[{"data":1,"prerenderedAt":11217},["ShallowReactive",2],{"post-\u002Fblog\u002F2024\u002F2024-09-21-the-37-rules-in-making-choice":3},{"id":4,"title":5,"body":6,"description":3110,"draft":11204,"enableComment":9708,"extension":11205,"image":11206,"meta":11207,"navigation":9708,"onday":11208,"path":11209,"seo":11210,"stem":11211,"summary":11212,"tags":11213,"__hash__":11216},"blog\u002Fblog\u002F2024\u002F2024-09-21-the-37-rules-in-making-choice.md","Socrates: The Story of Searching for the Largest Ear of Wheat",{"type":7,"value":8,"toc":11186},"minimark",[9,14,18,21,24,27,30,33,39,43,189,192,195,325,328,629,690,1015,1110,1143,1203,1410,1556,1859,1951,1985,1988,1992,1998,2001,2194,2316,2558,2652,2743,2747,2838,2843,3102,3105,3112,3115,3698,4000,4004,4007,4254,4415,5017,5072,5500,5503,5710,5714,5717,5936,6061,6217,6221,6443,6777,6780,7111,7316,7447,7450,7983,7986,8497,8529,8704,8707,8855,8858,8976,9289,9293,9296,9515,9617,9621,9624,9627,10144,10148,10216,10219,10230,10233,10236,10308,10312,11107,11111,11166,11169,11173,11179,11182],[10,11,13],"h2",{"id":12},"the-largest-ear-of-wheat","The Largest Ear of Wheat",[15,16,17],"p",{},"It is said that Socrates once led his students for a walk in a wheat field. His students asked him: how can one find the most suitable life partner?",[15,19,20],{},"Socrates looked at the wheat field and gave them a task: \"Go into the wheat field and find the largest ear of wheat, but as you walk, you can only move forward and cannot turn back.\"",[15,22,23],{},"The students began walking while choosing, and the field was full of large ears of wheat. But which one was the largest? They walked with their heads down, checking one ear of wheat after another. They shook their heads at this one; they shook their heads at that one. They always believed that the largest ear was still ahead. Although they picked a few, they were not satisfied and threw them away. They thought there were plenty of chances and no need to decide too early.",[15,25,26],{},"The students continued walking with their heads down, carefully selecting as they went, for quite a long time. Suddenly, they heard Socrates' deep, resounding voice: \"You have reached the end.\" At that moment, the students, empty-handed, realized what had happened and felt disheartened.",[15,28,29],{},"Socrates said to them: \"There is certainly a largest ear of wheat in this field, but you may not encounter it; and even if you do, you may not make the correct judgment. Therefore, the largest ear is the one you just picked.\"",[15,31,32],{},"This story tells us that life is like walking in a wheat field, also searching for the largest ear of wheat. Some people, upon seeing a full and ripe ear, seize the opportunity and pick it without hesitation; others keep looking around and miss their chances again and again. Of course, the goal should be the largest ear, but holding the one in hand is the most tangible reality.",[15,34,35],{},[36,37,38],"strong",{},"So, the question arises: Is there a method that can help us find the largest ear of wheat as efficiently as possible?",[10,40,42],{"id":41},"treasure-box-game","Treasure Box Game",[15,44,45,46,93,94,125,126,155,156,188],{},"Let’s consider another example. Imagine there is a gambling game with the following rules: There are ",[47,48,52,74],"span",{"className":49,"translate":51},[50],"katex","no",[47,53,56],{"className":54},[55],"katex-mathml",[57,58,60],"math",{"xmlns":59},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[61,62,63,70],"semantics",{},[64,65,66],"mrow",{},[67,68,69],"mi",{},"m",[71,72,69],"annotation",{"encoding":73},"application\u002Fx-tex",[47,75,79],{"className":76,"ariaHidden":78},[77],"katex-html","true",[47,80,83,88],{"className":81},[82],"base",[47,84],{"className":85,"style":87},[86],"strut","height:0.4306em;",[47,89,69],{"className":90},[91,92],"mord","mathnormal"," treasure boxes, each containing a random prize amount between ",[47,95,97,112],{"className":96,"translate":51},[50],[47,98,100],{"className":99},[55],[57,101,102],{"xmlns":59},[61,103,104,110],{},[64,105,106],{},[107,108,109],"mn",{},"0",[71,111,109],{"encoding":73},[47,113,115],{"className":114,"ariaHidden":78},[77],[47,116,118,122],{"className":117},[82],[47,119],{"className":120,"style":121},[86],"height:0.6444em;",[47,123,109],{"className":124},[91]," and ",[47,127,129,143],{"className":128,"translate":51},[50],[47,130,132],{"className":131},[55],[57,133,134],{"xmlns":59},[61,135,136,141],{},[64,137,138],{},[67,139,140],{},"x",[71,142,140],{"encoding":73},[47,144,146],{"className":145,"ariaHidden":78},[77],[47,147,149,152],{"className":148},[82],[47,150],{"className":151,"style":87},[86],[47,153,140],{"className":154},[91,92]," dollars, uniformly distributed. ",[36,157,158,159,187],{},"However, you don’t know what the maximum amount ",[47,160,162,175],{"className":161,"translate":51},[50],[47,163,165],{"className":164},[55],[57,166,167],{"xmlns":59},[61,168,169,173],{},[64,170,171],{},[67,172,140],{},[71,174,140],{"encoding":73},[47,176,178],{"className":177,"ariaHidden":78},[77],[47,179,181,184],{"className":180},[82],[47,182],{"className":183,"style":87},[86],[47,185,140],{"className":186},[91,92]," in the boxes is."," Once you open a treasure box and see the prize, you have two choices: either take the prize and stop, or discard the box and open another one. However, once discarded, a box cannot be chosen again.",[15,190,191],{},"Now the question arises: how should you strategize to get the most money possible? Naturally, the more money, the better.",[15,193,194],{},"What would happen if we didn’t use any strategy and randomly picked a treasure box?",[15,196,197,198,295,296,324],{},"Let’s assume the amounts in the treasure boxes range from ",[47,199,201,242],{"className":200,"translate":51},[50],[47,202,204],{"className":203},[55],[57,205,206],{"xmlns":59},[61,207,208,239],{},[64,209,210,213,217,220,222,225,227,231,233,235,237],{},[107,211,212],{},"1000",[214,215,216],"mo",{"separator":78},",",[107,218,219],{},"999",[214,221,216],{"separator":78},[107,223,224],{},"998",[214,226,216],{"separator":78},[67,228,230],{"mathvariant":229},"normal",".",[67,232,230],{"mathvariant":229},[67,234,230],{"mathvariant":229},[214,236,216],{"separator":78},[107,238,109],{},[71,240,241],{"encoding":73},"1000, 999, 998, ..., 0",[47,243,245],{"className":244,"ariaHidden":78},[77],[47,246,248,252,255,259,264,267,270,273,276,279,282,286,289,292],{"className":247},[82],[47,249],{"className":250,"style":251},[86],"height:0.8389em;vertical-align:-0.1944em;",[47,253,212],{"className":254},[91],[47,256,216],{"className":257},[258],"mpunct",[47,260],{"className":261,"style":263},[262],"mspace","margin-right:0.1667em;",[47,265,219],{"className":266},[91],[47,268,216],{"className":269},[258],[47,271],{"className":272,"style":263},[262],[47,274,224],{"className":275},[91],[47,277,216],{"className":278},[258],[47,280],{"className":281,"style":263},[262],[47,283,285],{"className":284},[91],"...",[47,287,216],{"className":288},[258],[47,290],{"className":291,"style":263},[262],[47,293,109],{"className":294},[91],", with