[{"data":1,"prerenderedAt":5464},["ShallowReactive",2],{"blog-post-en-what-is-a-probability-distribution":3,"locale-switcher-blog-en-what-is-a-probability-distribution":1631,"blog-siblings-en-what-is-a-probability-distribution":1633,"blog-series-nav-en-what-is-a-probability-distribution":1634,"post-authors":5452},{"id":4,"title":5,"_locale":6,"author":7,"body":8,"date":1604,"description":1605,"excerpt":1606,"extension":1616,"level":1617,"meta":1618,"navigation":1619,"path":1620,"published":1619,"seo":1621,"series":1622,"seriesOrder":1623,"slug":14,"stem":1624,"tags":1625,"__hash__":1630},"blog\u002Fblog\u002Fwhat-is-a-probability-distribution.en.md","What Is a Probability Distribution?","en",null,{"type":9,"value":10,"toc":1596},"minimal",[11,15,29,34,44,47,385,396,618,637,641,766,769,1084,1117,1390,1408,1521,1525,1542,1553,1557,1568,1571,1578,1588],[12,13,5],"h1",{"id":14},"what-is-a-probability-distribution",[16,17,18,19,23,24,28],"p",{},"Flip a coin once and the outcome is a coin toss. Flip it a hundred times and something more interesting appears: a ",[20,21,22],"em",{},"shape",". Count how often each result happens and you are no longer looking at luck — you are looking at a ",[25,26,27],"strong",{},"probability distribution",". This short article explains what that means, introduces the most famous distribution born from coin flips, and lets you watch one build itself, ball by ball, in an interactive Galton board.",[30,31,33],"h2",{"id":32},"a-distribution-is-just-a-map-of-how-likely","A distribution is just a map of \"how likely\"",[16,35,36,37,39,40,43],{},"A ",[25,38,27],{}," is a complete answer to the question ",[20,41,42],{},"\"what can happen, and how likely is each thing?\""," It pairs every possible outcome with its probability.",[16,45,46],{},"For a single fair coin flip, the distribution is tiny:",[16,48,51],{"className":49},[50],"math-block",[52,53,56,135],"span",{"className":54},[55],"katex",[52,57,60],{"className":58},[59],"katex-mathml",[61,62,64],"math",{"xmlns":63},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[65,66,67,130],"semantics",{},[68,69,70,74,79,83,86,89,103,107,111,113,115,118,120,122],"mrow",{},[71,72,73],"mi",{},"P",[75,76,78],"mo",{"stretchy":77},"false","(",[80,81,82],"mtext",{},"heads",[75,84,85],{"stretchy":77},")",[75,87,88],{},"=",[90,91,93],"mstyle",{"scriptlevel":92,"displaystyle":77},"0",[94,95,96,100],"mfrac",{},[97,98,99],"mn",{},"1",[97,101,102],{},"2",[75,104,106],{"separator":105},"true",",",[108,109],"mspace",{"width":110},"2em",[71,112,73],{},[75,114,78],{"stretchy":77},[80,116,117],{},"tails",[75,119,85],{"stretchy":77},[75,121,88],{},[90,123,124],{"scriptlevel":92,"displaystyle":77},[94,125,126,128],{},[97,127,99],{},[97,129,102],{},[131,132,134],"annotation",{"encoding":133},"application\u002Fx-tex","P(\\text{heads}) = \\tfrac{1}{2}, \\qquad P(\\text{tails}) = \\tfrac{1}{2}",[52,136,139,180,311],{"className":137,"ariaHidden":105},[138],"katex-html",[52,140,143,148,154,158,165,169,173,177],{"className":141},[142],"base",[52,144],{"className":145,"style":147},[146],"strut","height:1em;vertical-align:-0.25em;",[52,149,73],{"className":150,"style":153},[151,152],"mord","mathnormal","margin-right:0.1389em;",[52,155,78],{"className":156},[157],"mopen",[52,159,162],{"className":160},[151,161],"text",[52,163,82],{"className":164},[151],[52,166,85],{"className":167},[168],"mclose",[52,170],{"className":171,"style":172},[108],"margin-right:0.2778em;",[52,174,88],{"className":175},[176],"mrel",[52,178],{"className":179,"style":172},[108],[52,181,183,187,275,279,283,287,290,293,299,302,305,308],{"className":182},[142],[52,184],{"className":185,"style":186},[146],"height:1.1901em;vertical-align:-0.345em;",[52,188,190,194,272],{"className":189},[151],[52,191],{"className":192},[157,193],"nulldelimiter",[52,195,197],{"className":196},[94],[52,198,202,263],{"className":199},[200,201],"vlist-t","vlist-t2",[52,203,206,258],{"className":204},[205],"vlist-r",[52,207,211,232,243],{"className":208,"style":210},[209],"vlist","height:0.8451em;",[52,212,214,219],{"style":213},"top:-2.655em;",[52,215],{"className":216,"style":218},[217],"pstrut","height:3em;",[52,220,226],{"className":221},[222,223,224,225],"sizing","reset-size6","size3","mtight",[52,227,229],{"className":228},[151,225],[52,230,102],{"className":231},[151,225],[52,233,235,238],{"style":234},"top:-3.23em;",[52,236],{"className":237,"style":218},[217],[52,239],{"className":240,"style":242},[241],"frac-line","border-bottom-width:0.04em;",[52,244,246,249],{"style":245},"top:-3.394em;",[52,247],{"className":248,"style":218},[217],[52,250,252],{"className":251},[222,223,224,225],[52,253,255],{"className":254},[151,225],[52,256,99],{"className":257},[151,225],[52,259,262],{"className":260},[261],"vlist-s","​",[52,264,266],{"className":265},[205],[52,267,270],{"className":268,"style":269},[209],"height:0.345em;",[52,271],{},[52,273],{"className":274},[168,193],[52,276,106],{"className":277},[278],"mpunct",[52,280],{"className":281,"style":282},[108],"margin-right:2em;",[52,284],{"className":285,"style":286},[108],"margin-right:0.1667em;",[52,288,73],{"className":289,"style":153},[151,152],[52,291,78],{"className":292},[157],[52,294,296],{"className":295},[151,161],[52,297,117],{"className":298},[151],[52,300,85],{"className":301},[168],[52,303],{"className":304,"style":172},[108],[52,306,88],{"className":307},[176],[52,309],{"className":310,"style":172},[108],[52,312,314,317],{"className":313},[142],[52,315],{"className":316,"style":186},[146],[52,318,320,323,382],{"className":319},[151],[52,321],{"className":322},[157,193],[52,324,326],{"className":325},[94],[52,327,329,374],{"className":328},[200,201],[52,330,332,371],{"className":331},[205],[52,333,335,349,357],{"className":334,"style":210},[209],[52,336,337,340],{"style":213},[52,338],{"className":339,"style":218},[217],[52,341,343],{"className":342},[222,223,224,225],[52,344,346],{"className":345},[151,225],[52,347,102],{"className":348},[151,225],[52,350,351,354],{"style":234},[52,352],{"className":353,"style":218},[217],[52,355],{"className":356,"style":242},[241],[52,358,359,362],{"style":245},[52,360],{"className":361,"style":218},[217],[52,363,365],{"className":364},[222,223,224,225],[52,366,368],{"className":367},[151,225],[52,369,99],{"className":370},[151,225],[52,372,262],{"className":373},[261],[52,375,377],{"className":376},[205],[52,378,380],{"className":379,"style":269},[209],[52,381],{},[52,383],{"className":384},[168,193],[16,386,387,388,391,392,395],{},"Two outcomes, each with probability one-half. There is one rule a distribution can never break: ",[25,389,390],{},"all the probabilities have to add up to 1",", because ",[20,393,394],{},"something"," always happens.",[16,397,399],{"className":398},[50],[52,400,402,437],{"className":401},[55],[52,403,405],{"className":404},[59],[61,406,407],{"xmlns":63},[65,408,409,434],{},[68,410,411,419,422,430,432],{},[90,412,413],{"scriptlevel":92,"displaystyle":77},[94,414,415,417],{},[97,416,99],{},[97,418,102],{},[75,420,421],{},"+",[90,423,424],{"scriptlevel":92,"displaystyle":77},[94,425,426,428],{},[97,427,99],{},[97,429,102],{},[75,431,88],{},[97,433,99],{},[131,435,436],{"encoding":133},"\\tfrac{1}{2} + \\tfrac{1}{2} = 1",[52,438,440,525,608],{"className":439,"ariaHidden":105},[138],[52,441,443,446,514,518,522],{"className":442},[142],[52,444],{"className":445,"style":186},[146],[52,447,449,452,511],{"className":448},[151],[52,450],{"className":451},[157,193],[52,453,455],{"className":454},[94],[52,456,458,503],{"className":457},[200,201],[52,459,461,500],{"className":460},[205],[52,462,464,478,486],{"className":463,"style":210},[209],[52,465,466,469],{"style":213},[52,467],{"className":468,"style":218},[217],[52,470,472],{"className":471},[222,223,224,225],[52,473,475],{"className":474},[151,225],[52,476,102],{"className":477},[151,225],[52,479,480,483],{"style":234},[52,481],{"className":482,"style":218},[217],[52,484],{"className":485,"style":242},[241],[52,487,488,491],{"style":245},[52,489],{"className":490,"style":218},[217],[52,492,494],{"className":493},[222,223,224,225],[52,495,497],{"className":496},[151,225],[52,498,99],{"className":499},[151,225],[52,501,262],{"className":502},[261],[52,504,506],{"className":505},[205],[52,507,509],{"className":508,"style":269},[209],[52,510],{},[52,512],{"className":513},[168,193],[52,515],{"className":516,"style":517},[108],"margin-right:0.2222em;",[52,519,421],{"className":520},[521],"mbin",[52,523],{"className":524,"style":517},[108],[52,526,528,531,599,602,605],{"className":527},[142],[52,529],{"className":530,"style":186},[146],[52,532,534,537,596],{"className":533},[151],[52,535],{"className":536},[157,193],[52,538,540],{"className":539},[94],[52,541,543,588],{"className":542},[200,201],[52,544,546,585],{"className":545},[205],[52,547,549,563,571],{"className":548,"style":210},[209],[52,550,551,554],{"style":213},[52,552],{"className":553,"style":218},[217],[52,555,557],{"className":556},[222,223,224,225],[52,558,560],{"className":559},[151,225],[52,561,102],{"className":562},[151,225],[52,564,565,568],{"style":234},[52,566],{"className":567,"style":218},[217],[52,569],{"className":570,"style":242},[241],[52,572,573,576],{"style":245},[52,574],{"className":575,"style":218},[217],[52,577,579],{"className":578},[222,223,224,225],[52,580,582],{"className":581},[151,225],[52,583,99],{"className":584},[151,225],[52,586,262],{"className":587},[261],[52,589,591],{"className":590},[205],[52,592,594],{"className":593,"style":269},[209],[52,595],{},[52,597],{"className":598},[168,193],[52,600],{"className":601,"style":172},[108],[52,603,88],{"className":604},[176],[52,606],{"className":607,"style":172},[108],[52,609,611,615],{"className":610},[142],[52,612],{"className":613,"style":614},[146],"height:0.6444em;",[52,616,99],{"className":617},[151],[16,619,620,621,624,625,628,629,636],{},"That