":11330,"style":10179},[105],[33,11332],{},[33,11334,11336],{"className":11335,"style":10117},[173,118,7475,10116,121,1055],[33,11337,11339],{"className":11338},[1060,1061,121],[33,11340,1020],{"className":11341},[121],[33,11343,160],{"className":11344},[159],[33,11346,11348],{"className":11347},[101],[33,11349,11351],{"className":11350,"style":167},[105],[33,11352],{},[33,11354],{"className":11355},[173,89],[33,11357],{"className":11358,"style":2210},[364],[33,11360,7286],{"className":11361},[2214],[33,11363],{"className":11364,"style":2210},[364],[33,11366,11368,11372],{"className":11367},[75],[33,11369],{"className":11370,"style":11371},[79],"height:1.4213em;vertical-align:-0.4033em;",[33,11373,11375,11378,11487],{"className":11374},[84],[33,11376],{"className":11377},[88,89],[33,11379,11381],{"className":11380},[52],[33,11382,11384,11479],{"className":11383},[96,97],[33,11385,11387,11476],{"className":11386},[101],[33,11388,11391,11414,11422],{"className":11389,"style":11390},[105],"height:1.0179em;",[33,11392,11393,11396],{"style":109},[33,11394],{"className":11395,"style":114},[113],[33,11397,11399],{"className":11398},[118,119,120,121],[33,11400,11402,11405,11408,11411],{"className":11401},[84,121],[33,11403,402],{"className":11404},[84,121],[33,11406,2184],{"className":11407},[84,154,121],[33,11409,1299],{"className":11410},[2214,121],[33,11412,270],{"className":11413},[84,121],[33,11415,11416,11419],{"style":130},[33,11417],{"className":11418,"style":114},[113],[33,11420],{"className":11421,"style":138},[137],[33,11423,11424,11427],{"style":141},[33,11425],{"className":11426,"style":114},[113],[33,11428,11430],{"className":11429},[118,119,120,121],[33,11431,11433],{"className":11432},[84,121],[33,11434,11436,11439],{"className":11435},[84,121],[33,11437,11013],{"className":11438},[84,154,121],[33,11440,11442],{"className":11441},[1139],[33,11443,11445],{"className":11444},[96],[33,11446,11448],{"className":11447},[101],[33,11449,11452],{"className":11450,"style":11451},[105],"height:0.8913em;",[33,11453,11455,11458],{"style":11454},"top:-2.931em;margin-right:0.0714em;",[33,11456],{"className":11457,"style":7471},[113],[33,11459,11461],{"className":11460},[118,7475,1061,121],[33,11462,11464,11467,11470,11473],{"className":11463},[84,121],[33,11465,402],{"className":11466},[84,121],[33,11468,2184],{"className":11469},[84,154,121],[33,11471,1299],{"className":11472},[2214,121],[33,11474,270],{"className":11475},[84,121],[33,11477,160],{"className":11478},[159],[33,11480,11482],{"className":11481},[101],[33,11483,11485],{"className":11484,"style":3723},[105],[33,11486],{},[33,11488],{"className":11489},[173,89],[15,11491,11492,11493,11521,11522,11550,11551,174],{},"Look familiar? It is exactly our series, multiplied by ",[33,11494,11496,11509],{"className":11495},[36],[33,11497,11499],{"className":11498},[40],[42,11500,11501],{"xmlns":44},[46,11502,11503,11507],{},[49,11504,11505],{},[59,11506,402],{},[63,11508,402],{"encoding":65},[33,11510,11512],{"className":11511,"ariaHidden":71},[70],[33,11513,11515,11518],{"className":11514},[75],[33,11516],{"className":11517,"style":379},[79],[33,11519,402],{"className":11520},[84],", with ",[33,11523,11525,11538],{"className":11524},[36],[33,11526,11528],{"className":11527},[40],[42,11529,11530],{"xmlns":44},[46,11531,11532,11536],{},[49,11533,11534],{},[55,11535,11013],{},[63,11537,11013],{"encoding":65},[33,11539,11541],{"className":11540,"ariaHidden":71},[70],[33,11542,11544,11547],{"className":11543},[75],[33,11545],{"className":11546,"style":204},[79],[33,11548,11013],{"className":11549},[84,154]," instead of ",[33,11552,11554,11567],{"className":11553},[36],[33,11555,11557],{"className":11556},[40],[42,11558,11559],{"xmlns":44},[46,11560,11561,11565],{},[49,11562,11563],{},[59,11564,270],{},[63,11566,270],{"encoding":65},[33,11568,11570],{"className":11569,"ariaHidden":71},[70],[33,11571,11573,11576],{"className":11572},[75],[33,11574],{"className":11575,"style":379},[79],[33,11577,270],{"className":11578},[84],[15,11580,11581,11582,11633],{},"If we evaluate at ",[33,11583,11585,11603],{"className":11584},[36],[33,11586,11588],{"className":11587},[40],[42,11589,11590],{"xmlns":44},[46,11591,11592,11600],{},[49,11593,11594,11596,11598],{},[55,11595,11013],{},[274,11597,276],{},[59,11599,270],{},[63,11601,11602],{"encoding":65},"x = 1",[33,11604,11606,11624],{"className":11605,"ariaHidden":71},[70],[33,11607,11609,11612,11615,11618,11621],{"className":11608},[75],[33,11610],{"className":11611,"style":204},[79],[33,11613,11013],{"className":11614},[84,154],[33,11616],{"className":11617,"style":365},[364],[33,11619,276],{"className":11620},[369],[33,11622],{"className":11623,"style":365},[364],[33,11625,11627,11630],{"className":11626},[75],[33,11628],{"className":11629,"style":379},[79],[33,11631,270],{"className":11632},[84],":",[15,11635,11637],{"className":11636},[575],[33,11638,11640,11716],{"className":11639},[36],[33,11641,11643],{"className":11642},[40],[42,11644,11645],{"xmlns":44},[46,11646,11647,11713],{},[49,11648,11649,11651,11653,11655,11657,11659,11661,11675],{},[55,11650,11005],{},[274,11652,11008],{},[274,11654,1011],{"stretchy":6361},[59,11656,270],{},[274,11658,1020],{"stretchy":6361},[274,11660,276],{},[7245,11662,11663,11665,11673],{},[274,11664,7249],{},[49,11666,11667,11669,11671],{},[55,11668,2184],{},[274,11670,276],{},[59,11672,1401],{},[55,11674,7261],{"mathvariant":7260},[52,11676,11677,11693],{},[49,11678,11679,11681,11691],{},[274,11680,1011],{"fence":71},[52,11682,11683,11689],{"linethickness":4304},[49,11684,11685,11687],{},[59,11686,402],{},[55,11688,2184],{},[55,11690,2184],{},[274,11692,1020],{"fence":71},[49,11694,11695,11701,11703,11705,11707,11709,11711],{},[1005,11696,11697,11699],{},[59,11698,61],{},[55,11700,2184],{},[274,11702,1011],{"stretchy":6361},[59,11704,402],{},[55,11706,2184],{},[274,11708,1299],{},[59,11710,270],{},[274,11712,1020],{"stretchy":6361},[63,11714,11715],{"encoding":65},"\\arcsin(1) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n(2n+1)}",[33,11717,11719,11746],{"className":11718,"ariaHidden":71},[70],[33,11720,11722,11725,11728,11731,11734,11737,11740,11743],{"className":11721},[75],[33,11723],{"className":11724,"style":6384},[79],[33,11726,11005],{"className":11727},[7345],[33,11729,1011],{"className":11730},[88],[33,11732,270],{"className":11733},[84],[33,11735,1020],{"className":11736},[173],[33,11738],{"className":11739,"style":365},[364],[33,11741,276],{"className":11742},[369],[33,11744],{"className":11745,"style":365},[364],[33,11747,11749,11752,11815,11818],{"className":11748},[75],[33,11750],{"className":11751,"style":10380},[79],[33,11753,11755,11758],{"className":11754},[7345],[33,11756,7249],{"className":11757,"style":7351