the top prize being ",[47,297,299,312],{"className":298,"translate":51},[50],[47,300,302],{"className":301},[55],[57,303,304],{"xmlns":59},[61,305,306,310],{},[64,307,308],{},[107,309,212],{},[71,311,212],{"encoding":73},[47,313,315],{"className":314,"ariaHidden":78},[77],[47,316,318,321],{"className":317},[82],[47,319],{"className":320,"style":121},[86],[47,322,212],{"className":323},[91],". What is the probability of drawing the top prize?",[15,326,327],{},"Since each specific amount (including the top prize of $1000) has an equal chance of appearing, and the prize is uniformly distributed, the probability of drawing the top prize can be calculated as follows:",[47,329,332],{"className":330,"translate":51},[331],"katex-display",[47,333,335,409],{"className":334,"translate":51},[50],[47,336,338],{"className":337},[55],[57,339,341],{"xmlns":59,"display":340},"block",[61,342,343,406],{},[64,344,345,373,376],{},[346,347,348,351],"msub",{},[67,349,350],{},"P",[64,352,353,356,359,362,365,367,370],{},[67,354,355],{},"J",[67,357,358],{},"a",[67,360,361],{},"c",[67,363,364],{},"k",[67,366,15],{},[67,368,369],{},"o",[67,371,372],{},"t",[214,374,375],{},"=",[377,378,379,382],"mfrac",{},[107,380,381],{},"1",[64,383,384,388,392,395,398,401,403],{},[214,385,387],{"stretchy":386},"false","(",[389,390,391],"mtext",{},"MaxMoney",[214,393,394],{},"−",[389,396,397],{},"MinMoney",[214,399,400],{},"+",[107,402,381],{},[214,404,405],{"stretchy":386},")",[71,407,408],{"encoding":73},"P_{Jackpot} = \\frac{1}{(\\text{MaxMoney} - \\text{MinMoney} + 1)}",[47,410,412,510],{"className":411,"ariaHidden":78},[77],[47,413,415,419,499,503,507],{"className":414},[82],[47,416],{"className":417,"style":418},[86],"height:0.9694em;vertical-align:-0.2861em;",[47,420,422,426],{"className":421},[91],[47,423,350],{"className":424,"style":425},[91,92],"margin-right:0.13889em;",[47,427,430],{"className":428},[429],"msupsub",[47,431,435,490],{"className":432},[433,434],"vlist-t","vlist-t2",[47,436,439,485],{"className":437},[438],"vlist-r",[47,440,444],{"className":441,"style":443},[442],"vlist","height:0.3361em;",[47,445,447,452],{"style":446},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[47,448],{"className":449,"style":451},[450],"pstrut","height:2.7em;",[47,453,459],{"className":454},[455,456,457,458],"sizing","reset-size6","size3","mtight",[47,460,462,466,469,472,476,479,482],{"className":461},[91,458],[47,463,355],{"className":464,"style":465},[91,92,458],"margin-right:0.09618em;",[47,467,358],{"className":468},[91,92,458],[47,470,361],{"className":471},[91,92,458],[47,473,364],{"className":474,"style":475},[91,92,458],"margin-right:0.03148em;",[47,477,15],{"className":478},[91,92,458],[47,480,369],{"className":481},[91,92,458],[47,483,372],{"className":484},[91,92,458],[47,486,489],{"className":487},[488],"vlist-s","​",[47,491,493],{"className":492},[438],[47,494,497],{"className":495,"style":496},[442],"height:0.2861em;",[47,498],{},[47,500],{"className":501,"style":502},[262],"margin-right:0.2778em;",[47,504,375],{"className":505},[506],"mrel",[47,508],{"className":509,"style":502},[262],[47,511,513,517],{"className":512},[82],[47,514],{"className":515,"style":516},[86],"height:2.2574em;vertical-align:-0.936em;",[47,518,520,525,626],{"className":519},[91],[47,521],{"className":522},[523,524],"mopen","nulldelimiter",[47,526,528],{"className":527},[377],[47,529,531,617],{"className":530},[433,