is the whole idea. Distributions get interesting not when one event happens, but when we repeat an experiment many times and ask how the results spread out. (If the words ",[20,622,623],{},"probability"," and ",[20,626,627],{},"likelihood"," still feel slippery, start with ",[630,631,635],"a",{"href":632,"rel":633},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fprobability-basics\u002F",[634],"nofollow","Probability Basics",".)",[30,638,640],{"id":639},"the-binomial-distribution","The binomial distribution",[16,642,643,644,674,675,703,704,732,733,736,737,765],{},"Now flip a fair coin ",[52,645,647,661],{"className":646},[55],[52,648,650],{"className":649},[59],[61,651,652],{"xmlns":63},[65,653,654,659],{},[68,655,656],{},[71,657,658],{},"n",[131,660,658],{"encoding":133},[52,662,664],{"className":663,"ariaHidden":105},[138],[52,665,667,671],{"className":666},[142],[52,668],{"className":669,"style":670},[146],"height:0.4306em;",[52,672,658],{"className":673},[151,152]," times and count the heads. The number of heads can be anything from ",[52,676,678,691],{"className":677},[55],[52,679,681],{"className":680},[59],[61,682,683],{"xmlns":63},[65,684,685,689],{},[68,686,687],{},[97,688,92],{},[131,690,92],{"encoding":133},[52,692,694],{"className":693,"ariaHidden":105},[138],[52,695,697,700],{"className":696},[142],[52,698],{"className":699,"style":614},[146],[52,701,92],{"className":702},[151]," to ",[52,705,707,720],{"className":706},[55],[52,708,710],{"className":709},[59],[61,711,712],{"xmlns":63},[65,713,714,718],{},[68,715,716],{},[71,717,658],{},[131,719,658],{"encoding":133},[52,721,723],{"className":722,"ariaHidden":105},[138],[52,724,726,729],{"className":725},[142],[52,727],{"className":728,"style":670},[146],[52,730,658],{"className":731},[151,152],", and the probabilities of each count form the ",[25,734,735],{},"binomial distribution"," — the distribution of \"how many successes in ",[52,738,740,753],{"className":739},[55],[52,741,743],{"className":742},[59],[61,744,745],{"xmlns":63},[65,746,747,751],{},[68,748,749],{},[71,750,658],{},[131,752,658],{"encoding":133},[52,754,756],{"className":755,"ariaHidden":105},[138],[52,757,759,762],{"className":758},[142],[52,760],{"className":761,"style":670},[146],[52,763,658],{"className":764},[151,152]," independent yes\u002Fno trials\".",[16,767,768],{},"For a fair coin its formula is:",[16,770,772],{"className":771},[50],[52,773,775,838],{"className":774},[55],[52,776,778],{"className":777},[59],[61,779,780],{"xmlns":63},[65,781,782,835],{},[68,783,784,786,788,791,794,796,799,801,803,816],{},[71,785,73],{},[75,787,78],{"stretchy":77},[71,789,790],{},"k",[80,792,793],{}," heads in ",[71,795,658],{},[80,797,798],{}," flips",[75,800,85],{"stretchy":77},[75,802,88],{},[68,804,805,807,814],{},[75,806,78],{"fence":105},[94,808,810,812],{"linethickness":809},"0px",[71,811,658],{},[71,813,790],{},[75,815,85],{"fence":105},[817,818,819,833],"msup",{},[68,820,821,823,831],{},[75,822,78],{"fence":105},[90,824,825],{"scriptlevel":92,"displaystyle":77},[94,826,827,829],{},[97,828,99],{},[97,830,102],{},[75,832,85],{"fence":105},[71,834,658],{},[131,836,837],{"encoding":133},"P(k \\text{ heads in } n \\text{ flips}) = \\binom{n}{k}\\left(\\tfrac{1}{2}\\right)^{n}",[52,839,841,884],{"className":840,"ariaHidden":105},[138],[52,842,844,847,850,853,857,863,866,872,875,878,881],{"className":843},[142],[52,845],{"className":846,"style":147},[146],[52,848,73],{"className":849,"style":153},[151,152],[52,851,78],{"className":852},[157],[52,854,790],{"className":855,"style":856},[151,152],"margin-right:0.0315em;",[52,858,860],{"className":859},[151,161],[52,861,793],{"className":862},[151],[52,864,658],{"className":865},[151,152],[52,867,869],{"className":868},[151,161],[52,870,798],{"className":871},[151],[52,873,85],{"className":874},[168],[52,876],{"className":877,"style":172},[108],[52,879,88],{"className":880},[176],[52,882],{"className":883,"style":172},[108],[52,885,887,891,965,968],{"className":886},[142],[52,888],{"className":889,"style":890},[146],"height:1.2543em;vertical-align:-0.35em;",[52,892,894,904,959],{"className":893},[151],[52,895,899],{"className":896,"style":898},[157,897],"delimcenter","top:0em;",[52,900,78],{"className":901},[902,903],"delimsizing","size1",[52,905,907],{"className":906},[94],[52,908,910,951],{"className":909},[200,201],[52,911,913,948],{"className":912},[205],[52,914,917,933],{"className":915,"style":916},[209],"height:0.7454em;",[52,918,920,924],{"style":919},"top:-2.355em;",[52,921],{"className":922,"style":923},[217],"height:2.7em;",[52,925,927],{"className":926},[222,223,224,225],[52,928,930],{"className":929},[151,225],[52,931,790],{"className":932,"style":856},[151,152,225],[52,934,936,939],{"style":935},"top:-3.144em;",[52,937],{"className":938,"style":923},[217],[52,940,942],{"className":941},[222,223,224,225],[52,943,945],{"className":944},[151,225],[52,946,658],{"className":947},[151,152,225],[52,949,262],{"className":950},[261],[52,952,954],{"className":953},[205],[52,955,957],{"className":956,"style":269},[209],[52,958],{},[52,960,962],{"className":961,"style":898},[168,897],[52,963,85],{"className":964},[902,903],[52,966],{"className":967,"style":286},[108],[52,969,972,1055],{"className":970},[971],"minner",[52,973,975,981,1049],{"className":974},[971],[52,976,978],{"className":977,"style":898},[157,897],[52,979,78],{"className":980},[902,903],[52,982,984,987,1046],{"className":983},[151],[52,985],{"className":986},[157,193],[52,988,990],{"className":989},[94],[52,991,993,1038],{"className":992},[200,201],[52,994,996,1035],{"className":995},[205],[52,997,999,1013,1021],{"className":998,"style":210},[209],[52,1000,1001,1004],{"style":213},[52,1002],{"className":1003,"style":218},[217],[52,1005,1007],{"className":1006},[222,223,224,225],[52,1008,1010],{"className":1009},[151,225],[52,1011,102],{"className":1012},[151,225],[52,1014,1015,1018],{"style":234},[52,1016],{"className":1017,"style":218},[217],[52,1019],{"className":1020,"style":242},[241],[52,1022,1023,1026],{"style":245},[52,1024],{"className":1025,"style":218},[217],[52,1027,1029],{"className":1028},[222,223,224,225],[52,1030,1032],{"className":1031},[151,225],[52,1033,99],{"className":1034},[151,225],[52,1036,262],{"className":1037},[261],[52,1039,1041],{"className":1040},[205],[52,1042,1044],{"className":1043,"style":269},[209],[52,1045],{},[52,1047],{"className":1048},[168,193],[52,1050,1052],{"className":1051,"style":898},[168,897],[52,1053,85],{"className":1054},[902,903],[52,1056,1059],{"className":1057},[1058],"msupsub",[52,1060,1062],{"className":1061},[200],[52,1063,1065],{"className":1064},[205],[52,1066,1069],{"className":1067,"style":1068},[209],"height:0.9043em;",[52,1070,1072,1075],{"style":1071},"top:-3.3029em;margin-right:0.05em;",[52,1073],{"className":1074,"style":923},[217],[52,1076,1078],{"className":1077},[222,223,224,225],[52,1079,1081],{"className":1080},[151,225],[52,1082,658],{"className":1083},[151,152,225],[16,1085,1086,1087,1116],{},"and more generally, when \"heads\" has probability ",[52,1088,1090,1103],{"className":1089},[55],[52,1091,1093],{"className":1092},[59],[61,1094,1095],{"xmlns":63},[65,1096,1097,1101],{},[68,1098,1099],{},[71,1100,16],{},[131,1102,16],{"encoding":133},[52,1104,1106],{"className":1105,"ariaHidden":105},[138],[52,1107,1109,1113],{"className":1108},[142],[52,1110],{"className":1111,"style":1112},[146],"height:0.625em;vertical-align:-0.1944em;",[52,1114,16],{"className":1115},[151,152]," (not necessarily