[{"data":1,"prerenderedAt":2054},["ShallowReactive",2],{"blog-post-en-probability-basics":3,"locale-switcher-blog-en-probability-basics":2051,"blog-siblings-en-probability-basics":2053},{"id":4,"title":5,"_locale":6,"body":7,"date":2030,"description":2031,"extension":2032,"meta":2033,"navigation":2046,"path":2047,"published":2046,"seo":2048,"slug":13,"stem":2049,"__hash__":2050},"blog\u002Fblog\u002Fprobability-basics.en.md","Probability Basics","en",{"type":8,"value":9,"toc":2015},"minimal",[10,14,27,30,35,186,197,201,204,231,234,238,245,263,274,278,289,425,428,814,825,829,832,837,848,1109,1213,1217,1231,1727,1733,1938,1945,1949,1955,1968,1972,1975,2003,2006],[11,12,5],"h1",{"id":13},"probability-basics",[15,16,17,18,22,23,26],"p",{},"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 ",[19,20,21],"strong",{},"heads"," or ",[19,24,25],{},"tails",".",[15,28,29],{},"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.",[31,32,34],"h2",{"id":33},"the-basics","The Basics",[15,36,37,38,177,178,181,182,185],{},"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 ",[39,40,43,72],"span",{"className":41},[42],"katex",[39,44,47],{"className":45},[46],"katex-mathml",[48,49,51],"math",{"xmlns":50},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[52,53,54,67],"semantics",{},[55,56,57],"mrow",{},[58,59,60,64],"mfrac",{},[61,62,63],"mn",{},"1",[61,65,66],{},"2",[68,69,71],"annotation",{"encoding":70},"application\u002Fx-tex","\\frac{1}{2}",[39,73,77],{"className":74,"ariaHidden":76},[75],"katex-html","true",[39,78,81,86],{"className":79},[80],"base",[39,82],{"className":83,"style":85},[84],"strut","height:1.1901em;vertical-align:-0.345em;",[39,87,90,95,173],{"className":88},[89],"mord",[39,91],{"className":92},[93,94],"mopen","nulldelimiter",[39,96,98],{"className":97},[58],[39,99,103,164],{"className":100},[101,102],"vlist-t","vlist-t2",[39,104,107,159],{"className":105},[106],"vlist-r",[39,108,112,133,144],{"className":109,"style":111},[110],"vlist","height:0.8451em;",[39,113,115,120],{"style":114},"top:-2.655em;",[39,116],{"className":117,"style":119},[118],"pstrut","height:3em;",[39,121,127],{"className":122},[123,124,125,126],"sizing","reset-size6","size3","mtight",[39,128,130],{"className":129},[89,126],[39,131,66],{"className":132},[89,126],[39,134,136,139],{"style":135},"top:-3.23em;",[39,137],{"className":138,"style":119},[118],[39,140],{"className":141,"style":143},[142],"frac-line","border-bottom-width:0.04em;",[39,145,147,150],{"style":146},"top:-3.394em;",[39,148],{"className":149,"style":119},[118],[39,151,153],{"className":152},[123,124,125,126],[39,154,156],{"className":155},[89,126],[39,157,63],{"className":158},[89,126],[39,160,163],{"className":161},[162],"vlist-s","​",[39,165,167],{"className":166},[106],[39,168,171],{"className":169,"style":170},[110],"height:0.345em;",[39,172],{},[39,174],{"className":175},[176,94],"mclose",", which is the same as ",[19,179,180],{},"0.5",", or also ",[19,183,184],{},"50%",". The three representations are equivalent and can be used interchangeably.",[15,187,188,189,196],{},"As we will see in ",[190,191,195],"a",{"href":192,"rel":193},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fthe-unexpected-statistics-of-coin-flips",[194],"nofollow","The Unexpected Statistics of Coin Flips blog post",", the probability of a coin landing is not exactly 50-50 every time.",[31,198,200],{"id":199},"outcomes-and-events","Outcomes and events",[15,202,203],{},"Before we go further, two small pieces of vocabulary:",[205,206,207,225],"ul",{},[208,209,210,211,214,215,218,219,221,222,26],"li",{},"An ",[19,212,213],{},"outcome"," is one of the possible things that can happen. When you flip a coin, the outcomes are ",[216,217,21],"em",{},", ",[216,220,25],{},", and (very rarely) ",[216,223,224],{},"edge",[208,226,210,227,230],{},[19,228,229],{},"event"," is a question we ask about the outcomes, like \"did it land on heads?\" or \"did it land on a face at all?