434],[47,532,534,614],{"className":533},[438],[47,535,538,591,602],{"className":536,"style":537},[442],"height:1.3214em;",[47,539,541,545],{"style":540},"top:-2.314em;",[47,542],{"className":543,"style":544},[450],"height:3em;",[47,546,548,551,558,562,566,569,575,578,581,584,587],{"className":547},[91],[47,549,387],{"className":550},[523],[47,552,555],{"className":553},[91,554],"text",[47,556,391],{"className":557},[91],[47,559],{"className":560,"style":561},[262],"margin-right:0.2222em;",[47,563,394],{"className":564},[565],"mbin",[47,567],{"className":568,"style":561},[262],[47,570,572],{"className":571},[91,554],[47,573,397],{"className":574},[91],[47,576],{"className":577,"style":561},[262],[47,579,400],{"className":580},[565],[47,582],{"className":583,"style":561},[262],[47,585,381],{"className":586},[91],[47,588,405],{"className":589},[590],"mclose",[47,592,594,597],{"style":593},"top:-3.23em;",[47,595],{"className":596,"style":544},[450],[47,598],{"className":599,"style":601},[600],"frac-line","border-bottom-width:0.04em;",[47,603,605,608],{"style":604},"top:-3.677em;",[47,606],{"className":607,"style":544},[450],[47,609,611],{"className":610},[91],[47,612,381],{"className":613},[91],[47,615,489],{"className":616},[488],[47,618,620],{"className":619},[438],[47,621,624],{"className":622,"style":623},[442],"height:0.936em;",[47,625],{},[47,627],{"className":628},[590,524],[15,630,631,632,660,661,689],{},"In this case, the maximum prize is ",[47,633,635,648],{"className":634,"translate":51},[50],[47,636,638],{"className":637},[55],[57,639,640],{"xmlns":59},[61,641,642,646],{},[64,643,644],{},[107,645,212],{},[71,647,212],{"encoding":73},[47,649,651],{"className":650,"ariaHidden":78},[77],[47,652,654,657],{"className":653},[82],[47,655],{"className":656,"style":121},[86],[47,658,212],{"className":659},[91]," and the minimum is ",[47,662,664,677],{"className":663,"translate":51},[50],[47,665,667],{"className":666},[55],[57,668,669],{"xmlns":59},[61,670,671,675],{},[64,672,673],{},[107,674,109],{},[71,676,109],{"encoding":73},[47,678,680],{"className":679,"ariaHidden":78},[77],[47,681,683,686],{"className":682},[82],[47,684],{"className":685,"style":121},[86],[47,687,109],{"className":688},[91],", so:",[47,691,693],{"className":692,"translate":51},[331],[47,694,696,759],{"className":695,"translate":51},[50],[47,697,699],{"className":698},[55],[57,700,701],{"xmlns":59,"display":340},[61,702,703,756],{},[64,704,705,725,727,747,749],{},[346,706,707,709],{},[67,708,350],{},[64,710,711,713,715,717,719,721,723],{},[67,712,355],{},[67,714,358],{},[67,716,361],{},[67,718,364],{},[67,720,15],{},[67,722,369],{},[67,724,372],{},[214,726,375],{},[377,728,729,731],{},[107,730,381],{},[64,732,733,735,737,739,741,743,745],{},[214,734,387],{"stretchy":386},[107,736,212],{},[214,738,394],{},[107,740,109],{},[214,742,400],{},[107,744,381],{},[214,746,405],{"stretchy":386},[214,748,375],{},[377,750,751,753],{},[107,752,381],{},[107,754,755],{},"1001",[71,757,758],{"encoding":73},"P_{Jackpot} = \\frac{1}{(1000 - 0 + 