},[7345,7349,7350],[33,11759,11761],{"className":11760},[1139],[33,11762,11764,11807],{"className":11763},[96,97],[33,11765,11767,11804],{"className":11766},[101],[33,11768,11770,11790],{"className":11769,"style":7364},[105],[33,11771,11772,11775],{"style":7367},[33,11773],{"className":11774,"style":1156},[113],[33,11776,11778],{"className":11777},[118,119,120,121],[33,11779,11781,11784,11787],{"className":11780},[84,121],[33,11782,2184],{"className":11783},[84,154,121],[33,11785,276],{"className":11786},[369,121],[33,11788,1401],{"className":11789},[84,121],[33,11791,11792,11795],{"style":7388},[33,11793],{"className":11794,"style":1156},[113],[33,11796,11798],{"className":11797},[118,119,120,121],[33,11799,11801],{"className":11800},[84,121],[33,11802,7261],{"className":11803},[84,121],[33,11805,160],{"className":11806},[159],[33,11808,11810],{"className":11809},[101],[33,11811,11813],{"className":11812,"style":7410},[105],[33,11814],{},[33,11816],{"className":11817,"style":1439},[364],[33,11819,11821,11824,11999],{"className":11820},[84],[33,11822],{"className":11823},[88,89],[33,11825,11827],{"className":11826},[52],[33,11828,11830,11991],{"className":11829},[96,97],[33,11831,11833,11988],{"className":11832},[101],[33,11834,11836,11894,11902],{"className":11835,"style":10066},[105],[33,11837,11838,11841],{"style":109},[33,11839],{"className":11840,"style":114},[113],[33,11842,11844],{"className":11843},[118,119,120,121],[33,11845,11847,11876,11879,11882,11885,11888,11891],{"className":11846},[84,121],[33,11848,11850,11853],{"className":11849},[84,121],[33,11851,61],{"className":11852},[84,121],[33,11854,11856],{"className":11855},[1139],[33,11857,11859],{"className":11858},[96],[33,11860,11862],{"className":11861},[101],[33,11863,11865],{"className":11864,"style":10843},[105],[33,11866,11867,11870],{"style":7467},[33,11868],{"className":11869,"style":7471},[113],[33,11871,11873],{"className":11872},[118,7475,1061,121],[33,11874,2184],{"className":11875},[84,154,121],[33,11877,1011],{"className":11878},[88,121],[33,11880,402],{"className":11881},[84,121],[33,11883,2184],{"className":11884},[84,154,121],[33,11886,1299],{"className":11887},[2214,121],[33,11889,270],{"className":11890},[84,121],[33,11892,1020],{"className":11893},[173,121],[33,11895,11896,11899],{"style":130},[33,11897],{"className":11898,"style":114},[113],[33,11900],{"className":11901,"style":138},[137],[33,11903,11904,11907],{"style":10100},[33,11905],{"className":11906,"style":114},[113],[33,11908,11910],{"className":11909},[118,119,120,121],[33,11911,11913],{"className":11912},[84,121],[33,11914,11916,11925,11979],{"className":11915},[84,121],[33,11917,11919],{"className":11918,"style":10117},[88,118,7475,10116,121,1055],[33,11920,11922],{"className":11921},[1060,1061,121],[33,11923,1011],{"className":11924},[121],[33,11926,11928],{"className":11927},[52],[33,11929,11931,11971],{"className":11930},[96,97],[33,11932,11934,11968],{"className":11933},[101],[33,11935,11937,11951],{"className":11936,"style":10136},[105],[33,11938,11939,11942],{"style":10139},[33,11940],{"className":11941,"style":7471},[113],[33,11943,11945],{"className":11944},[118,7475,1061,121],[33,11946,11948],{"className":11947},[84,121],[33,11949,2184],{"className":11950},[84,154,121],[33,11952,11953,11956],{"style":10154},[33,11954],{"className":11955,"style":7471},[113],[33,11957,11959