\".",[15,232,233],{},"Probability is the tool we use to put a number on how likely an event is.",[31,235,237],{"id":236},"the-probability-scale","The probability scale",[15,239,240,241,244],{},"Probabilities always live on a scale from ",[19,242,243],{},"0 to 1",":",[205,246,247,253,258],{},[208,248,249,252],{},[19,250,251],{},"0"," means the event is impossible. A coin turning into a pigeon mid-air has probability 0.",[208,254,255,257],{},[19,256,63],{}," means the event is certain. A flipped coin eventually coming back down, thanks to gravity, has probability 1.",[208,259,260,261,26],{},"Anything in between is \"somewhat likely\". A fair coin landing on heads sits right in the middle, at ",[19,262,180],{},[15,264,265,266,269,270,273],{},"In percentage terms, that same scale goes from ",[19,267,268],{},"0%"," to ",[19,271,272],{},"100%",". Same idea, expressed in a different way.",[31,275,277],{"id":276},"the-complement-rule","The complement rule",[15,279,280,281,284,285,288],{},"Every event has a mirror image called its ",[19,282,283],{},"complement",": the event of it ",[216,286,287],{},"not"," happening. And because something either happens or it doesn't, the two probabilities always add up to 1. In symbols:",[15,290,293],{"className":291},[292],"math-block",[39,294,296,343],{"className":295},[42],[39,297,299],{"className":298},[46],[48,300,301],{"xmlns":50},[52,302,303,340],{},[55,304,305,309,314,318,321,324,327,329,332,334,336,338],{},[306,307,308],"mi",{},"P",[310,311,313],"mo",{"stretchy":312},"false","(",[315,316,317],"mtext",{},"not ",[306,319,320],{},"A",[310,322,323],{"stretchy":312},")",[310,325,326],{},"=",[61,328,63],{},[310,330,331],{},"−",[306,333,308],{},[310,335,313],{"stretchy":312},[306,337,320],{},[310,339,323],{"stretchy":312},[68,341,342],{"encoding":70},"P(\\text{not } A) = 1 - P(A)",[39,344,346,386,407],{"className":345,"ariaHidden":76},[75],[39,347,349,353,358,361,368,371,374,379,383],{"className":348},[80],[39,350],{"className":351,"style":352},[84],"height:1em;vertical-align:-0.25em;",[39,354,308],{"className":355,"style":357},[89,356],"mathnormal","margin-right:0.1389em;",[39,359,313],{"className":360},[93],[39,362,365],{"className":363},[89,364],"text",[39,366,317],{"className":367},[89],[39,369,320],{"className":370},[89,356],[39,372,323],{"className":373},[176],[39,375],{"className":376,"style":378},[377],"mspace","margin-right:0.2778em;",[39,380,326],{"className":381},[382],"mrel",[39,384],{"className":385,"style":378},[377],[39,387,389,393,396,400,404],{"className":388},[80],[39,390],{"className":391,"style":392},[84],"height:0.7278em;vertical-align:-0.0833em;",[39,394,63],{"className":395},[89],[39,397],{"className":398,"style":399},[377],"margin-right:0.2222em;",[39,401,331],{"className":402},[403],"mbin",[39,405],{"className":406,"style":399},[377],[39,408,410,413,416,419,422],{"className":409},[80],[39,411],{"className":412,"style":352},[84],[39,414,308],{"className":415,"style":357},[89,356],[39,417,313],{"className":418},[93],[39,420,320],{"className":421},[89,356],[39,423,323],{"className":424},[176],[15,426,427],{},"For a fair coin:",[205,429,430,617],{},[208,431,432,502,503,26],{},[39,433,435,459],{"className":434},[42],[39,436,438],{"className":437},[46],[48,439,440],{"xmlns":50},[52,441,442,456],{},[55,443,444,446,448,450,452,454],{},[306,445,308],{},[310,447,313],{"stretchy":312},[315,449,21],{},[310,451,323],{"stretchy":312},[310,453,326],{},[61,455,180],{},[68,457,458],{"encoding":70},"P(\\text{heads}) = 0.5",[39,460,462,492],{"className":461,"ariaHidden":76},[75],[39,463,465,468,471,474,480,483,486,489],{"className":464},[80],[39,466],{"className":467,"style":352},[84],[39,469,308],{"className":470,"style":357},[89,356],[39,472,313],{"className":473},[93],[39,475,477],{"className":476},[89,364],[39,478,21],{"className":479},[89],[39,481,323],{"className":482},[176],[39,484],{"className":485,"style":378},[377],[39,487,326],{"className":488},[382],[39,490],{"className":491,"style":378},[377],[39,493,495,499],{"className":494},[80],[39,496],{"className":497,"style":498},[84],"height:0.6444em;",[39,500,180],{"className":501},[89],", so ",[39,504,506,539],{"className":505},[42],[39,507,509],{"className":508},[46],[48,510,511],{"xmlns":50},[52,512,513,536],{},[55,514,515,517,519,522,524,526,528,530,532,534],{},[306,516,308],{},[310,518,313],{"stretchy":312},[315,520,521],{},"not heads",[310,523,323],{"stretchy":312},[310,525,326],{},[61,527,63],{},[310,529,331],{},[61,531,180],{},[310,533,326],{},[61,535,180],{},[68,537,538],{"encoding":70},"P(\\text{not heads}) = 1 - 0.5 = 0.5",[39,540,542,572,590,608],{"className":541,"ariaHidden":76},[75],[39,543,545,548,551,554,560,563,566,569],{"className":544},[80],[39,546],{"className":547,"style":352},[84],[39,549,308],{"className":550,"style":357},[89,356],[39,552,313],{"className":553},[93],[39,555,557],{"className":556},[89,364],[39,558,521],{"className":559},[89],[39,561,323],{"className":562},[176],[39,564],{"className":565,"style":378},[377],[39,567,326],{"className":568},[382],[39,570],{"className":571,"style":378},[377],[39,573,575,578,581,584,587],{"className":574},[80],[39,576],{"className":577,"style":392},[84],[39,579,63],{"className":580},[89],[39,582],{"className":583,"style":399},[377],[39,585,331],{"className":586},[403],[39,588],{"className":589,"style":399},[377],[39,591,593,596,599,602,605],{"className":592},[80],[39,594],{"className":595,"style":498},[84],[39,597,180],{"className":598},[89],[39,600],{"className":601,"style":378},[377],[39,603,326],{"className":604},[382],[39,606],{"className":607,"style":378},[377],[39,609,611,614],{"className":610},[80],[39,612],{"className":613,"style":498},[84],[39,615,180],{"className":616},[89],[208,618,619,808,809,813],{},[39,620,622,665],{"className":621},[42],[39,623,625],{"className":624},[46],[48,626,627],{"xmlns":50},[52,628,629,662],{},[55,630,631,633,635,638,640,643,645,647,657,659],{},[306,632,308],{},[310,634,313],{"stretchy":312},[315,636,637],{},"face",[310,639,323],{"stretchy":312},[310,641,642],{},"≈",[61,644,63],{},[310,646,331],{},[648,649,650],"mstyle",{"scriptlevel":251,"displaystyle":312},[58,651,652,654],{},[61,653,63],{},[61,655,656],{},"6000",[310,658,642],{},[61,660,661],{},"0.99983",[68,663,664],{"encoding":70},"P(\\text{face}) \\approx 1 - \\tfrac{1}{6000} \\approx 0.99983",[39,666,668,698,716,799],{"className":667,"ariaHidden":76},[75],[39,669,671,674,677,680,686,689,692,695],{"className":670},[80],[39,672],{"className":673,"style":352},[84],[39,675,308],{"className":676,"style":357},[89,356],[39,678,313],{"className":679},[93],[39,681,683],{"className":682},[89,364],[39,684,637],{"className":685},[89],[39,687,323],{"className":688},[176],[39,690],{"className":691,"style":378},[377],[39,693,642],{"className":694},[382],[39,696],{"className":697,"style":378},[377],[39,699,701,704,707,710,713],{"className":700},[80],[39,702],{"className":703,"style":392},[84],[39,705,63],{"className":706},[89],[39,708],{"className":709,"style":399},[377],[39,711,331],{"className":712},[403],[39,714],{"className":715,"style":399},[377],[39,717,719,722,790,793,796],{"className":718},[80],[39,720],{"className":721,"style":85},[84],[39,723,725,728,787],{"className":724},[89],[39,726],{"className":727},[93,94],[39,729,731],{"className":730},[58],[39,732,734,779],{"className":733},[101,102],[39,735,737,776],{"className":736},[106],[39,738,740,754,762],{"className":739,"style":111},[110],[39,741,742,745],{"style":114},[39,743],{"className":744,"style":119},[118],[39,746,748],{"className":747},[123,124,125,126],[39,749,751],{"className":750},[89,126],[39,752,656],{"className":753},[89,126],[39,755,756,759],{"style":135},[39,757],{"className":758,"style":119},[118],[39,760],{"className":761,"style":143},[142],[39,763,764,767],{"style":146},[39,765],{"className":766,"style":119},[118],[39,768,770],{"className":769},[123,124,125,126],[39,771,773],{"className":772},[89,126],[39,774,63],{"className":775},[89,126],[39,777,163],{"className":778},[162],[39,780,782],{"className":781},[106],[39,783,785],{"className":784,"style":170},[110],[39,786],{},[39,788],{"className":789},[176,94],[39,791],{"className":792,"style":378},[377],[39,794,642],{"className":795},[382],[39,797],{"className":798,"style":378},[377],[39,800,802,805],{"className":801},[80],[39,803],{"className":804,"style":498},[84],[39,806,661],{"className":807},[89],", because