one-half):",[16,1118,1120],{"className":1119},[50],[52,1121,1123,1187],{"className":1122},[55],[52,1124,1126],{"className":1125},[59],[61,1127,1128],{"xmlns":63},[65,1129,1130,1184],{},[68,1131,1132,1134,1136,1138,1140,1142,1154,1157,1163,1165,1167,1170,1172],{},[71,1133,73],{},[75,1135,78],{"stretchy":77},[71,1137,790],{},[75,1139,85],{"stretchy":77},[75,1141,88],{},[68,1143,1144,1146,1152],{},[75,1145,78],{"fence":105},[94,1147,1148,1150],{"linethickness":809},[71,1149,658],{},[71,1151,790],{},[75,1153,85],{"fence":105},[80,1155,1156],{}," ",[817,1158,1159,1161],{},[71,1160,16],{},[71,1162,790],{},[75,1164,78],{"stretchy":77},[97,1166,99],{},[75,1168,1169],{},"−",[71,1171,16],{},[817,1173,1174,1176],{},[75,1175,85],{"stretchy":77},[68,1177,1178,1180,1182],{},[71,1179,658],{},[75,1181,1169],{},[71,1183,790],{},[131,1185,1186],{"encoding":133},"P(k) = \\binom{n}{k}\\, p^{k} (1-p)^{n-k}",[52,1188,1190,1217,1342],{"className":1189,"ariaHidden":105},[138],[52,1191,1193,1196,1199,1202,1205,1208,1211,1214],{"className":1192},[142],[52,1194],{"className":1195,"style":147},[146],[52,1197,73],{"className":1198,"style":153},[151,152],[52,1200,78],{"className":1201},[157],[52,1203,790],{"className":1204,"style":856},[151,152],[52,1206,85],{"className":1207},[168],[52,1209],{"className":1210,"style":172},[108],[52,1212,88],{"className":1213},[176],[52,1215],{"className":1216,"style":172},[108],[52,1218,1220,1224,1290,1293,1327,1330,1333,1336,1339],{"className":1219},[142],[52,1221],{"className":1222,"style":1223},[146],"height:1.2em;vertical-align:-0.35em;",[52,1225,1227,1233,1284],{"className":1226},[151],[52,1228,1230],{"className":1229,"style":898},[157,897],[52,1231,78],{"className":1232},[902,903],[52,1234,1236],{"className":1235},[94],[52,1237,1239,1276],{"className":1238},[200,201],[52,1240,1242,1273],{"className":1241},[205],[52,1243,1245,1259],{"className":1244,"style":916},[209],[52,1246,1247,1250],{"style":919},[52,1248],{"className":1249,"style":923},[217],[52,1251,1253],{"className":1252},[222,223,224,225],[52,1254,1256],{"className":1255},[151,225],[52,1257,790],{"className":1258,"style":856},[151,152,225],[52,1260,1261,1264],{"style":935},[52,1262],{"className":1263,"style":923},[217],[52,1265,1267],{"className":1266},[222,223,224,225],[52,1268,1270],{"className":1269},[151,225],[52,1271,658],{"className":1272},[151,152,225],[52,1274,262],{"className":1275},[261],[52,1277,1279],{"className":1278},[205],[52,1280,1282],{"className":1281,"style":269},[209],[52,1283],{},[52,1285,1287],{"className":1286,"style":898},[168,897],[52,1288,85],{"className":1289},[902,903],[52,1291],{"className":1292,"style":286},[108],[52,1294,1296,1299],{"className":1295},[151],[52,1297,16],{"className":1298},[151,152],[52,1300,1302],{"className":1301},[1058],[52,1303,1305],{"className":1304},[200],[52,1306,1308],{"className":1307},[205],[52,1309,1312],{"className":1310,"style":1311},[209],"height:0.8491em;",[52,1313,1315,1318],{"style":1314},"top:-3.063em;margin-right:0.05em;",[52,1316],{"className":1317,"style":923},[217],[52,1319,1321],{"className":1320},[222,223,224,225],[52,1322,1324],{"className":1323},[151,225],[52,1325,790],{"className":1326,"style":856},[151,152,225],[52,1328,78],{"className":1329},[157],[52,1331,99],{"className":1332},[151],[52,1334],{"className":1335,"style":517},[108],[52,1337,1169],{"className":1338},[521],[52,1340],{"className":1341,"style":517},[108],[52,1343,1345,1349,1352],{"className":1344},[142],[52,1346],{"className":1347,"style":1348},[146],"height:1.0991em;vertical-align:-0.25em;",[52,1350,16],{"className":1351},[151,152],[52,1353,1355,1358],{"className":1354},[168],[52,1356,85],{"className":1357},[168],[52,1359,1361],{"className":1360},[1058],[52,1362,1364],{"className":1363},[200],[52,1365,1367],{"className":1366},[205],[52,1368,1370],{"className":1369,"style":1311},[209],[52,1371,1372,1375],{"style":1314},[52,1373],{"className":1374,"style":923},[217],[52,1376,1378],{"className":1377},[222,223,224,225],[52,1379,1381,1384,1387],{"className":1380},[151,225],[52,1382,658],{"className":1383},[151,152,225],[52,1385,1169],{"className":1386},[521,225],[52,1388,790],{"className":1389,"style":856},[151,152,225],[16,1391,1392,1393,1395,1396,1399,1400,1403,1404,1407],{},"The interesting part is the ",[25,1394,22],{},". Extreme results — all heads or all tails — are rare, because there is only ",[20,1397,1398],{},"one"," way to get each of them. Middle results are common, because there are ",[20,1401,1402],{},"many"," different orderings that land there. Plot the probabilities and you get a hump that is tall in the centre and thin at the edges: the beginnings of the famous ",[25,1405,1406],{},"bell curve",".",[16,1409,1410,1411,1511,1512,1517,1518],{},"Why exactly those ",[52,1412,1414,1436],{"className":1413},[55],[52,1415,1417],{"className":1416},[59],[61,1418,1419],{"xmlns":63},[65,1420,1421,1433],{},[68,1422,1423,1425,1431],{},[75,1424,78],{"fence":105},[94,1426,1427,1429],{"linethickness":809},[71,1428,658],{},[71,1430,790],{},[75,1432,85],{"fence":105},[131,1434,1435],{"encoding":133},"\\binom{n}{k}",[52,1437,1439],{"className":1438,"ariaHidden":105},[138],[52,1440,1442,1445],{"className":1441},[142],[52,1443],{"className":1444,"style":1223},[146],[52,1446,1448,1454,1505],{"className":1447},[151],[52,1449,1451],{"className":1450,"style":898},[157,897],[52,1452,78],{"className":1453},[902,903],[52,1455,1457],{"className":1456},[94],[52,1458,1460,1497],{"className":1459},[200,201],[52,1461,1463,1494],{"className":1462},[205],[52,1464,1466,1480],{"className":1465,"style":916},[209],[52,1467,1468,1471],{"style":919},[52,1469],{"className":1470,"style":923},[217],[52,1472,1474],{"className":1473},[222,223,224,225],[52,1475,1477],{"className":1476},[151,225],[52,1478,790],{"className":1479,"style":856},[151,152,225],[52,1481,1482,1485],{"style":935},[52,1483],{"className":1484,"style":923},[217],[52,1486,1488],{"className":1487},[222,223,224,225],[52,1489,1491],{"className":1490},[151,225],[52,1492,658],{"className":1493},[151,152,225],[52,1495,262],{"className":1496},[261],[52,1498,1500],{"className":1499},[205],[52,1501,1503],{"className":1502,"style":269},[209],[52,1504],{},[52,1506,1508],{"className":1507,"style":898},[168,897],[52,1509,85],{"className":1510},[902,903]," counts appear — and why they are the rows of Pascal's triangle — is a story in itself, told in ",[630,1513,1516],{"href":1514,"rel":1515},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fpascals-triangle-combinatorics-and-coin-flips\u002F",[634],"What Do Pascal's Triangle and a Coin Have in Common?",". Here we just need the punchline: ",[25,1519,1520],{},"add up enough fair, independent choices and the results pile up into a bell.",[30,1522,1524],{"id":1523},"who-was-francis-galton","Who was Francis Galton?",[16,1526,1527,1528,1531,1532,1535,1536,624,1539,1407],{},"The most charming way to ",[20,1529,1530],{},"see"," a binomial distribution was dreamed up by ",[25,1533,1534],{},"Sir Francis Galton"," (1822–1911), a restless Victorian polymath and a half-cousin of Charles Darwin (they shared a grandfather, Erasmus Darwin). Galton explored south-west Africa as a young man, then spent the rest of his life measuring and counting almost everything he could — weather (he drew the first newspaper weather map), fingerprints, even the supposed \"efficacy\" of prayer. Along the way he pioneered statistical ideas we still use daily, including ",[25,1537,1538],{},"correlation",[25,1540,1541],{},"regression toward the mean",[16,1543,1544,1545,1548,1549,1552],{},"To demonstrate how order emerges from randomness, Galton built a device he called the ",[25,1546,1547],{},"quincunx"," — today we call it the ",[25,1550,1551],{},"Galton board",". Drop a ball into a triangular array of pegs. At every peg it bounces left or right, more or less at random — exactly like a coin flip. After many rows of pegs, each ball lands in a bin near the bottom. A single ball is unpredictable. But pour in hundreds, and the bins fill up into the unmistakable bell shape of the binomial distribution. Galton had turned an abstract formula into something you can watch with your own eyes.",[30,1554,1556],{"id":1555},"see-it-happen","See it happen",[16,1558,1559,1560,1563,1564,1567],{},"Press ",[25,1561,1562],{},"Drop balls"," and watch the pile build. Each ball is one little experiment; the growing stacks are the distribution drawing itself. Press ",[25,1565,1566],{},"Reset"," to start over.",[1569,1570],"galton-board",{},[16,1572,1573,1574,1577],{},"Notice how no single ball \"knows\" where it should go, yet the crowd reliably sketches the same bell every time. That is the quiet magic of probability distributions: individual events stay unpredictable, but their ",[20,1575,1576],{},"collective"," behaviour is remarkably orderly.",[16,1579,1580,1581,1584,1585,1587],{},"So how does this connect to a single coin? ",[25,1582,1583],{},"Each left-or-right bounce in the board is one coin flip"," — and every ball that reaches the bottom is one whole experiment of several flips, landing in the bin that matches how many times it went \"heads\". That is exactly why a single toss can never reveal a curve: a curve is a pattern that lives in ",[20,1586,1402],{}," results, not in one. Flip once and you simply get heads or tails. Flip again and again, keep count, and those counts slowly trace the same bell you see above.",[16,1589,1590,1595],{},[630,1591,1594],{"href":1592,"rel":1593},"https:\u002F\u002Fflipthecoin.app\u002Fplay\u002F",[634],"Flip a coin online"," and watch your own flip history grow — the distribution only takes shape once the tosses pile up, just like the