1)} = \\frac{1}{1001}",[47,760,762,838,945],{"className":761,"ariaHidden":78},[77],[47,763,765,768,829,832,835],{"className":764},[82],[47,766],{"className":767,"style":418},[86],[47,769,771,774],{"className":770},[91],[47,772,350],{"className":773,"style":425},[91,92],[47,775,777],{"className":776},[429],[47,778,780,821],{"className":779},[433,434],[47,781,783,818],{"className":782},[438],[47,784,786],{"className":785,"style":443},[442],[47,787,788,791],{"style":446},[47,789],{"className":790,"style":451},[450],[47,792,794],{"className":793},[455,456,457,458],[47,795,797,800,803,806,809,812,815],{"className":796},[91,458],[47,798,355],{"className":799,"style":465},[91,92,458],[47,801,358],{"className":802},[91,92,458],[47,804,361],{"className":805},[91,92,458],[47,807,364],{"className":808,"style":475},[91,92,458],[47,810,15],{"className":811},[91,92,458],[47,813,369],{"className":814},[91,92,458],[47,816,372],{"className":817},[91,92,458],[47,819,489],{"className":820},[488],[47,822,824],{"className":823},[438],[47,825,827],{"className":826,"style":496},[442],[47,828],{},[47,830],{"className":831,"style":502},[262],[47,833,375],{"className":834},[506],[47,836],{"className":837,"style":502},[262],[47,839,841,844,936,939,942],{"className":840},[82],[47,842],{"className":843,"style":516},[86],[47,845,847,850,933],{"className":846},[91],[47,848],{"className":849},[523,524],[47,851,853],{"className":852},[377],[47,854,856,925],{"className":855},[433,434],[47,857,859,922],{"className":858},[438],[47,860,862,903,911],{"className":861,"style":537},[442],[47,863,864,867],{"style":540},[47,865],{"className":866,"style":544},[450],[47,868,870,873,876,879,882,885,888,891,894,897,900],{"className":869},[91],[47,871,387],{"className":872},[523],[47,874,212],{"className":875},[91],[47,877],{"className":878,"style":561},[262],[47,880,394],{"className":881},[565],[47,883],{"className":884,"style":561},[262],[47,886,109],{"className":887},[91],[47,889],{"className":890,"style":561},[262],[47,892,400],{"className":893},[565],[47,895],{"className":896,"style":561},[262],[47,898,381],{"className":899},[91],[47,901,405],{"className":902},[590],[47,904,905,908],{"style":593},[47,906],{"className":907,"style":544},[450],[47,909],{"className":910,"style":601},[600],[47,912,913,916],{"style":604},[47,914],{"className":915,"style":544},[450],[47,917,919],{"className":918},[91],[47,920,381],{"className":921},[91],[47,923,489],{"className":924},[488],[47,926,928],{"className":927},[438],[47,929,931],{"className":930,"style":623},[442],[47,932],{},[47,934],{"className":935},[590,524],[47,937],{"className":938,"style":502},[262],[47,940,375],{"className":941},[506],[47,943],{"className":944,"style":502},[262],[47,946,948,952],{"className":947},[82],[47,949],{"className":950,"style":951},[86],"height:2.0074em;vertical-align:-0.686em;",[47,953,955,958,1012],{"className":954},[91],[47,956],{"className":957},[523,524],[47,959,961],{"className":960},[377],[47,962,964,1003],{"className":963},[433,434],[47,965,967,1000],{"className":966},[438],[47,968,970,981,989],{"className":969,"style":537},[442],[47,971,972,975],{"style":540},[47,973],{"className":974,"style":544},[450],[47,976,978],{"className":977},[91],[47,979,755],{"className":980},[91],[47,982,983,986],{"style":593},[47,984],{"className":985,"style":544},[450],[47,987],{"className":988,"style":601},[600],[47,990,991,994],{"style":604},[47,992],{"className":993,"style":544},[450],[47,995,997],{"className":996},[91],[47,998,381],{"className":999},[91],[47,1001,489],{"className":1002},[488],[47,1004,1006],{"className":1005},[438],[47,1007,1010],{"className":1008,"style":1009},[442],"height:0.686em;",[47,1011],{},[47,1013],{"className":1014},[590,524],[15,1016,1017,1018,1046,1047,1109],{},"Therefore, the probability of randomly picking a treasure box containing the top prize of ",[47,1019,1021,1034],{"className":1020,"translate":51},[50],[47,1022,1024],{"className":1023},[55],[57,1025,1026],{"xmlns":59},[61,1027,1028,1032],{},[64,1029,1030],{},[107,1031,212],{},[71,1033,212],{"encoding":73},[47,1035,1037],{"className":1036,"ariaHidden":78},[77],[47,1038,1040,1043],{"className":1039},[82],[47,1041],{"className":1042,"style":121},[86],[47,1044,212],{"className":1045},[91]," is ",[47,1048,1050,1076],{"className":1049,"translate":51},[50],[47,1051,1053],{"className":1052},[55],[57,1054,1055],{"xmlns":59},[61,1056,1057,1073],{},[64,1058,1059,1062,1065,1068,1071],{},[107,1060,1061],{},"0.0999",[67,1063,1064],{"mathvariant":229},"%",[214,1066,1067],{},"≈",[107,1069,1070],{},"0.1",[67,1072,1064],{"mathvariant":229},[71,1074,1075],{"encoding":73},"0.0999\\% ≈ 0.1\\%",[47,1077,1079,1099],{"className":1078,"ariaHidden":78},[77],[47,1080,1082,1086,1090,1093,1096],{"className":1081},[82],[47,1083],{"className":1084,"style":1085},[86],"height:0.8056em;vertical-align:-0.0556em;",[47,1087,1089],{"className":1088},[91],"0.0999%",[47,1091],{"className":1092,"style":502},[262],[47,1094,1067],{"className":1095},[506],[47,1097],{"className":1098,"style":502},[262],[47,1100,1102,1105],{"className":1101},[82],[47,1103],{"className":1104,"style":1085},[86],[47,1106,1108],{"className":1107},[91],"0.1%",". The odds are very low indeed.",[15,1111,1112,1113,1142],{},"Since chasing the jackpot is unrealistic, let’s lower our expectations a bit. What if we aim for a prize of at least ",[47,1114,1116,1130],{"className":1115,"translate":51},[50],[47,1117,1119],{"className":1118},[55],[57,1120,1121],{"xmlns":59},[61,1122,1123,1128],{},[64,1124,1125],{},[107,1126,1127],{},"800",[71,1129,1127],{"encoding":73},[47,1131,1133],{"className":1132,"ariaHidden":78},[77],[47,1134,1136,1139],{"className":1135},[82],[47,1137],{"className":1138,"style":121},[86],[47,1140,1127],{"className":1141},[91],"? What is the probability of winning that?",[15,1144,1145,1146,125,1174,1202],{},"We can use the formula for uniform distribution. For a prize uniformly distributed between ",[47,1147,1149,1162],{"className":1148,"translate":51},[50],[47,1150,1152],{"className":1151},[55],[57,1153,1154],{"xmlns":59},[61,1155,1156,1160],{},[64,1157,1158],{},[107,1159,109],{},[71,1161,109],{"encoding":73},[47,1163,1165],{"className":1164,"ariaHidden":78},[77],[47,1166,1168,1171],{"className":1167},[82],[47,1169],{"className":1170,"style":121},[86],[47,1172,109],{"className":1173},[91],[47,1175,1177,1190],{"className":1176,"translate":51},[50],[47,1178,1180],{"className":1179},[55],[57,1181,1182],{"xmlns":59},[61,1183,1184,1188],{},[64,1185,1186],{},[107,1187,212],{},[71,1189,212],{"encoding":73},[47,1191,1193],{"className":1192,"ariaHidden":78},[77],[47,1194,1196,1199],{"className":1195},[82],[47,1197],{"className":1198,"style":121},[86],[47,1200,212],{"className":1201},[91],", the probability density function is constant. The formula for probability is:",[47,1204,1206],{"className":1205,"translate":51},[331],[47,1207,1209,1255],{"className":1208,"translate":51},[50],[47,1210,1212],{"className":1211},[55],[57,1213,1214],{"xmlns":59,"display":340},[61,1215,1216,1252],{},[64,1217,1218,1220,1222,1224,1227,1230,1232,1235,1237,1239],{},[67,1219,350],{},[214,1221,387],{"stretchy":386},[67,1223,358],{},[214,1225,1226],{},"≤",[67,1228,1229],{},"X",[214,1231,1226],{},[67,1233,1234],{},"b",[214,1236,405],{"stretchy":386},[214,1238,375],{},[377,1240,1241,1249],{},[64,1242,1243,1245,1247],{},[67,1244,1234],{},[214,1246,394],{},[67,1248,358],{},[389,1250,1251],{},"range",[71,1253,1254],{"encoding":73},"P(a \\leq X \\leq