the only other option is the rare \"edge\" landing. (More on that 1-in-6000 number in ",[190,810,812],{"href":192,"rel":811},[194],"The Unexpected Statistics of Coin Flips",".)",[15,815,816,817,820,821,824],{},"The complement rule is often the easiest way to compute a probability: instead of counting every way something ",[216,818,819],{},"can"," happen, count the one way it ",[216,822,823],{},"can't"," and subtract from 1.",[31,826,828],{"id":827},"combining-events-and-vs-or","Combining events: \"and\" vs \"or\"",[15,830,831],{},"Things get interesting when we ask about more than one flip at a time. Two rules cover most of what you'll ever need:",[833,834,836],"h3",{"id":835},"and-both-things-happen","\"AND\" — both things happen",[15,838,839,840,843,844,847],{},"Coin flips don't influence each other (they are ",[19,841,842],{},"independent","). In this case, you ",[19,845,846],{},"multiply"," their probabilities:",[205,849,850,969],{},[208,851,852,968],{},[39,853,855,890],{"className":854},[42],[39,856,858],{"className":857},[46],[48,859,860],{"xmlns":50},[52,861,862,887],{},[55,863,864,866,868,871,873,875,877,880,882,884],{},[306,865,308],{},[310,867,313],{"stretchy":312},[315,869,870],{},"OO",[310,872,323],{"stretchy":312},[310,874,326],{},[61,876,180],{},[310,878,879],{},"×",[61,881,180],{},[310,883,326],{},[61,885,886],{},"0.25",[68,888,889],{"encoding":70},"P(\\text{OO}) = 0.5 \\times 0.5 = 0.25",[39,891,893,923,941,959],{"className":892,"ariaHidden":76},[75],[39,894,896,899,902,905,911,914,917,920],{"className":895},[80],[39,897],{"className":898,"style":352},[84],[39,900,308],{"className":901,"style":357},[89,356],[39,903,313],{"className":904},[93],[39,906,908],{"className":907},[89,364],[39,909,870],{"className":910},[89],[39,912,323],{"className":913},[176],[39,915],{"className":916,"style":378},[377],[39,918,326],{"className":919},[382],[39,921],{"className":922,"style":378},[377],[39,924,926,929,932,935,938],{"className":925},[80],[39,927],{"className":928,"style":392},[84],[39,930,180],{"className":931},[89],[39,933],{"className":934,"style":399},[377],[39,936,879],{"className":937},[403],[39,939],{"className":940,"style":399},[377],[39,942,944,947,950,953,956],{"className":943},[80],[39,945],{"className":946,"style":498},[84],[39,948,180],{"className":949},[89],[39,951],{"className":952,"style":378},[377],[39,954,326],{"className":955},[382],[39,957],{"className":958,"style":378},[377],[39,960,962,965],{"className":961},[80],[39,963],{"className":964,"style":498},[84],[39,966,886],{"className":967},[89],", or 25%.",[208,970,971,1108],{},[39,972,974,1012],{"className":973},[42],[39,975,977],{"className":976},[46],[48,978,979],{"xmlns":50},[52,980,981,1009],{},[55,982,983,985,987,990,992,994,996,998,1000,1002,1004,1006],{},[306,984,308],{},[310,986,313],{"stretchy":312},[315,988,989],{},"OOO",[310,991,323],{"stretchy":312},[310,993,326],{},[61,995,180],{},[310,997,879],{},[61,999,180],{},[310,1001,879],{},[61,1003,180],{},[310,1005,326],{},[61,1007,1008],{},"0.125",[68,1010,1011],{"encoding":70},"P(\\text{OOO}) = 0.5 \\times 0.5 \\times 0.5 = 0.125",[39,1013,1015,1045,1063,1081,1099],{"className":1014,"ariaHidden":76},[75],[39,1016,1018,1021,1024,1027,1033,1036,1039,1042],{"className":1017},[80],[39,1019],{"className":1020,"style":352},[84],[39,1022,308],{"className":1023,"style":357},[89,356],[39,1025,313],{"className":1026},[93],[39,1028,1030],{"className":1029},[89,364],[39,1031,989],{"className":1032},[89],[39,1034,323],{"className":1035},[176],[39,1037],{"className":1038,"style":378},[377],[39,1040,326],{"className":1041},[382],[39,1043],{"className":1044,"style":378},[377],[39,1046,1048,1051,1054,1057,1060],{"className":1047},[80],[39,1049],{"className":1050,"style":392},[84],[39,1052,180],{"className":1053