balls.",{"title":1597,"searchDepth":1598,"depth":1598,"links":1599},"",2,[1600,1601,1602,1603],{"id":32,"depth":1598,"text":33},{"id":639,"depth":1598,"text":640},{"id":1523,"depth":1598,"text":1524},{"id":1555,"depth":1598,"text":1556},"2026-06-12T10:00:00Z","A gentle introduction to probability distributions and the binomial distribution — with an interactive Galton board you can play with.",{"type":9,"value":1607},[1608,1610],[12,1609,5],{"id":14},[16,1611,18,1612,23,1614,28],{},[20,1613,22],{},[25,1615,27],{},"md","beginner",{},true,"\u002Fblog\u002Fwhat-is-a-probability-distribution.en",{"title":5,"description":1605},"probability-basics",3,"blog\u002Fwhat-is-a-probability-distribution.en",[1626,1627,1628,1629,1551],"Probability distribution","Binomial distribution","Francis Galton","Bell curve","NPAVeI_57gw1jMCTERyw0iCILFd-WRe92OdRbgRwtDU",{"en":14,"es":1632},"que-es-una-distribucion-de-probabilidad",{"en":14,"es":1632},{"members":1635,"meta":5442,"seriesSlug":1622},[1636,3600,4029],{"id":1637,"title":635,"_locale":6,"author":7,"body":1638,"date":3580,"description":3581,"excerpt":3582,"extension":1616,"level":1617,"meta":3594,"navigation":1619,"path":3595,"published":1619,"seo":3596,"series":1622,"seriesOrder":3597,"slug":1622,"stem":3598,"tags":7,"__hash__":3599},"blog\u002Fblog\u002Fprobability-basics.en.md",{"type":9,"value":1639,"toc":3568},[1640,1642,1650,1653,1657,1767,1776,1780,1783,1809,1812,1816,1823,1840,1850,1854,1865,1982,1985,2368,2379,2383,2386,2391,2402,2663,2767,2771,2785,3280,3286,3491,3498,3502,3508,3521,3525,3528,3556,3559],[12,1641,635],{"id":1622},[16,1643,1644,1645,1647,1648,1407],{},"Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is a way of quantifying uncertainty and measuring the likelihood of different outcomes. In the context of coin flips, probability is used to determine the likelihood of the coin landing on ",[25,1646,82],{}," or ",[25,1649,117],{},[16,1651,1652],{},"This article is an introduction to the topic of probability. We will stick to plain language and use a coin as the only example. More advanced statistics will come in future articles.",[30,1654,1656],{"id":1655},"the-basics","The Basics",[16,1658,1659,1660,1758,1759,1762,1763,1766],{},"When we talk about probability, this is a mathematical concept that can be represented in a few different ways. The most common way to represent probability is as a fraction, decimal, or percentage. For example, if you flip a fair coin, the probability of it landing on heads is ",[52,1661,1663,1681],{"className":1662},[55],[52,1664,1666],{"className":1665},[59],[61,1667,1668],{"xmlns":63},[65,1669,1670,1678],{},[68,1671,1672],{},[94,1673,1674,1676],{},[97,1675,99],{},[97,1677,102],{},[131,1679,1680],{"encoding":133},"\\frac{1}{2}",[52,1682,1684],{"className":1683,"ariaHidden":105},[138],[52,1685,1687,1690],{"className":1686},[142],[52,1688],{"className":1689,"style":186},[146],[52,1691,1693,1696,1755],{"className":1692},[151],[52,1694],{"className":1695},[157,193],[52,1697,1699],{"className":1698},[94],[52,1700,1702,1747],{"className":1701},[200,201],[52,1703,1705,1744],{"className":1704},[205],[52,1706,1708,1722,1730],{"className":1707,"style":210},[209],[52,1709,1710,1713],{"style":213},[52,1711],{"className":1712,"style":218},[217],[52,1714,1716],{"className":1715},[222,223,224,225],[52,1717,1719],{"className":1718},[151,225],[52,1720,102],{"className":1721},[151,225],[52,1723,1724,1727],{"style":234},[52,1725],{"className":1726,"style":218},[217],[52,1728],{"className":1729,"style":242},[241],[52,1731,1732,1735],{"style":245},[52,1733],{"className":1734,"style":218},[217],[52,1736,1738],{"className":1737},[222,223,224,225],[52,1739,1741],{"className":1740},[151,225],[52,1742,99],{"className":1743},[151,225],[52,1745,262],{"className":1746},[261],[52,1748,1750],{"className":1749},[205],[52,1751,1753],{"className":1752,"style":269},[209],[52,1754],{},[52,1756],{"className":1757},[168,193],", which is the same as ",[25,1760,1761],{},"0.5",", or also ",[25,1764,1765],{},"50%",". The three representations are equivalent and can be used interchangeably.",[16,1768,1769,1770,1775],{},"As we will see in ",[630,1771,1774],{"href":1772,"rel":1773},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fthe-unexpected-statistics-of-coin-flips",[634],"The Unexpected Statistics of Coin Flips blog post",", the probability of a coin landing is not exactly 50-50 every time.",[30,1777,1779],{"id":1778},"outcomes-and-events","Outcomes and events",[16,1781,1782],{},"Before we go further, two small pieces of vocabulary:",[1784,1785,1786,1803],"ul",{},[1787,1788,1789,1790,1793,1794,1796,1797,1799,1800,1407],"li",{},"An ",[25,1791,1792],{},"outcome"," is one of the possible things that can happen. When you flip a coin, the outcomes are ",[20,1795,82],{},", ",[20,1798,117],{},", and (very rarely) ",[20,1801,1802],{},"edge",[1787,1804,1789,1805,1808],{},[25,1806,1807],{},"event"," is a question we ask about the outcomes, like \"did it land on heads?\" or \"did it land on a face at all?\".",[16,1810,1811],{},"Probability is the tool we use to put a number on how likely an event is.",[30,1813,1815],{"id":1814},"the-probability-scale","The probability scale",[16,1817,1818,1819,1822],{},"Probabilities always live on a scale from ",[25,1820,1821],{},"0 to 1",":",[1784,1824,1825,1830,1835],{},[1787,1826,1827,1829],{},[25,1828,92],{}," means the event is impossible. A coin turning into a pigeon mid-air has probability 0.",[1787,1831,1832,1834],{},[25,1833,99],{}," means the event is certain. A flipped coin eventually coming back down, thanks to gravity, has probability 1.",[1787,1836,1837,1838,1407],{},"Anything in between is \"somewhat likely\". A fair coin landing on heads sits right in the middle, at ",[25,1839,1761],{},[16,1841,1842,1843,703,1846,1849],{},"In percentage terms, that same scale goes from ",[25,1844,1845],{},"0%",[25,1847,1848],{},"100%",". Same idea, expressed in a different way.",[30,1851,1853],{"id":1852},"the-complement-rule","The complement rule",[16,1855,1856,1857,1860,1861,1864],{},"Every event has a mirror image called its ",[25,1858,1859],{},"complement",": the event of it ",[20,1862,1863],{},"not"," happening. And because something either happens or it doesn't, the two probabilities always add up to 1. In symbols:",[16,1866,1868],{"className":1867},[50],[52,1869,1871,1909],{"className":1870},[55],[52,1872,1874],{"className":1873},[59],[61,1875,1876],{"xmlns":63},[65,1877,1878,1906],{},[68,1879,1880,1882,1884,1887,1890,1892,1894,1896,1898,1900,1902,1904],{},[71,1881,73],{},[75,1883,78],{"stretchy":77},[80,1885,1886],{},"not ",[71,1888,1889],{},"A",[75,1891,85],{"stretchy":77},[75,1893,88],{},[97,1895,99],{},[75,1897,1169],{},[71,1899,73],{},[75,1901,78],{"stretchy":77},[71,1903,1889],{},[75,1905,85],{"stretchy":77},[131,1907,1908],{"encoding":133},"P(\\text{not } A) = 1 - P(A)",[52,1910,1912,1945,1964],{"className":1911,"ariaHidden":105},[138],[52,1913,1915,1918,1921,1924,1930,1933,1936,1939,1942],{"className":1914},[142],[52,1916],{"className":1917,"style":147},[146],[52,1919,73],{"className":1920,"style":153},[151,152],[52,1922,78],{"className":1923},[157],[52,1925,1927],{"className":1926},[151,161],[52,1928,1886],{"className":1929},[151],[52,1931,1889],{"className":1932},[151,152],[52,1934,85],{"className":1935},[168],[52,1937],{"className":1938,"style":172},[108],[52,1940,88],{"className":1941},[176],[52,1943],{"className":1944,"style":172},[108],[52,1946,1948,1952,1955,1958,1961],{"className":1947},[142],[52,1949],{"className":1950,"style":1951},[146],"height:0.7278em;vertical-align:-0.0833em;",[52,1953,99],{"className":1954},[151],[52,1956],{"className":1957,"style":517},[108],[52,1959,1169],{"className":1960},[521],[52,1962],{"className":1963,"style":517},[108],[52,1965,1967,1970,1973,1976,1979],{"className":1966},[142],[52,1968],{"className":1969,"style":147},[146],[52,1971,73],{"className":1972,"style":153},[151,152],[52,1974,78],{"className":1975},[157],[52,1977,1889],{"className":1978},[151,152],[52,1980,85],{"className":1981},[168],[16,1983,1984],{},"For a fair coin:",[1784,1986,1987,2173],{},[1787,1988,1989,2058,2059,1407],{},[52,1990,1992,2016],{"className":1991},[55],[52,1993,1995],{"className":1994},[59],[61,1996,1997],{"xmlns":63},[65,1998,1999,2013],{},[68,2000,2001,2003,2005,2007,2009,2011],{},[71,2002,73],{},[75,2004,78],{"stretchy":77},[80,2006,82],{},[75,2008,85],{"stretchy":77},[75,2010,88],{},[97,2012,1761],{},[131,2014,2015],{"encoding":133},"P(\\text{heads}) = 0.5",[52,2017,2019,2049],{"className":2018,"ariaHidden":105},[138],[52,2020,2022,2025,2028,2031,2037,2040,2043,2046],{"className":2021},[142],[52,2023],{"className":2024,"style":147},[146],[52,2026,73],{"className":2027,"style":153},[151,152],[52,2029,78],{"className":2030},[157],[52,2032,2034],{"className":2033},[151,161],[52,2035,82],{"className":2036},[151],[52,2038,85],{"className":2039},[168],[52,2041],{"className":2042,"style":172},[108],[52,2044,88],{"className":2045},[176],[52,2047],{"className":2048,"style":172},[108],[52,2050,2052,2055],{"className":2051},[142],[52,2053],{"className":2054,"style":614},[146],[52,2056,1761],{"className":2057},[151],", so ",[52,2060,2062,2095],{"className":2061},[55],[52,2063,2065],{"className":2064},[59],[61,2066,2067],{"xmlns":63},[65,2068,2069,2092],{},[68,2070,2071,2073,2075,2078,2080,2082,2084,2086,2088,2090],{},[71,2072,73],{},[75,2074,78],{"stretchy":77},[80,2076,2077],{},"not heads",[75,2079,85],{"stretchy":77},[75,2081,88],{},[97,2083,99],{},[75,2085,1169],{},[97,2087,1761],{},[75,2089,88],{},[97,2091,1761],{},[131,2093,2094],{"encoding":133},"P(\\text{not heads}) = 1 - 0.5 = 0.5",[52,2096,2098,2128,2146,2164],{"className":2097,"ariaHidden":105},[138],[52,2099,2101,2104,2107,2110,2116,2119,2122,2125],{"className":2100},[142],[52,2102],{"className":2103,"style":147},[146],[52,2105,73],{"className":2106,"style":153},[151,152],[52,2108,78],{"className":2109},[157],[52,2111,2113],{"className":2112},[151,161],[52,2114,2077],{"className":2115},[151],[52,2117,85],{"className":2118},[168],[52,2120],{"className":2121,"style":172},[108],[52,2123,88],{"className":2124},[176],[52,2126],{"className":2127,"style":172},[108],[52,2129,2131,2134,2137,2140,2143],{"className":2130},[142],[52,2132],{"className":2133,"style":1951},[146],[52,2135,99],{"className":2136},[151],[52,2138],{"className":2139,"style":517},[108],[52,2141,1169],{"className":2142},[521],[52,2144],{"className":2145,"style":517},[108],[52,2147,2149,2152,2155,2158,2161],{"className":2148},[142],[52,2150],{"className":2151,"style":614},[146],[52,2153,1761],{"className":2154},[151],[52,2156],{"className":2157,"style":172},[108],[52,2159,88],{"className":2160},[176],[52,2162],{"className":2163,"style":172},[108],[52,2165,2167,2170],{"className":2166},[142],[52,2168],{"className":2169,"style":614},[146],[52,2171,1761],{"className":2172},[151],[1787,2174,2175,2363,2364,636],{},[52,2176,2178,2220],{"className":2177},[55],[52,2179,2181],{"className":2180},[59],[61,2182,2183],{"xmlns":63},[65,2184,2185,2217],{},[68,2186,2187,2189,2191,2194,2196,2199,2201,2203,2212,2214],{},[71,2188,73],{},[75,2190,78],{"stretchy":77},[80,2192,2193],{},"face",[75,2195,85],{"stretchy":77},[75,2197,2198],{},"≈",[97,2200,99],{},[75,2202,1169],{},[90,2204,2205],{"scriptlevel":92,"displaystyle":77},[94,2206,2207,2209],{},[97,2208,99],{},[97,2210,2211],{},"6000",[75,2213,2198],{},[97,2215,2216],{},"0.99983",[131,2218,2219],{"encoding":133},"P(\\text{face}) \\approx 1 - \\tfrac{1}{6000} \\approx 0.99983",[52,2221,2223,2253,2271,2354],{"className":2222,"ariaHidden":105},[138],[52,2224,2226,2229,2232,2235,2241,2244,2247,2250],{"className":2225},[142],[52,2227],{"className":2228,"style":147},[146],[52,2230,73],{"className":2231,"style":153},[151,152],[52,2233,78],{"className":2234},[157],[52,2236,2238],{"className":2237},[151,161],[52,2239,2193],{"className":2240},[151],[52,2242,85],{"className":2243},[168],[52,2245],{"className":2246,"style":172},[108],[52,2248,2198],{"className":2249},[176],[52,2251],{"className":2252,"style":172},[108],[52,2254,2256,2259,2262,2265,2268],{"className":2255},[142],[52,2257],{"className":2258,"style":1951},[146],[52,2260,99],{"className":2261},[151],[52,2263],{"className":2264,"style":517},[108],[52,2266,1169],{"className":2267},[521],[52,2269],{"className":2270,"style":517},[108],[52,2272,2274,2277,2345,2348,2351],{"className":2273},[142],[52,2275],{"className":2276,"style":186},[146],[52,2278,2280,2283,2342],{"className":2279},[151],[52,2281],{"className":2282},[157,193],[52,2284,2286],{"className":2285},[94],[52,2287,2289,2334],{"className":2288},[200,201],[52,2290,2292,2331],{"className":2291},[205],[52,2293,2295,2309,2317],{"className":2294,"style":210},[209],[52,2296,2297,2300],{"style":213},[52,2298],{"className":2299,"style":218},[217],[52,2301,2303],{"className":2302},[222,223,224,225],[52,2304,2306],{"className":2305},[151,225],[52,2307,2211],{"className":2308},[151,225],[52,2310,2311,2314],{"style":234},[52,2312],{"className":2313,"style":218},[217],[52,2315],{"className":2316,"style":242},[241],[52,2318,2319,2322],{"style":245},[52,2320],{"className":2321,"style":218},[217],[52,2323,2325],{"className":2324},[222,223,224,225],[52,2326,2328],{"className":2327},[151,225],[52,2329,99],{"className":2330},[151,225],[52,2332,262],{"className":2333},[261],[52,2335,2337],{"className":2336},[205],[52,2338,2340],{"className":2339,"style":269},[209],[52,2341],{},[52,2343],{"className":2344},[168,193],[52,2346],{"className":2347,"style":172},[108],[52,2349,2198],{"className":2350},[176],[52,2352],{"className":2353,"style":172},[108],[52,2355,2357,2360],{"className":2356},[142],[52,2358],{"className":2359,"style":614},[146],[52,2361,2216],{"className":2362},[151],", because the only other option is the rare \"edge\" landing. (More on that 1-in-6000 number in ",[630,2365,2367],{"href":1772,"rel":2366},[634],"The Unexpected Statistics of Coin Flips",[16,2369,2370,2371,2374,2375,2378],{},"The complement rule is often the easiest way to compute a probability: instead of counting every way something ",[20,2372,2373],{},"can"," happen, count the one way it ",[20,2376,2377],{},"can't"," and subtract from 1.",[30,2380,2382],{"id":2381},"combining-events-and-vs-or","Combining events: \"and\" vs \"or\"",[16,2384,2385],{},"Things get interesting when we ask about more than one flip at a time. Two rules cover most of what you'll ever need:",[2387,2388,2390],"h3",{"id":2389},"and-both-things-happen","\"AND\" — both things happen",[16,2392,2393,2394,2397,2398,2401],{},"Coin flips don't influence each other (they are ",[25,2395,2396],{},"independent","). In this case, you ",[25,2399,2400],{},"multiply"," their probabilities:",[1784,2403,2404,2523],{},[1787,2405,2406,2522],{},[52,2407,2409,2444],{"className":2408},[55],[52,2410,2412],{"className":2411},[59],[61,2413,2414],{"xmlns":63},[65,2415,2416,2441],{},[68,2417,2418,2420,2422,2425,2427,2429,2431,2434,2436,2438],{},[71,2419,73],{},[75,2421,78],{"stretchy":77},[80,2423,2424],{},"OO",[75,2426,85],{"stretchy":77},[75,2428,88],{},[97,2430,1761],{},[75,2432,2433],{},"×",[97,2435,1761],{},[75,2437,88],{},[97,2439,2440],{},"0.25",[131,2442,2443],{"encoding":133},"P(\\text{OO}) = 0.5 \\times 0.5 = 0.25",[52,2445,2447,2477,2495,2513],{"className":2446,"ariaHidden":105},[138],[52,2448,2450,2453,2456,2459,2465,2468,2471,2474],{"className":2449},[142],[52,2451],{"className":2452,"style":147},[146],[52,2454,73],{"className":2455,"style":153},[151,152],[52,2457,78],{"className":2458},[157],[52,2460,2462],{"className":2461},[151,161],[52,2463,2424],{"className":2464},[151],[52,2466,85],{"className":2467},[168],[52,2469],{"className":2470,"style":172},[108],[52,2472,88],{"className":2473},[176],[52,2475],{"className":2476,"style":172},[108],[52,2478,2480,2483,2486,2489,2492],{"className":2479},[142],[52,2481],{"className":2482,"style":1951},[146],[52,2484,1761],{"className":2485},[151],[52,2487],{"className":2488,"style":517},[108],[52,2490,2433],{"className":2491},[521],[52,2493],{"className":2494,"style":517},[108],[52,2496,2498,2501,2504,2507,2510],{"className":2497},[142],[52,2499],{"className":2500,"style":614},[146],[52,2502,1761],{"className":2503},[151],[52,2505],{"className":2506,"style":172},[108],[52,2508,88],{"className":2509},[176],[52,2511],{"className":2512,"style":172},[108],[52,2514,2516,2519],{"className":2515},[142],[52,2517],{"className":2518,"style":614},[146],[52,2520,2440],{"className":2521},[151],", or 25%.",[1787,2524,2525,2662],{},[52,2526,2528,2566],{"className":2527},[55],[52,2529,2531],{"className":2530},[59],[61,2532,2533],{"xmlns":63},[65,2534,2535,2563],{},[68,2536,2537,2539,2541,2544,2546,2548,2550,2552,2554,2556,2558,2560],{},[71,2538,73],{},[75,2540,78],{"stretchy":77},[80,2542,2543],{},"OOO",[75,2545,85],{"stretchy":77},[75,2547,88],{},[97,2549,1761],{},[75,2551,2433],{},[97,2553,1761],{},[75,2555,2433],{},[97,2557,1761],{},[75,2559,88],{},[97,2561,2562],{},"0.125",[131,2564,2565],{"encoding":133},"P(\\text{OOO}) = 0.5 \\times 0.5 \\times 0.5 = 0.125",[52,2567,2569,2599,2617,2635,2653],{"className":2568,"ariaHidden":105},[138],[52,2570,2572,2575,2578,2581,2587,2590,2593,2596],{"className":2571},[142],[52,2573],{"className":2574,"style":147},[146],[52,2576,73],{"className":2577,"style":153},[151,152],[52,2579,78],{"className":2580},[157],[52,2582,2584],{"className":2583},[151,161],[52,2585,2543],{"className":2586},[151],[52,2588,85],{"className":2589},[168],[52,2591],{"className":2592,"style":172},[108],[52,2594,88],{"className":2595},[176],[52,2597],{"className":2598,"style":172},[108],[52,2600,2602,2605,2608,2611,2614],{"className":2601},[142],[52,2603],{"className":2604,"style":1951},[146],[52,2606,1761],{"className":2607},[151],[52,2609],{"className":2610,"style":517},[108],[52,2612,2433],{"className":2613},[521],[52,2615],{"className":2616,"style":517},[108],[52,2618,2620,2623,2626,2629,2632],{"className":2619},[142],[52,2621],{"className":2622,"style":1951},[146],[52,2624,1761],{"className":2625},[151],[52,2627],{"className":2628,"style":517},[108],[52,2630,2433],{"className":2631},[521],[52,2633],{"className":2634,"style":517},[108],[52,2636,2638,2641,2644,2647,2650],{"className":2637},[142],[52,2639],{"className":2640,"style":614},[146],[52,2642,1761],{"className":2643},[151],[52,2645],{"className":2646,"style":172},[108],[52,2648,88],{"className":2649},[176],[52,2651],{"className":2652,"style":172},[108],[52,2654,2656,2659],{"className":2655},[142],[52,2657],{"className":2658,"style":614},[146],[52,2660,2562],{"className":2661},[151],", or 