],{"className":11958},[118,7475,1061,121],[33,11960,11962,11965],{"className":11961},[84,121],[33,11963,402],{"className":11964},[84,121],[33,11966,2184],{"className":11967},[84,154,121],[33,11969,160],{"className":11970},[159],[33,11972,11974],{"className":11973},[101],[33,11975,11977],{"className":11976,"style":10179},[105],[33,11978],{},[33,11980,11982],{"className":11981,"style":10117},[173,118,7475,10116,121,1055],[33,11983,11985],{"className":11984},[1060,1061,121],[33,11986,1020],{"className":11987},[121],[33,11989,160],{"className":11990},[159],[33,11992,11994],{"className":11993},[101],[33,11995,11997],{"className":11996,"style":10638},[105],[33,11998],{},[33,12000],{"className":12001},[173,89],[15,12003,12004,12005,12051,12052,12080,12081,11633],{},"And what is ",[33,12006,12008,12030],{"className":12007},[36],[33,12009,12011],{"className":12010},[40],[42,12012,12013],{"xmlns":44},[46,12014,12015,12027],{},[49,12016,12017,12019,12021,12023,12025],{},[55,12018,11005],{},[274,12020,11008],{},[274,12022,1011],{"stretchy":6361},[59,12024,270],{},[274,12026,1020],{"stretchy":6361},[63,12028,12029],{"encoding":65},"\\arcsin(1)",[33,12031,12033],{"className":12032,"ariaHidden":71},[70],[33,12034,12036,12039,12042,12045,12048],{"className":12035},[75],[33,12037],{"className":12038,"style":6384},[79],[33,12040,11005],{"className":12041},[7345],[33,12043,1011],{"className":12044},[88],[33,12046,270],{"className":12047},[84],[33,12049,1020],{"className":12050},[173],"? It is the angle whose sine is ",[33,12053,12055,12068],{"className":12054},[36],[33,12056,12058],{"className":12057},[40],[42,12059,12060],{"xmlns":44},[46,12061,12062,12066],{},[49,12063,12064],{},[59,12065,270],{},[63,12067,270],{"encoding":65},[33,12069,12071],{"className":12070,"ariaHidden":71},[70],[33,12072,12074,12077],{"className":12073},[75],[33,12075],{"className":12076,"style":379},[79],[33,12078,270],{"className":12079},[84],", that is, ",[33,12082,12084,12102],{"className":12083},[36],[33,12085,12087],{"className":12086},[40],[42,12088,12089],{"xmlns":44},[46,12090,12091,12099],{},[49,12092,12093,12096],{},[59,12094,12095],{},"90",[55,12097,12098],{"mathvariant":7260},"°",[63,12100,12101],{"encoding":65},"90°",[33,12103,12105],{"className":12104,"ariaHidden":71},[70],[33,12106,12108,12112],{"className":12107},[75],[33,12109],{"className":12110,"style":12111},[79],"height:0.6944em;",[33,12113,12101],{"className":12114},[84],[15,12116,12118],{"className":12117},[575],[33,12119,12121,12151],{"className":12120},[36],[33,12122,12124],{"className":12123},[40],[42,12125,12126],{"xmlns":44},[46,12127,12128,12148],{},[49,12129,12130,12132,12134,12136,12138,12140,12142],{},[55,12131,11005],{},[274,12133,11008],{},[274,12135,1011],{"stretchy":6361},[59,12137,270],{},[274,12139,1020],{"stretchy":6361},[274,12141,276],{},[52,12143,12144,12146],{},[55,12145,57],{},[59,12147,402],{},[63,12149,12150],{"encoding":65},"\\arcsin(1) = \\frac{\\pi}{2}",[33,12152,12154,12181],{"className":12153,"ariaHidden":71},[70],[33,12155,12157,12160,12163,12166,12169,12172,12175,12178],{"className":12156},[75],[33,12158],{"className":12159,"style":6384},[79],[33,12161,11005],{"className":12162},[7345],[33,12164,1011],{"className":12165},[88],[33,12167,270],{"className":12168},[84],[33,12170,1020],{"className":12171},[173],[33,12173],{"className":12174,"style":365},[364],[33,12176,276