},[89],[39,1055],{"className":1056,"style":399},[377],[39,1058,879],{"className":1059},[403],[39,1061],{"className":1062,"style":399},[377],[39,1064,1066,1069,1072,1075,1078],{"className":1065},[80],[39,1067],{"className":1068,"style":392},[84],[39,1070,180],{"className":1071},[89],[39,1073],{"className":1074,"style":399},[377],[39,1076,879],{"className":1077},[403],[39,1079],{"className":1080,"style":399},[377],[39,1082,1084,1087,1090,1093,1096],{"className":1083},[80],[39,1085],{"className":1086,"style":498},[84],[39,1088,180],{"className":1089},[89],[39,1091],{"className":1092,"style":378},[377],[39,1094,326],{"className":1095},[382],[39,1097],{"className":1098,"style":378},[377],[39,1100,1102,1105],{"className":1101},[80],[39,1103],{"className":1104,"style":498},[84],[39,1106,1008],{"className":1107},[89],", or 12.5%.",[15,1110,1111,1112,1212],{},"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 ",[39,1113,1115,1135],{"className":1114},[42],[39,1116,1118],{"className":1117},[46],[48,1119,1120],{"xmlns":50},[52,1121,1122,1132],{},[55,1123,1124],{},[648,1125,1126],{"scriptlevel":251,"displaystyle":312},[58,1127,1128,1130],{},[61,1129,63],{},[61,1131,66],{},[68,1133,1134],{"encoding":70},"\\tfrac{1}{2}",[39,1136,1138],{"className":1137,"ariaHidden":76},[75],[39,1139,1141,1144],{"className":1140},[80],[39,1142],{"className":1143,"style":85},[84],[39,1145,1147,1150,1209],{"className":1146},[89],[39,1148],{"className":1149},[93,94],[39,1151,1153],{"className":1152},[58],[39,1154,1156,1201],{"className":1155},[101,102],[39,1157,1159,1198],{"className":1158},[106],[39,1160,1162,1176,1184],{"className":1161,"style":111},[110],[39,1163,1164,1167],{"style":114},[39,1165],{"className":1166,"style":119},[118],[39,1168,1170],{"className":1169},[123,124,125,126],[39,1171,1173],{"className":1172},[89,126],[39,1174,66],{"className":1175},[89,126],[39,1177,1178,1181],{"style":135},[39,1179],{"className":1180,"style":119},[118],[39,1182],{"className":1183,"style":143},[142],[39,1185,1186,1189],{"style":146},[39,1187],{"className":1188,"style":119},[118],[39,1190,1192],{"className":1191},[123,124,125,126],[39,1193,1195],{"className":1194},[89,126],[39,1196,63],{"className":1197},[89,126],[39,1199,163],{"className":1200},[162],[39,1202,1204],{"className":1203},[106],[39,1205,1207],{"className":1206,"style":170},[110],[39,1208],{},[39,1210],{"className":1211},[176,94]," onto the product.",[833,1214,1216],{"id":1215},"or-at-least-one-of-several-things-happens","\"OR\" — at least one of several things happens",[15,1218,1219,1220,1223,1224,1227,1228,847],{},"If two outcomes ",[19,1221,1222],{},"can't happen at the same time"," (they're ",[19,1225,1226],{},"mutually exclusive","), you ",[19,1229,1230],{},"add",[205,1232,1233,1378],{},[208,1234,1235,1377],{},[39,1236,1238,1278],{"className":1237},[42],[39,1239,1241],{"className":1240},[46],[48,1242,1243],{"xmlns":50},[52,1244,1245,1275],{},[55,1246,1247,1249,1251,1254,1257,1260,1262,1264,1266,1269,1271,1273],{},[306,1248,308],{},[310,1250,313],{"stretchy":312},[315,1252,1253],{},"O",[310,1255,1256],{},"∪",[315,1258,1259],{},"X",[310,1261,323],{"stretchy":312},[310,1263,326],{},[61,1265,180],{},[310,1267,1268],{},"+",[61,1270,180],{},[310,1272,326],{},[61,1274,63],{},[68,1276,1277],{"encoding":70},"P(\\text{O} \\cup \\text{X}) = 0.5 + 0.5 = 1",[39,1279,1281,1308,1332,1350,1368],{"className":1280,"ariaHidden":76},[75],[39,1282,1284,1287,1290,1293,1299,1302,1305],{"className":1283},[80],[39,1285],{"className":1286,"style":352},[84],[39,1288,308],{"className":1289,"style":357},[89,356],[39,1291,313],{"className":1292},[93],[39,1294,1296],{"className":1295},[89,364],[39,1297,1253],{"className":1298},[89],[39,1300],{"className":1301,"style":399},[377],[39,1303,1256],{"className":1304},[403],[39,1306],{"className":1307,"style":399},[377],[39,1309,1311,1314,1320,1323,1326,1329],{"className":1310},[80],[39,1312],{"className":1313,"style":352},[84],[39,1315,1317],{"className":1316},[89,364],[39,1318,1259],{"className":1319},[89],[39,1321,323],{"className