12.5%.",[16,2664,2665,2666,2766],{},"Each extra flip halves the probability. That's why long streaks feel surprising — not because the coin is \"due\" for the other side, but because each new flip piles another ",[52,2667,2669,2689],{"className":2668},[55],[52,2670,2672],{"className":2671},[59],[61,2673,2674],{"xmlns":63},[65,2675,2676,2686],{},[68,2677,2678],{},[90,2679,2680],{"scriptlevel":92,"displaystyle":77},[94,2681,2682,2684],{},[97,2683,99],{},[97,2685,102],{},[131,2687,2688],{"encoding":133},"\\tfrac{1}{2}",[52,2690,2692],{"className":2691,"ariaHidden":105},[138],[52,2693,2695,2698],{"className":2694},[142],[52,2696],{"className":2697,"style":186},[146],[52,2699,2701,2704,2763],{"className":2700},[151],[52,2702],{"className":2703},[157,193],[52,2705,2707],{"className":2706},[94],[52,2708,2710,2755],{"className":2709},[200,201],[52,2711,2713,2752],{"className":2712},[205],[52,2714,2716,2730,2738],{"className":2715,"style":210},[209],[52,2717,2718,2721],{"style":213},[52,2719],{"className":2720,"style":218},[217],[52,2722,2724],{"className":2723},[222,223,224,225],[52,2725,2727],{"className":2726},[151,225],[52,2728,102],{"className":2729},[151,225],[52,2731,2732,2735],{"style":234},[52,2733],{"className":2734,"style":218},[217],[52,2736],{"className":2737,"style":242},[241],[52,2739,2740,2743],{"style":245},[52,2741],{"className":2742,"style":218},[217],[52,2744,2746],{"className":2745},[222,223,224,225],[52,2747,2749],{"className":2748},[151,225],[52,2750,99],{"className":2751},[151,225],[52,2753,262],{"className":2754},[261],[52,2756,2758],{"className":2757},[205],[52,2759,2761],{"className":2760,"style":269},[209],[52,2762],{},[52,2764],{"className":2765},[168,193]," onto the product.",[2387,2768,2770],{"id":2769},"or-at-least-one-of-several-things-happens","\"OR\" — at least one of several things happens",[16,2772,2773,2774,2777,2778,2781,2782,2401],{},"If two outcomes ",[25,2775,2776],{},"can't happen at the same time"," (they're ",[25,2779,2780],{},"mutually exclusive","), you ",[25,2783,2784],{},"add",[1784,2786,2787,2931],{},[1787,2788,2789,2930],{},[52,2790,2792,2831],{"className":2791},[55],[52,2793,2795],{"className":2794},[59],[61,2796,2797],{"xmlns":63},[65,2798,2799,2828],{},[68,2800,2801,2803,2805,2808,2811,2814,2816,2818,2820,2822,2824,2826],{},[71,2802,73],{},[75,2804,78],{"stretchy":77},[80,2806,2807],{},"O",[75,2809,2810],{},"∪",[80,2812,2813],{},"X",[75,2815,85],{"stretchy":77},[75,2817,88],{},[97,2819,1761],{},[75,2821,421],{},[97,2823,1761],{},[75,2825,88],{},[97,2827,99],{},[131,2829,2830],{"encoding":133},"P(\\text{O} \\cup \\text{X}) = 0.5 + 0.5 = 1",[52,2832,2834,2861,2885,2903,2921],{"className":2833,"ariaHidden":105},[138],[52,2835,2837,2840,2843,2846,2852,2855,2858],{"className":2836},[142],[52,2838],{"className":2839,"style":147},[146],[52,2841,73],{"className":2842,"style":153},[151,152],[52,2844,78],{"className":2845},[157],[52,2847,2849],{"className":2848},[151,161],[52,2850,2807],{"className":2851},[151],[52,2853],{"className":2854,"style":517},[108],[52,2856,2810],{"className":2857},[521],[52,2859],{"className":2860,"style":517},[108],[52,2862,2864,2867,2873,2876,2879,2882],{"className":2863},[142],[52,2865],{"className":2866,"style":147},[146],[52,2868,2870],{"className":2869},[151,161],[52,2871,2813],{"className":2872},[151],[52,2874,85],{"className":2875},[168],[52,2877],{"className":2878,"style":172},[108],[52,2880,88],{"className":2881},[176],[52,2883],{"className":2884,"style":172},[108],[52,2886,2888,2891,2894,2897,2900],{"className":2887},[142],[52,2889],{"className":2890,"style":1951},[146],[52,2892,1761],{"className":2893},[151],[52,2895],{"className":2896,"style":517},[108],[52,2898,421],{"className":2899},[521],[52,2901],{"className":2902,"style":517},[108],[52,2904,2906,2909,2912,2915,2918],{"className":2905},[142],[52,2907],{"className":2908,"style":614},[146],[52,2910,1761],{"className":2911},[151],[52,2913],{"className":2914,"style":172},[108],[52,2916,88],{"className":2917},[176],[52,2919],{"className":2920,"style":172},[108],[52,2922,2924,2927],{"className":2923},[142],[52,2925],{"className":2926,"style":614},[146],[52,2928,99],{"className":2929},[151],". Unsurprising — that's every non-edge outcome.",[1787,2932,2933,3279],{},[52,2934,2936,2991],{"className":2935},[55],[52,2937,2939],{"className":2938},[59],[61,2940,2941],{"xmlns":63},[65,2942,2943,2988],{},[68,2944,2945,2947,2949,2951,2953,2955,2957,2959,2968,2970,2978,2980],{},[71,2946,73],{},[75,2948,78],{"stretchy":77},[97,2950,99],{},[75,2952,2810],{},[97,2954,102],{},[75,2956,85],{"stretchy":77},[75,2958,88],{},[90,2960,2961],{"scriptlevel":92,"displaystyle":77},[94,2962,2963,2965],{},[97,2964,99],{},[97,2966,2967],{},"6",[75,2969,421],{},[90,2971,2972],{"scriptlevel":92,"displaystyle":77},[94,2973,2974,2976],{},[97,2975,99],{},[97,2977,2967],{},[75,2979,88],{},[90,2981,2982],{"scriptlevel":92,"displaystyle":77},[94,2983,2984,2986],{},[97,2985,102],{},[97,2987,2967],{},[131,2989,2990],{"encoding":133},"P(1 \\cup 2) = \\tfrac{1}{6} + \\tfrac{1}{6} = \\tfrac{2}{6}",[52,2992,2994,3018,3039,3122,3205],{"className":2993,"ariaHidden":105},[138],[52,2995,2997,3000,3003,3006,3009,3012,3015],{"className":2996},[142],[52,2998],{"className":2999,"style":147},[146],[52,3001,73],{"className":3002,"style":153},[151,152],[52,3004,78],{"className":3005},[157],[52,3007,99],{"className":3008},[151],[52,3010],{"className":3011,"style":517},[108],[52,3013,2810],{"className":3014},[521],[52,3016],{"className":3017,"style":517},[108],[52,3019,3021,3024,3027,3030,3033,3036],{"className":3020},[142],[52,3022],{"className":3023,"style":147},[146],[52,3025,102],{"className":3026},[151],[52,3028,85],{"className":3029},[168],[52,3031],{"className":3032,"style":172},[108],[52,3034,88],{"className":3035},[176],[52,3037],{"className":3038,"style":172},[108],[52,3040,3042,3045,3113,3116,3119],{"className":3041},[142],[52,3043],{"className":3044,"style":186},[146],[52,3046,3048,3051,3110],{"className":3047},[151],[52,3049],{"className":3050},[157,193],[52,3052,3054],{"className":3053},[94],[52,3055,3057,3102],{"className":3056},[200,201],[52,3058,3060,3099],{"className":3059},[205],[52,3061,3063,3077,3085],{"className":3062,"style":210},[209],[52,3064,3065,3068],{"style":213},[52,3066],{"className":3067,"style":218},[217],[52,3069,3071],{"className":3070},[222,223,224,225],[52,3072,3074],{"className":3073},[151,225],[52,3075,2967],{"className":3076},[151,225],[52,3078,3079,3082],{"style":234},[52,3080],{"className":3081,"style":218},[217],[52,3083],{"className":3084,"style":242},[241],[52,3086,3087,3090],{"style":245},[52,3088],{"className":3089,"style":218},[217],[52,3091,3093],{"className":3092},[222,223,224,225],[52,3094,3096],{"className":3095},[151,225],[52,3097,99],{"className":3098},[151,225],[52,3100,262],{"className":3101},[261],[52,3103,3105],{"className":3104},[205],[52,3106,3108],{"className":3107,"style":269},[209],[52,3109],{},[52,3111],{"className":3112},[168,193],[52,3114],{"className":3115,"style":517},[108],[52,3117,421],{"className":3118},[521],[52,3120],{"className":3121,"style":517},[108],[52,3123,3125,3128,3196,3199,3202],{"className":3124},[142],[52,3126],{"className":3127,"style":186},[146],[52,3129,3131,3134,3193],{"className":3130},[151],[52,3132],{"className":3133},[157,193],[52,3135,3137],{"className":3136},[94],[52,3138,3140,3185],{"className":3139},[200,201],[52,3141,3143,3182],{"className":3142},[205],[52,3144,3146,3160,3168],{"className":3145,"style":210},[209],[52,3147,3148,3151],{"style":213},[52,3149],{"className":3150,"style":218},[217],[52,3152,3154],{"className":3153},[222,223,224,225],[52,3155,3157],{"className":3156},[151,225],[52,3158,2967],{"className":3159},[151,225],[52,3161,3162,3165],{"style":234},[52,3163],{"className":3164,"style":218},[217],[52,3166],{"className":3167,"style":242},[241],[52,3169,3170,3173],{"style":245},[52,3171],{"className":3172,"style":218},[217],[52,3174,3176],{"className":3175},[222,223,224,225],[52,3177,3179],{"className":3178},[151,225],[52,3180,99],{"className":3181},[151,225],[52,3183,262],{"className":3184},[261],[52,3186,3188],{"className":3187},[205],[52,3189,3191],{"className":3190,"style":269},[209],[52,3192],{},[52,3194],{"className":3195},[168,193],[52,3197],{"className":3198,"style":172},[108],[52,3200,88],{"className":3201},[176],[52,3203],{"className":3204,"style":172},[108],[52,3206,3208,3211],{"className":3207},[142],[52,3209],{"className":3210,"style":186},[146],[52,3212,3214,3217,3276],{"className":3213},[151],[52,3215],{"className":3216},[157,193],[52,3218,3220],{"className":3219},[94],[52,3221,3223,3268],{"className":3222},[200,201],[52,3224,3226,3265],{"className":3225},[205],[52,3227,3229,3243,3251],{"className":3228,"style":210},[209],[52,3230,3231,3234],{"style":213},[52,3232],{"className":3233,"style":218},[217],[52,3235,3237],{"className":3236},[222,223,224,225],[52,3238,3240],{"className":3239},[151,225],[52,3241,2967],{"className":3242},[151,225],[52,3244,3245,3248],{"style":234},[52,3246],{"className":3247,"style":218},[217],[52,3249],{"className":3250,"style":242},[241],[52,3252,3253,3256],{"style":245},[52,3254],{"className":3255,"style":218},[217],[52,3257,3259],{"className":3258},[222,223,224,225],[52,3260,3262],{"className":3261},[151,225],[52,3263,102],{"className":3264},[151,225],[52,3266,262],{"className":3267},[261],[52,3269,3271],{"className":3270},[205],[52,3272,3274],{"className":3273,"style":269},[209],[52,3275],{},[52,3277],{"className":3278},[168,193]," (rolling