],{"className":12177},[369],[33,12179],{"className":12180,"style":365},[364],[33,12182,12184,12187],{"className":12183},[75],[33,12185],{"className":12186,"style":80},[79],[33,12188,12190,12193,12252],{"className":12189},[84],[33,12191],{"className":12192},[88,89],[33,12194,12196],{"className":12195},[52],[33,12197,12199,12244],{"className":12198},[96,97],[33,12200,12202,12241],{"className":12201},[101],[33,12203,12205,12219,12227],{"className":12204,"style":106},[105],[33,12206,12207,12210],{"style":109},[33,12208],{"className":12209,"style":114},[113],[33,12211,12213],{"className":12212},[118,119,120,121],[33,12214,12216],{"className":12215},[84,121],[33,12217,402],{"className":12218},[84,121],[33,12220,12221,12224],{"style":130},[33,12222],{"className":12223,"style":114},[113],[33,12225],{"className":12226,"style":138},[137],[33,12228,12229,12232],{"style":141},[33,12230],{"className":12231,"style":114},[113],[33,12233,12235],{"className":12234},[118,119,120,121],[33,12236,12238],{"className":12237},[84,121],[33,12239,57],{"className":12240,"style":155},[84,154,121],[33,12242,160],{"className":12243},[159],[33,12245,12247],{"className":12246},[101],[33,12248,12250],{"className":12249,"style":167},[105],[33,12251],{},[33,12253],{"className":12254},[173,89],[15,12256,12257],{},"Therefore:",[15,12259,12261],{"className":12260},[575],[33,12262,12264,12302],{"className":12263},[36],[33,12265,12267],{"className":12266},[40],[42,12268,12269],{"xmlns":44},[46,12270,12271,12299],{},[49,12272,12273,12275,12277,12283,12285,12291,12293],{},[589,12274,7241],{},[274,12276,276],{},[52,12278,12279,12281],{},[59,12280,270],{},[59,12282,402],{},[274,12284,7286],{},[52,12286,12287,12289],{},[55,12288,57],{},[59,12290,402],{},[274,12292,276],{},[52,12294,12295,12297],{},[55,12296,57],{},[59,12298,61],{},[63,12300,12301],{"encoding":65},"\\text{Average} = \\frac{1}{2} \\cdot \\frac{\\pi}{2} = \\frac{\\pi}{4}",[33,12303,12305,12326,12409,12492],{"className":12304,"ariaHidden":71},[70],[33,12306,12308,12311,12317,12320,12323],{"className":12307},[75],[33,12309],{"className":12310,"style":7319},[79],[33,12312,12314],{"className":12313},[84,608],[33,12315,7241],{"className":12316},[84],[33,12318],{"className":12319,"style":365},[364],[33,12321,276],{"className":12322},[369],[33,12324],{"className":12325,"style":365},[364],[33,12327,12329,12332,12400,12403,12406],{"className":12328},[75],[33,12330],{"className":12331,"style":291},[79],[33,12333,12335,12338,12397],{"className":12334},[84],[33,12336],{"className":12337},[88,89],[33,12339,12341],{"className":12340},[52],[33,12342,12344,12389],{"className":12343},[96,97],[33,12345,12347,12386],{"className":12346},[101],[33,12348,12350,12364,12372],{"className":12349,"style":310},[105],[33,12351,12352,12355],{"style":109},[33,12353],{"className":12354,"style":114},[113],[33,12356,12358],{"className":12357},[118,119,120,121],[33,12359,12361],{"className":12360},[84,121],[33,12362,402],{"className":12363},[84,121],[33,12365,12366,12369],{"style":130},[33,12367],{"className":12368,"style":114},[113],[33,12370],{"className":12371,"style":138},[137],[33,12373,12374,12377],{"style":141},[33,12375],{"className":12376,"style":114},[113],[33,12378,12380],{"className":12379},[118,119,120,121],[33,12381,12383],{"className":12382},[84,121],[33,12384,270],{"className":12385},[84,121],[33,12387,160],{"className":12388