":1322},[176],[39,1324],{"className":1325,"style":378},[377],[39,1327,326],{"className":1328},[382],[39,1330],{"className":1331,"style":378},[377],[39,1333,1335,1338,1341,1344,1347],{"className":1334},[80],[39,1336],{"className":1337,"style":392},[84],[39,1339,180],{"className":1340},[89],[39,1342],{"className":1343,"style":399},[377],[39,1345,1268],{"className":1346},[403],[39,1348],{"className":1349,"style":399},[377],[39,1351,1353,1356,1359,1362,1365],{"className":1352},[80],[39,1354],{"className":1355,"style":498},[84],[39,1357,180],{"className":1358},[89],[39,1360],{"className":1361,"style":378},[377],[39,1363,326],{"className":1364},[382],[39,1366],{"className":1367,"style":378},[377],[39,1369,1371,1374],{"className":1370},[80],[39,1372],{"className":1373,"style":498},[84],[39,1375,63],{"className":1376},[89],". Unsurprising — that's every non-edge outcome.",[208,1379,1380,1726],{},[39,1381,1383,1438],{"className":1382},[42],[39,1384,1386],{"className":1385},[46],[48,1387,1388],{"xmlns":50},[52,1389,1390,1435],{},[55,1391,1392,1394,1396,1398,1400,1402,1404,1406,1415,1417,1425,1427],{},[306,1393,308],{},[310,1395,313],{"stretchy":312},[61,1397,63],{},[310,1399,1256],{},[61,1401,66],{},[310,1403,323],{"stretchy":312},[310,1405,326],{},[648,1407,1408],{"scriptlevel":251,"displaystyle":312},[58,1409,1410,1412],{},[61,1411,63],{},[61,1413,1414],{},"6",[310,1416,1268],{},[648,1418,1419],{"scriptlevel":251,"displaystyle":312},[58,1420,1421,1423],{},[61,1422,63],{},[61,1424,1414],{},[310,1426,326],{},[648,1428,1429],{"scriptlevel":251,"displaystyle":312},[58,1430,1431,1433],{},[61,1432,66],{},[61,1434,1414],{},[68,1436,1437],{"encoding":70},"P(1 \\cup 2) = \\tfrac{1}{6} + \\tfrac{1}{6} = \\tfrac{2}{6}",[39,1439,1441,1465,1486,1569,1652],{"className":1440,"ariaHidden":76},[75],[39,1442,1444,1447,1450,1453,1456,1459,1462],{"className":1443},[80],[39,1445],{"className":1446,"style":352},[84],[39,1448,308],{"className":1449,"style":357},[89,356],[39,1451,313],{"className":1452},[93],[39,1454,63],{"className":1455},[89],[39,1457],{"className":1458,"style":399},[377],[39,1460,1256],{"className":1461},[403],[39,1463],{"className":1464,"style":399},[377],[39,1466,1468,1471,1474,1477,1480,1483],{"className":1467},[80],[39,1469],{"className":1470,"style":352},[84],[39,1472,66],{"className":1473},[89],[39,1475,323],{"className":1476},[176],[39,1478],{"className":1479,"style":378},[377],[39,1481,326],{"className":1482},[382],[39,1484],{"className":1485,"style":378},[377],[39,1487,1489,1492,1560,1563,1566],{"className":1488},[80],[39,1490],{"className":1491,"style":85},[84],[39,1493,1495,1498,1557],{"className":1494},[89],[39,1496],{"className":1497},[93,94],[39,1499,1501],{"className":1500},[58],[39,1502,1504,1549],{"className":1503},[101,102],[39,1505,1507,1546],{"className":1506},[106],[39,1508,1510,1524,1532],{"className":1509,"style":111},[110],[39,1511,1512,1515],{"style":114},[39,1513],{"className":1514,"style":119},[118],[39,1516,1518],{"className":1517},[123,124,125,126],[39,1519,1521],{"className":1520},[89,126],[39,1522,1414],{"className":1523},[89,126],[39,1525,1526,1529],{"style":135},[39,1527],{"className":1528,"style":119},[118],[39,1530],{"className":1531,"style":143},[142],[39,1533,1534,1537],{"style":146},[39,1535],{"className":1536,"style":119},[118],[39,1538,1540],{"className":1539},[123,124,125,126],[39,1541,1543],{"className":1542},[89,126],[39,1544,63],{"className":1545},[89,126],[39,1547,163],{"className":1548},[162],[39,1550,1552],{"className":1551},[106],[39,1553,1555],{"className":1554,"style":170},[110],[39,1556],{},[39,1558],{"className":1559},[176,94],[39,1561],{"className":1562,"style":399},[377],[39,1564,1268],{"className":1565},[403],[39,1567],{"className":1568,"style":399},[377],[39,1570,1572,1575,1643,1646,1649],{"className":1571},[80],[39,1573],{"className":1574,"style":85},[84],[39,1576,1578,1581,1640],{"className":1577},[89],[39,1579],{"className":1580},[93,94],[39,1582,1584],{"className":1583},[58],[39,1585,1587,1632],{"className":1586},[101,102],