a 1 or a 2 on a six-sided die).",[16,3281,3282,3283,3285],{},"If the events ",[20,3284,2373],{}," overlap, simple addition double-counts the overlap and you have to subtract it back out:",[16,3287,3289],{"className":3288},[50],[52,3290,3292,3352],{"className":3291},[55],[52,3293,3295],{"className":3294},[59],[61,3296,3297],{"xmlns":63},[65,3298,3299,3349],{},[68,3300,3301,3303,3305,3307,3309,3312,3314,3316,3318,3320,3322,3324,3326,3328,3330,3332,3334,3336,3338,3340,3342,3345,3347],{},[71,3302,73],{},[75,3304,78],{"stretchy":77},[71,3306,1889],{},[75,3308,2810],{},[71,3310,3311],{},"B",[75,3313,85],{"stretchy":77},[75,3315,88],{},[71,3317,73],{},[75,3319,78],{"stretchy":77},[71,3321,1889],{},[75,3323,85],{"stretchy":77},[75,3325,421],{},[71,3327,73],{},[75,3329,78],{"stretchy":77},[71,3331,3311],{},[75,3333,85],{"stretchy":77},[75,3335,1169],{},[71,3337,73],{},[75,3339,78],{"stretchy":77},[71,3341,1889],{},[75,3343,3344],{},"∩",[71,3346,3311],{},[75,3348,85],{"stretchy":77},[131,3350,3351],{"encoding":133},"P(A \\cup B) = P(A) + P(B) - P(A \\cap B)",[52,3353,3355,3379,3401,3428,3455,3479],{"className":3354,"ariaHidden":105},[138],[52,3356,3358,3361,3364,3367,3370,3373,3376],{"className":3357},[142],[52,3359],{"className":3360,"style":147},[146],[52,3362,73],{"className":3363,"style":153},[151,152],[52,3365,78],{"className":3366},[157],[52,3368,1889],{"className":3369},[151,152],[52,3371],{"className":3372,"style":517},[108],[52,3374,2810],{"className":3375},[521],[52,3377],{"className":3378,"style":517},[108],[52,3380,3382,3385,3389,3392,3395,3398],{"className":3381},[142],[52,3383],{"className":3384,"style":147},[146],[52,3386,3311],{"className":3387,"style":3388},[151,152],"margin-right:0.0502em;",[52,3390,85],{"className":3391},[168],[52,3393],{"className":3394,"style":172},[108],[52,3396,88],{"className":3397},[176],[52,3399],{"className":3400,"style":172},[108],[52,3402,3404,3407,3410,3413,3416,3419,3422,3425],{"className":3403},[142],[52,3405],{"className":3406,"style":147},[146],[52,3408,73],{"className":3409,"style":153},[151,152],[52,3411,78],{"className":3412},[157],[52,3414,1889],{"className":3415},[151,152],[52,3417,85],{"className":3418},[168],[52,3420],{"className":3421,"style":517},[108],[52,3423,421],{"className":3424},[521],[52,3426],{"className":3427,"style":517},[108],[52,3429,3431,3434,3437,3440,3443,3446,3449,3452],{"className":3430},[142],[52,3432],{"className":3433,"style":147},[146],[52,3435,73],{"className":3436,"style":153},[151,152],[52,3438,78],{"className":3439},[157],[52,3441,3311],{"className":3442,"style":3388},[151,152],[52,3444,85],{"className":3445},[168],[52,3447],{"className":3448,"style":517},[108],[52,3450,1169],{"className":3451},[521],[52,3453],{"className":3454,"style":517},[108],[52,3456,3458,3461,3464,3467,3470,3473,3476],{"className":3457},[142],[52,3459],{"className":3460,"style":147},[146],[52,3462,73],{"className":3463,"style":153},[151,152],[52,3465,78],{"className":3466},[157],[52,3468,1889],{"className":3469},[151,152],[52,3471],{"className":3472,"style":517},[108],[52,3474,3344],{"className":3475},[521],[52,3477],{"className":3478,"style":517},[108],[52,3480,3482,3485,3488],{"className":3481},[142],[52,3483],{"className":3484,"style":147},[146],[52,3486,3311],{"className":3487,"style":3388},[151,152],[52,3489,85],{"className":3490},[168],[16,3492,3493,3494,3497],{},"That's the ",[25,3495,3496],{},"inclusion–exclusion principle",", and the deeper version is a topic for a future article.",[30,3499,3501],{"id":3500},"a-quick-note-on-independence","A quick note on independence",[16,3503,3504,3505,3507],{},"Two events are ",[25,3506,2396],{}," when the outcome of one doesn't change the probability of the other. Coin flips are the textbook example: the coin has no memory, so a run of ten heads in a row doesn't make tails any more likely on the eleventh flip.",[16,3509,3510,3511,3514,3515,3520],{},"This is the setup behind the famous ",[25,3512,3513],{},"Gambler's Fallacy",", which I unpack in ",[630,3516,3519],{"href":3517,"rel":3518},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fthe-unexpected-statistics-of-coin-flips#the-gamblers-fallacy",[634],"the statistics article",". For now, the short version: each flip is a fresh start.",[30,3522,3524],{"id":3523},"conclusion","Conclusion",[16,3526,3527],{},"That's really it for the basics:",[3529,3530,3531,3534,3544,3553],"ol",{},[1787,3532,3533],{},"Probabilities live between 0 and 1.",[1787,3535,3536,3537,3539,3540,3543],{},"Use the ",[25,3538,1859],{}," (1 − P) when it's easier to count what ",[20,3541,3542],{},"doesn't"," happen.",[1787,3545,3546,3549,3550,3552],{},[25,3547,3548],{},"Multiply"," for \"and\", ",[25,3551,2784],{}," for mutually exclusive \"or\".",[1787,3554,3555],{},"Independent events don't care about each other's history.",[16,3557,3558],{},"With just those four ideas you can reason about almost any coin-flip question — and most dice, cards, and lottery-style problems too.",[16,3560,3561,3562,3567],{},"Put the theory to work: ",[630,3563,3566],{"href":3564,"rel":3565},"https:\u002F\u002Fflipthecoin.app\u002Fplay",[634],"flip a coin online"," and track your own streak.",{"title":1597,"searchDepth":1598,"depth":1598,"links":3569},[3570,3571,3572,3573,3574,3578,3579],{"id":1655,"depth":1598,"text":1656},{"id":1778,"depth":1598,"text":1779},{"id":1814,"depth":1598,"text":1815},{"id":1852,"depth":1598,"text":1853},{"id":2381,"depth":1598,"text":2382,"children":3575},[3576,3577],{"id":2389,"depth":1623,"text":2390},{"id":2769,"depth":1623,"text":2770},{"id":3500,"depth":1598,"text":3501},{"id":3523,"depth":1598,"text":3524},"2026-04-23T01:00:00Z","Learn the basics of probability and how it relates to coin flips.",{"type":9,"value":3583},[3584,3586,3592],[12,3585,635],{"id":1622},[16,3587,1644,3588,1647,3590,1407],{},[25,3589,82],{},[25,3591,117],{},[16,3593,1652],{},{},"\u002Fblog\u002Fprobability-basics.en",{"title":635,"description":3581},1,"blog\u002Fprobability-basics.en","rosCytsdKmrJokXaRgGPdvFI-t2qar8ER8kBFoOSAFU",{"id":3601,"title":3602,"_locale":6,"author":7,"body":3603,"date":4012,"description":4013,"excerpt":4014,"extension":1616,"level":1617,"meta":4020,"navigation":1619,"path":4021,"published":1619,"seo":4022,"series":1622,"seriesOrder":1598,"slug":4023,"stem":4024,"tags":4025,"__hash__":4028},"blog\u002Fblog\u002Fheads-or-tails.en.md","Heads or Tails — The Complete Guide to the Coin Toss",{"type":9,"value":3604,"toc":4004},[3605,3608,3611,3615,3621,3631,3638,3642,3649,3736,3821,3894,3904,3921,3925,3932,3943,3946,3950,3956,3962,3968,3974,3978,3985,3988,3991,3993,3996],[12,3606,3602],{"id":3607},"heads-or-tails-the-complete-guide-to-the-coin-toss",[16,3609,3610],{},"\"Heads or tails?\" Three words that have settled disputes, started games, and made decisions for thousands of years. But where does that phrase actually come from? And is the coin really fair?",[30,3612,3614],{"id":3613},"where-do-heads-and-tails-come-from","Where do \"heads\" and \"tails\" come from?",[16,3616,3617,3618,3620],{},"The word ",[25,3619,82],{}," is straightforward: coins have featured portraits since ancient times. Greek and Roman coins showed the profile of a ruler, deity, or emperor — a literal head.",[16,3622,3623,3626,3627,3630],{},[25,3624,3625],{},"Tails"," is trickier. The most common explanation is that the reverse side often depicted animals — particularly eagles or lions whose tails were visible. In Roman times the phrase was ",[20,3628,3629],{},"navia aut caput"," (\"ship or head\"), because the reverse of the coin showed a ship's prow. English later substituted \"tails\" as the standard reverse-side term, possibly because the reverse of early British coins featured the cross of the national emblem, whose lower point could be read as a stylised tail.",[16,3632,3633,3634,3637],{},"The modern pairing \"heads or tails\" became widespread in English by the 17th century. Julius Caesar's name appears in a variant: in Rome, the game was sometimes called ",[20,3635,3636],{},"capita aut navia"," (\"head or ship\") in his honour after coins bearing his