},[159],[33,12390,12392],{"className":12391},[101],[33,12393,12395],{"className":12394,"style":167},[105],[33,12396],{},[33,12398],{"className":12399},[173,89],[33,12401],{"className":12402,"style":2210},[364],[33,12404,7286],{"className":12405},[2214],[33,12407],{"className":12408,"style":2210},[364],[33,12410,12412,12415,12483,12486,12489],{"className":12411},[75],[33,12413],{"className":12414,"style":80},[79],[33,12416,12418,12421,12480],{"className":12417},[84],[33,12419],{"className":12420},[88,89],[33,12422,12424],{"className":12423},[52],[33,12425,12427,12472],{"className":12426},[96,97],[33,12428,12430,12469],{"className":12429},[101],[33,12431,12433,12447,12455],{"className":12432,"style":106},[105],[33,12434,12435,12438],{"style":109},[33,12436],{"className":12437,"style":114},[113],[33,12439,12441],{"className":12440},[118,119,120,121],[33,12442,12444],{"className":12443},[84,121],[33,12445,402],{"className":12446},[84,121],[33,12448,12449,12452],{"style":130},[33,12450],{"className":12451,"style":114},[113],[33,12453],{"className":12454,"style":138},[137],[33,12456,12457,12460],{"style":141},[33,12458],{"className":12459,"style":114},[113],[33,12461,12463],{"className":12462},[118,119,120,121],[33,12464,12466],{"className":12465},[84,121],[33,12467,57],{"className":12468,"style":155},[84,154,121],[33,12470,160],{"className":12471},[159],[33,12473,12475],{"className":12474},[101],[33,12476,12478],{"className":12477,"style":167},[105],[33,12479],{},[33,12481],{"className":12482},[173,89],[33,12484],{"className":12485,"style":365},[364],[33,12487,276],{"className":12488},[369],[33,12490],{"className":12491,"style":365},[364],[33,12493,12495,12498],{"className":12494},[75],[33,12496],{"className":12497,"style":80},[79],[33,12499,12501,12504,12563],{"className":12500},[84],[33,12502],{"className":12503},[88,89],[33,12505,12507],{"className":12506},[52],[33,12508,12510,12555],{"className":12509},[96,97],[33,12511,12513,12552],{"className":12512},[101],[33,12514,12516,12530,12538],{"className":12515,"style":106},[105],[33,12517,12518,12521],{"style":109},[33,12519],{"className":12520,"style":114},[113],[33,12522,12524],{"className":12523},[118,119,120,121],[33,12525,12527],{"className":12526},[84,121],[33,12528,61],{"className":12529},[84,121],[33,12531,12532,12535],{"style":130},[33,12533],{"className":12534,"style":114},[113],[33,12536],{"className":12537,"style":138},[137],[33,12539,12540,12543],{"style":141},[33,12541],{"className":12542,"style":114},[113],[33,12544,12546],{"className":12545},[118,119,120,121],[33,12547,12549],{"className":12548},[84,121],[33,12550,57],{"className":12551,"style":155},[84,154,121],[33,12553,160],{"className":12554},[159],[33,12556,12558],{"className":12557},[101],[33,12559,12561],{"className":12560,"style":167},[105],[33,12562],{},[33,12564],{"className":12565},[173,89],[15,12567,12568,12569,12577,12578,12606,12607,12610],{},"This connection was ",[19,12570,12573,12574],{"href":12571,"rel":12572},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.14487",[23],"discovered in 2025 by the mathematician ",[237,12575,12576],{},"Jim Propp",". And what makes it extraordinary is not that ",[33,12579,12581,12594],{"className":12580},[36],[33,12582,12584],{"className":12583},[40],[42,12585,12586],{"xmlns":44},[46,12587,12588,12592],{},[49,12589,12590],{},[55,12591,57],{},[63,12593,194],{"encoding":65},[33,12595,12597],{"className":12596,"ariaHidden":71},[70],[33,12598,12600,12603],{"className":12599},[75],[33,12601],{"className":12602,"style":204},[79],[33,12604,57],{"className":12605,"style":155},[84,154]," 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 ",[210,12608,12609],{},"exactly"," the same series that defines a trigonometric function.",[15,12612,12613],{},"It is not an approximation or a numerical coincidence — it is an exact mathematical identity.",[15,12615,12616,12617,12645,12646,11633],{},"Of course, that does not make it a practical way to compute ",[33,12618,12620,12633],{"className":12619},[36],[33,12621,12623],{"className":12622},[40],[42,12624,12625],{"xmlns":44},[46,12626,12627,12631],{},[49,12628,12629],{},[55,12630,57],{},[63,12632,194],{"encoding":65},[33,12634,12636],{"className":12635,"ariaHidden":71},[70],[33,12637,12639,12642],{"className":12638},[75],[33,12640],{"className":12641,"style":204},[79],[33,12643,57],{"className":12644,"style":155},[84,154],". As Propp himself ",[19,12647,12649],{"href":12571,"rel":12648},[23],"told in the Arxiv paper itself",[12651,12652,12653],"blockquote",{},[15,12654,12655],{},"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.",[544,12657,12659],{"id":12658},"final-reflection","Final reflection",[15,12661,12662],{},"In the previous article, we discovered that Pascal's triangle connected combinatorics, algebra, and probability. Three languages for the same story.",[15,12664,12665,12666,174],{},"Today we have added a fourth character to the plot: the number ",[33,12667,12669,12682],{"className":12668},[36],[33,12670,12672],{"className":12671},[40],[42,12673,12674],{"xmlns":44},[46,12675,12676,12680],{},[49,12677,12678],{},[55,12679,57],{},[63,12681,194],{"encoding":65},[33,12683,12685],{"className":12684,"ariaHidden":71},[70],[33,12686,12688,12691],{"className":12687},[75],[33,12689],{"className":12690,"style":204},[79],[33,12692,57],{"className":12693,"style":155},[84,154],[15,12695,12696,12697,12725],{},"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 ",[33,12698,12700,12713],{"className":12699},[36],[33,12701,12703],{"className":12702},[40],[42,12704,12705],{"xmlns":44},[46,12706,12707,12711],{},[49,12708,12709],{},[55,12710,57],{},[63,12712,194],{"encoding":65},[33,12714,12716],{"className":12715,"ariaHidden":71},[70],[33,12717,12719,12722],{"className":12718},[75],[33,12720],{"className":12721,"style":204},[79],[33,12723,57],{"className":12724,"style":155},[84,154]," seems to belong exclusively to geometry — to circles, to curves, to the continuous world.",[15,12727,12728],{},"And yet, they all end up connected in a single equation.",[15,12730,12731,12732,12760],{},"There is something deeply satisfying about discovering that a new fact about ",[33,12733,12735,12748],{"className":12734},[36],[33,12736,12738],{"className":12737},[40],[42,12739,12740],{"xmlns":44},[46,12741,12742,12746],{},[49,12743,12744],{},[55,12745,57],{},[63,12747,194],{"encoding":65},[33,12749,12751],{"className":12750,"ariaHidden":71},[70],[33,12752,12754,12757],{"className":12753},[75],[33,12755],{"className":12756,"style":204},[79],[33,12758,57],{"className":12759,"style":155},[84,154]," was hiding, all along, behind something as mundane as flipping a coin. And that nobody noticed until 2025.",[15,12762,12763],{},"Mathematics still has secrets. And sometimes, they