[39,1588,1590,1629],{"className":1589},[106],[39,1591,1593,1607,1615],{"className":1592,"style":111},[110],[39,1594,1595,1598],{"style":114},[39,1596],{"className":1597,"style":119},[118],[39,1599,1601],{"className":1600},[123,124,125,126],[39,1602,1604],{"className":1603},[89,126],[39,1605,1414],{"className":1606},[89,126],[39,1608,1609,1612],{"style":135},[39,1610],{"className":1611,"style":119},[118],[39,1613],{"className":1614,"style":143},[142],[39,1616,1617,1620],{"style":146},[39,1618],{"className":1619,"style":119},[118],[39,1621,1623],{"className":1622},[123,124,125,126],[39,1624,1626],{"className":1625},[89,126],[39,1627,63],{"className":1628},[89,126],[39,1630,163],{"className":1631},[162],[39,1633,1635],{"className":1634},[106],[39,1636,1638],{"className":1637,"style":170},[110],[39,1639],{},[39,1641],{"className":1642},[176,94],[39,1644],{"className":1645,"style":378},[377],[39,1647,326],{"className":1648},[382],[39,1650],{"className":1651,"style":378},[377],[39,1653,1655,1658],{"className":1654},[80],[39,1656],{"className":1657,"style":85},[84],[39,1659,1661,1664,1723],{"className":1660},[89],[39,1662],{"className":1663},[93,94],[39,1665,1667],{"className":1666},[58],[39,1668,1670,1715],{"className":1669},[101,102],[39,1671,1673,1712],{"className":1672},[106],[39,1674,1676,1690,1698],{"className":1675,"style":111},[110],[39,1677,1678,1681],{"style":114},[39,1679],{"className":1680,"style":119},[118],[39,1682,1684],{"className":1683},[123,124,125,126],[39,1685,1687],{"className":1686},[89,126],[39,1688,1414],{"className":1689},[89,126],[39,1691,1692,1695],{"style":135},[39,1693],{"className":1694,"style":119},[118],[39,1696],{"className":1697,"style":143},[142],[39,1699,1700,1703],{"style":146},[39,1701],{"className":1702,"style":119},[118],[39,1704,1706],{"className":1705},[123,124,125,126],[39,1707,1709],{"className":1708},[89,126],[39,1710,66],{"className":1711},[89,126],[39,1713,163],{"className":1714},[162],[39,1716,1718],{"className":1717},[106],[39,1719,1721],{"className":1720,"style":170},[110],[39,1722],{},[39,1724],{"className":1725},[176,94]," (rolling a 1 or a 2 on a six-sided die).",[15,1728,1729,1730,1732],{},"If the events ",[216,1731,819],{}," overlap, simple addition double-counts the overlap and you have to subtract it back out:",[15,1734,1736],{"className":1735},[292],[39,1737,1739,1799],{"className":1738},[42],[39,1740,1742],{"className":1741},[46],[48,1743,1744],{"xmlns":50},[52,1745,1746,1796],{},[55,1747,1748,1750,1752,1754,1756,1759,1761,1763,1765,1767,1769,1771,1773,1775,1777,1779,1781,1783,1785,1787,1789,1792,1794],{},[306,1749,308],{},[310,1751,313],{"stretchy":312},[306,1753,320],{},[310,1755,1256],{},[306,1757,1758],{},"B",[310,1760,323],{"stretchy":312},[310,1762,326],{},[306,1764,308],{},[310,1766,313],{"stretchy":312},[306,1768,320],{},[310,1770,323],{"stretchy":312},[310,1772,1268],{},[306,1774,308],{},[310,1776,313],{"stretchy":312},[306,1778,1758],{},[310,1780,323],{"stretchy":312},[310,1782,331],{},[306,1784,308],{},[310,1786,313],{"stretchy":312},[306,1788,320],{},[310,1790,1791],{},"∩",[306,1793,1758],{},[310,1795,323],{"stretchy":312},[68,1797,1798],{"encoding":70},"P(A \\cup B) = P(A) + P(B) - P(A \\cap B)",[39,1800,1802,1826,1848,1875,1902,1926],{"className":1801,"ariaHidden":76},[75],[39,1803,1805,1808,1811,1814,1817,1820,1823],{"className":1804},[80],[39,1806],{"className":1807,"style":352},[84],[39,1809,308],{"className":1810,"style":357},[89,356],[39,1812,313],{"className":1813},[93],[39,1815,320],{"className":1816},[89,356],[39,1818],{"className":1819,"style":399},[377],[39,1821,1256],{"className":1822},[403],[39,1824],{"className":1825,"style":399},[377],[39,1827,1829,1832,1836,1839,1842,1845],{"className":1828},[80],[39,1830],{"className":1831,"style":352},[84],[39,1833,1758],{"className":1834,"style":1835},[89,356],"margin-right:0.0502em;",[39,1837,323],{"className":1838},[176],[39,1840],{"className":1841,"style":378},[377],[39,1843,326],{"className":1844},[382],[39,1846],{"className":1847,"style":378},[377],[39,