portrait became common.",[30,3639,3641],{"id":3640},"the-mathematics-of-a-coin-toss","The mathematics of a coin toss",[16,3643,3644,3645,3648],{},"In probability theory, a fair coin toss is the textbook example of a ",[25,3646,3647],{},"Bernoulli trial",": a random experiment with exactly two equally-likely outcomes (1 or 0). The probability of heads is 0.5, the probability of tails is 0.5, and each toss is independent of every other. In mathematical notation, it is represented as:",[16,3650,3652],{"className":3651},[50],[52,3653,3655,3684],{"className":3654},[55],[52,3656,3658],{"className":3657},[59],[61,3659,3660],{"xmlns":63},[65,3661,3662,3681],{},[68,3663,3664,3666,3668,3670,3672,3674,3676,3678],{},[71,3665,73],{},[75,3667,78],{"stretchy":77},[80,3669,82],{},[75,3671,85],{"stretchy":77},[75,3673,88],{},[97,3675,92],{},[75,3677,106],{"separator":105},[97,3679,3680],{},"5",[131,3682,3683],{"encoding":133},"P(\\text{heads}) = 0,5",[52,3685,3687,3717],{"className":3686,"ariaHidden":105},[138],[52,3688,3690,3693,3696,3699,3705,3708,3711,3714],{"className":3689},[142],[52,3691],{"className":3692,"style":147},[146],[52,3694,73],{"className":3695,"style":153},[151,152],[52,3697,78],{"className":3698},[157],[52,3700,3702],{"className":3701},[151,161],[52,3703,82],{"className":3704},[151],[52,3706,85],{"className":3707},[168],[52,3709],{"className":3710,"style":172},[108],[52,3712,88],{"className":3713},[176],[52,3715],{"className":3716,"style":172},[108],[52,3718,3720,3724,3727,3730,3733],{"className":3719},[142],[52,3721],{"className":3722,"style":3723},[146],"height:0.8389em;vertical-align:-0.1944em;",[52,3725,92],{"className":3726},[151],[52,3728,106],{"className":3729},[278],[52,3731],{"className":3732,"style":286},[108],[52,3734,3680],{"className":3735},[151],[16,3737,3739],{"className":3738},[50],[52,3740,3742,3770],{"className":3741},[55],[52,3743,3745],{"className":3744},[59],[61,3746,3747],{"xmlns":63},[65,3748,3749,3767],{},[68,3750,3751,3753,3755,3757,3759,3761,3763,3765],{},[71,3752,73],{},[75,3754,78],{"stretchy":77},[80,3756,117],{},[75,3758,85],{"stretchy":77},[75,3760,88],{},[97,3762,92],{},[75,3764,106],{"separator":105},[97,3766,3680],{},[131,3768,3769],{"encoding":133},"P(\\text{tails}) = 0,5",[52,3771,3773,3803],{"className":3772,"ariaHidden":105},[138],[52,3774,3776,3779,3782,3785,3791,3794,3797,3800],{"className":3775},[142],[52,3777],{"className":3778,"style":147},[146],[52,3780,73],{"className":3781,"style":153},[151,152],[52,3783,78],{"className":3784},[157],[52,3786,3788],{"className":3787},[151,161],[52,3789,117],{"className":3790},[151],[52,3792,85],{"className":3793},[168],[52,3795],{"className":3796,"style":172},[108],[52,3798,88],{"className":3799},[176],[52,3801],{"className":3802,"style":172},[108],[52,3804,3806,3809,3812,3815,3818],{"className":3805},[142],[52,3807],{"className":3808,"style":3723},[146],[52,3810,92],{"className":3811},[151],[52,3813,106],{"className":3814},[278],[52,3816],{"className":3817,"style":286},[108],[52,3819,3680],{"className":3820},[151],[16,3822,3824],{"className":3823},[50],[52,3825,3827,3852],{"className":3826},[55],[52,3828,3830],{"className":3829},[59],[61,3831,3832],{"xmlns":63},[65,3833,3834,3849],{},[68,3835,3836,3838,3840,3843,3845,3847],{},[71,3837,73],{},[75,3839,78],{"stretchy":77},[80,3841,3842],{},"heads or tails",[75,3844,85],{"stretchy":77},[75,3846,88],{},[97,3848,99],{},[131,3850,3851],{"encoding":133},"P(\\text{heads or tails}) = 1",[52,3853,3855,3885],{"className":3854,"ariaHidden":105},[138],[52,3856,3858,3861,3864,3867,3873,3876,3879,3882],{"className":3857},[142],[52,3859],{"className":3860,"style":147},[146],[52,3862,73],{"className":3863,"style":153},[151,152],[52,3865,78],{"className":3866},[157],[52,3868,3870],{"className":3869},[151,161],[52,3871,3842],{"className":3872},[151],[52,3874,85],{"className":3875},[168],[52,3877],{"className":3878,"style":172},[108],[52,3880,88],{"className":3881},[176],[52,3883],{"className":3884,"style":172},[108],[52,3886,3888,3891],{"className":3887},[142],[52,3889],{"className":3890,"style":614},[146],[52,3892,99],{"className":3893},[151],[16,3895,3896,3897,3899,3900,1407],{},"That independence has a famous implication: if you've just flipped ten heads in a row, the eleventh flip is still 50\u002F50. The coin has no memory. Believing otherwise is the ",[25,3898,3513],{}," — ",[630,3901,3903],{"href":1772,"rel":3902},[634],"explored in depth here",[16,3905,3906,3907,3909,3910,3912,3913,624,3915,3917,3918,3920],{},"The number of heads you'd expect in ",[20,3908,658],{}," tosses follows a ",[25,3911,735],{}," with parameters ",[20,3914,658],{},[20,3916,16],{}," = 0.5. For large ",[20,3919,658],{},", this approximates a normal distribution — which is why you can use coin flips to teach the Central Limit Theorem—which explains that the sum of many independent random variables follows a normal distribution (bell curve).",[30,3922,3924],{"id":3923},"is-a-coin-toss-actually-fair","Is a coin toss actually fair?",[16,3926,3927,3928,3931],{},"Not quite. The same study by Stanford mathematician Persi Diaconis mentioned across this blog showed that a naturally-tossed coin lands on the same face it started on approximately ",[25,3929,3930],{},"51% of the time",". The bias comes from:",[1784,3933,3934,3937,3940],{},[1787,3935,3936],{},"The coin spending slightly more time rotating in its initial orientation before gravity takes over",[1787,3938,3939],{},"Thumb-push mechanics that consistently favour one direction",[1787,3941,3942],{},"Air resistance affecting the face that starts uppermost",[16,3944,3945],{},"A 1% bias sounds small. Over a single toss, it's irrelevant. Over thousands of tosses in a tournament or a statistical study, it matters. Although it seems that online physics simulators don't inherit the human throwing mechanics, they carry part of this bias and cannot guarantee a 50\u002F50 result, even if the toss is repeated many many times.",[30,3947,3949],{"id":3948},"famous-coin-tosses-in-history","Famous coin tosses in history",[16,3951,3952,3955],{},[25,3953,3954],{},"The 1998 Orange Bowl",": Nebraska vs. Tennessee. Nebraska called tails, the coin landed tails, Nebraska got the ball — and won 42–17.",[16,3957,3958,3961],{},[25,3959,3960],{},"The 1903 Wright Brothers moment",": Orville and Wilbur Wright reportedly flipped a coin to decide who would attempt the first powered flight at Kitty Hawk. Wilbur won the toss but his first attempt failed. Orville's second attempt, three days later, succeeded.",[16,3963,3964,3967],{},[25,3965,3966],{},"The 2015 Iowa Democratic Caucus",": Several precincts were tied between Hillary Clinton and Bernie Sanders. The tiebreaker? A coin toss. Clinton reportedly won all six flips — a sequence with a probability of about 1.6%, unlikely but entirely possible.",[16,3969,3970,3973],{},[25,3971,3972],{},"Football",": FIFA mandates coin tosses before international matches. The losing team picks the end they'll defend; the winning captain either takes first kick-off or lets the other team do so.",[30,3975,3977],{"id":3976},"why-we-trust-coin-tosses","Why we trust coin tosses",[16,3979,3980,3981,3984],{},"There's something philosophically interesting about the coin toss: it's accepted as fair precisely because ",[20,3982,3983],{},"nobody controls it",". The coin is an arbitrator that has no stake in the outcome.",[16,3986,3987],{},"This is why coin tosses are used to settle disputes rather than, say, rock-paper-scissors (which can be influenced by psychology and tells). The random element removes human agency — and with it, the possibility of complaint.",[16,3989,3990],{},"An online coin flip serves the same purpose. When both parties can see the flip happen in real time, the result carries the same social weight as a physical toss.",[30,3992,3524],{"id":3523},[16,3994,3995],{},"Heads or tails is one of the oldest randomisation tools in human history — simple, trusted, and, as it turns out, not quite perfectly fair in the real world. An online physics simulator gets you closer to the theoretical ideal than a human throw ever could.",[16,3997,3998,3999,4003],{},"Ready to flip? ",[630,4000,4002],{"href":3564,"rel":4001},[634],"Try heads or tails online"," — real physics, instant result.",{"title":1597,"searchDepth":1598,"depth":1598,"links":4005},[4006,4007,4008,4009,4010,4011],{"id":3613,"depth":1598,"text":3614},{"id":3640,"depth":1598,"text":3641},{"id":3923,"depth":1598,"text":3924},{"id":3948,"depth":1598,"text":3949},{"id":3976,"depth":1598,"text":3977},{"id":3523,"depth":1598,"text":3524},"2026-05-22T12:00:00Z","Why do we call it heads or tails? A look at the history, mathematics, and surprising science of the coin toss — from Julius Caesar to modern physics.",{"type":9,"value":4015},[4016,4018],[12,4017,3602],{"id":3607},[16,4019,3610],{},{},"\u002Fblog\u002Fheads-or-tails.en",{"title":3602,"description":4013},"heads-or-tails","blog\u002Fheads-or-tails.en",[4026,4027],"coin toss","probability 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