are closer than we think.",[15,12765,12766,12767,12771],{},"Put the theory into practice: ",[19,12768,12770],{"href":561,"rel":12769},[23],"flip a coin online"," and watch the patterns unfold.",{"title":1274,"searchDepth":12773,"depth":12773,"links":12774},2,[12775,12783,12784,12785,12786,12787],{"id":546,"depth":12773,"text":547,"children":12776},[12777,12779,12780,12781,12782],{"id":567,"depth":12778,"text":568},3,{"id":835,"depth":12778,"text":836},{"id":1249,"depth":12778,"text":1250},{"id":2142,"depth":12778,"text":2143},{"id":2152,"depth":12778,"text":2153},{"id":4061,"depth":12773,"text":4062},{"id":6136,"depth":12773,"text":6137},{"id":6342,"depth":12773,"text":6343},{"id":9156,"depth":12773,"text":9157},{"id":12658,"depth":12773,"text":12659},"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.","md",{"excerpt":12792},{"type":8,"value":12793},[12794,12796,12901,12933],[11,12795,5],{"id":13},[15,12797,17,12798,25,12801,31,12804,174],{},[19,12799,24],{"href":21,"rel":12800},[23],[19,12802,30],{"href":28,"rel":12803},[23],[33,12805,12807,12824],{"className":12806},[36],[33,12808,12810],{"className":12809},[40],[42,12811,12812],{"xmlns":44},[46,12813,12814,12822],{},[49,12815,12816],{},[52,12817,12818,12820],{},[55,12819,57],{},[59,12821,61],{},[63,12823,66],{"encoding":65},[33,12825,12827],{"className":12826,"ariaHidden":71},[70],[33,12828,12830,12833],{"className":12829},[75],[33,12831],{"className":12832,"style":80},[79],[33,12834,12836,12839,12898],{"className":12835},[84],[33,12837],{"className":12838},[88,89],[33,12840,12842],{"className":12841},[52],[33,12843,12845,12890],{"className":12844},[96,97],[33,12846,12848,12887],{"className":12847},[101],[33,12849,12851,12865,12873],{"className":12850,"style":106},[105],[33,12852,12853,12856],{"style":109},[33,12854],{"className":12855,"style":114},[113],[33,12857,12859],{"className":12858},[118,119,120,121],[33,12860,12862],{"className":12861},[84,121],[33,12863,61],{"className":12864},[84,121],[33,12866,12867,12870],{"style":130},[33,12868],{"className":12869,"style":114},[113],[33,12871],{"className":12872,"style":138},[137],[33,12874,12875,12878],{"style":141},[33,12876],{"className":12877,"style":114},[113],[33,12879,12881],{"className":12880},[118,119,120,121],[33,12882,12884],{"className":12883},[84,121],[33,12885,57],{"className":12886,"style":155},[84,154,121],[33,12888,160],{"className":12889},[159],[33,12891,12893],{"className":12892},[101],[33,12894,12896],{"className":12895,"style":167},[105],[33,12897],{},[33,12899],{"className":12900},[173,89],[15,12902,177,12903,208,12931,213],{},[33,12904,12906,12919],{"className":12905},[36],[33,12907,12909],{"className":12908},[40],[42,12910,12911],{"xmlns":44},[46,12912,12913,12917],{},[49,12914,12915],{},[55,12916,57],{},[63,12918,194],{"encoding":65},[33,12920,12922],{"className":12921,"ariaHidden":71},[70],[33,12923,12925,12928],{"className":12924},[75],[33,12926],{"className":12927,"style":204},[79],[33,12929,57],{"className":12930,"style":155},[84,154],[210,12932,212],{},[15,12934,216,12935,222],{},[19,12936,221],{"href":219,"rel":12937},[23],true,"\u002Fblog\u002Fpi-coin-flips-catalan-numbers.en",{"title":5,"description":12789},"blog\u002Fpi-coin-flips-catalan-numbers.en","yLAG06Mz86DGlgmAadvrR3wuMIGXlrhmsAYvHgh8-5Y",{"en":13,"es":12944},"lanzando-una-moneda-hasta-obtener-pi",{"en":13,"es":12944},1779926068279]