1849,1851,1854,1857,1860,1863,1866,1869,1872],{"className":1850},[80],[39,1852],{"className":1853,"style":352},[84],[39,1855,308],{"className":1856,"style":357},[89,356],[39,1858,313],{"className":1859},[93],[39,1861,320],{"className":1862},[89,356],[39,1864,323],{"className":1865},[176],[39,1867],{"className":1868,"style":399},[377],[39,1870,1268],{"className":1871},[403],[39,1873],{"className":1874,"style":399},[377],[39,1876,1878,1881,1884,1887,1890,1893,1896,1899],{"className":1877},[80],[39,1879],{"className":1880,"style":352},[84],[39,1882,308],{"className":1883,"style":357},[89,356],[39,1885,313],{"className":1886},[93],[39,1888,1758],{"className":1889,"style":1835},[89,356],[39,1891,323],{"className":1892},[176],[39,1894],{"className":1895,"style":399},[377],[39,1897,331],{"className":1898},[403],[39,1900],{"className":1901,"style":399},[377],[39,1903,1905,1908,1911,1914,1917,1920,1923],{"className":1904},[80],[39,1906],{"className":1907,"style":352},[84],[39,1909,308],{"className":1910,"style":357},[89,356],[39,1912,313],{"className":1913},[93],[39,1915,320],{"className":1916},[89,356],[39,1918],{"className":1919,"style":399},[377],[39,1921,1791],{"className":1922},[403],[39,1924],{"className":1925,"style":399},[377],[39,1927,1929,1932,1935],{"className":1928},[80],[39,1930],{"className":1931,"style":352},[84],[39,1933,1758],{"className":1934,"style":1835},[89,356],[39,1936,323],{"className":1937},[176],[15,1939,1940,1941,1944],{},"That's the ",[19,1942,1943],{},"inclusion–exclusion principle",", and the deeper version is a topic for a future article.",[31,1946,1948],{"id":1947},"a-quick-note-on-independence","A quick note on independence",[15,1950,1951,1952,1954],{},"Two events are ",[19,1953,842],{}," 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.",[15,1956,1957,1958,1961,1962,1967],{},"This is the setup behind the famous ",[19,1959,1960],{},"Gambler's Fallacy",", which I unpack in ",[190,1963,1966],{"href":1964,"rel":1965},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fthe-unexpected-statistics-of-coin-flips#the-gamblers-fallacy",[194],"the statistics article",". For now, the short version: each flip is a fresh start.",[31,1969,1971],{"id":1970},"conclusion","Conclusion",[15,1973,1974],{},"That's really it for the basics:",[1976,1977,1978,1981,1991,2000],"ol",{},[208,1979,1980],{},"Probabilities live between 0 and 1.",[208,1982,1983,1984,1986,1987,1990],{},"Use the ",[19,1985,283],{}," (1 − P) when it's easier to count what ",[216,1988,1989],{},"doesn't"," happen.",[208,1992,1993,1996,1997,1999],{},[19,1994,1995],{},"Multiply"," for \"and\", ",[19,1998,1230],{}," for mutually exclusive \"or\".",[208,2001,2002],{},"Independent events don't care about each other's history.",[15,2004,2005],{},"With just those four ideas you can reason about almost any coin-flip question — and most dice, cards, and lottery-style problems too.",[15,2007,2008,2009,2014],{},"Put the theory to work: ",[190,2010,2013],{"href":2011,"rel":2012},"https:\u002F\u002Fflipthecoin.app\u002Fplay",[194],"flip a coin online"," and track your own streak.",{"title":2016,"searchDepth":2017,"depth":2017,"links":2018},"",2,[2019,2020,2021,2022,2023,2028,2029],{"id":33,"depth":2017,"text":34},{"id":199,"depth":2017,"text":200},{"id":236,"depth":2017,"text":237},{"id":276,"depth":2017,"text":277},{"id":827,"depth":2017,"text":828,"children":2024},[2025,2027],{"id":835,"depth":2026,"text":836},3,{"id":1215,"depth":2026,"text":1216},{"id":1947,"depth":2017,"text":1948},{"id":1970,"depth":2017,"text":1971},"2026-04-23T01:00:00Z","Learn the basics of probability and how it relates to coin flips.","md",{"excerpt":2034},{"type":8,"value":2035},[2036,2038,2044],[11,2037,5],{"id":13},[15,2039,17,2040,22,2042,26],{},[19,2041,21],{},[19,2043,25],{},[15,2045,29],{},true,"\u002Fblog\u002Fprobability-basics.en",{"title":5,"description":2031},"blog\u002Fprobability-basics.en","WJNn5BZd0K6VkTZrvq51xtk2P14CdrrltmD84BBmhW8",{"en":13,"es":2052},"fundamentos-de-probabilidad",{"en":13,"es":2052},1779926069799]