b) = \\frac{b - a}{\\text{range}}",[47,1256,1258,1283,1303,1324],{"className":1257,"ariaHidden":78},[77],[47,1259,1261,1265,1268,1271,1274,1277,1280],{"className":1260},[82],[47,1262],{"className":1263,"style":1264},[86],"height:1em;vertical-align:-0.25em;",[47,1266,350],{"className":1267,"style":425},[91,92],[47,1269,387],{"className":1270},[523],[47,1272,358],{"className":1273},[91,92],[47,1275],{"className":1276,"style":502},[262],[47,1278,1226],{"className":1279},[506],[47,1281],{"className":1282,"style":502},[262],[47,1284,1286,1290,1294,1297,1300],{"className":1285},[82],[47,1287],{"className":1288,"style":1289},[86],"height:0.8193em;vertical-align:-0.136em;",[47,1291,1229],{"className":1292,"style":1293},[91,92],"margin-right:0.07847em;",[47,1295],{"className":1296,"style":502},[262],[47,1298,1226],{"className":1299},[506],[47,1301],{"className":1302,"style":502},[262],[47,1304,1306,1309,1312,1315,1318,1321],{"className":1305},[82],[47,1307],{"className":1308,"style":1264},[86],[47,1310,1234],{"className":1311},[91,92],[47,1313,405],{"className":1314},[590],[47,1316],{"className":1317,"style":502},[262],[47,1319,375],{"className":1320},[506],[47,1322],{"className":1323,"style":502},[262],[47,1325,1327,1331],{"className":1326},[82],[47,1328],{"className":1329,"style":1330},[86],"height:2.2519em;vertical-align:-0.8804em;",[47,1332,1334,1337,1407],{"className":1333},[91],[47,1335],{"className":1336},[523,524],[47,1338,1340],{"className":1339},[377],[47,1341,1343,1398],{"className":1342},[433,434],[47,1344,1346,1395],{"className":1345},[438],[47,1347,1350,1364,1372],{"className":1348,"style":1349},[442],"height:1.3714em;",[47,1351,1352,1355],{"style":540},[47,1353],{"className":1354,"style":544},[450],[47,1356,1358],{"className":1357},[91],[47,1359,1361],{"className":1360},[91,554],[47,1362,1251],{"className":1363},[91],[47,1365,1366,1369],{"style":593},[47,1367],{"className":1368,"style":544},[450],[47,1370],{"className":1371,"style":601},[600],[47,1373,1374,1377],{"style":604},[47,1375],{"className":1376,"style":544},[450],[47,1378,1380,1383,1386,1389,1392],{"className":1379},[91],[47,1381,1234],{"className":1382},[91,92],[47,1384],{"className":1385,"style":561},[262],[47,1387,394],{"className":1388},[565],[47,1390],{"className":1391,"style":561},[262],[47,1393,358],{"className":1394},[91,92],[47,1396,489],{"className":1397},[488],[47,1399,1401],{"className":1400},[438],[47,1402,1405],{"className":1403,"style":1404},[442],"height:0.8804em;",[47,1406],{},[47,1408],{"className":1409},[590,524],[15,1411,1412,1413,1441,1442,1470,1471,1499,1500,1441,1528,689],{},"In this case, the range is from ",[47,1414,1416,1429],{"className":1415,"translate":51},[50],[47,1417,1419],{"className":1418},[55],[57,1420,1421],{"xmlns":59},[61,1422,1423,1427],{},[64,1424,1425],{},[107,1426,109],{},[71,1428,109],{"encoding":73},[47,1430,1432],{"className":1431,"ariaHidden":78},[77],[47,1433,1435,1438],{"className":1434},[82],[47,1436],{"className":1437,"style":121},[86],[47,1439,109],{"className":1440},[91]," to ",[47,1443,1445,1458],{"className":1444,"translate":51},[50],[47,1446,1448],{"className":1447},[55],[57,1449,1450],{"xmlns":59},[61,1451,1452,1456],{},[64,1453,1454],{},[107,1455,212],{},[71,1457,212],{"encoding":73},[47,1459,1461],{"className":1460,"ariaHidden":78},[77],[47,1462,1464,1467],{"className":1463},[82],[47,1465],{"className":1466,"style":121},[86],[47,1468,212],{"className":1469},[91],", so the total range is ",[47,1472,1474,1487],{"className":1473,"translate":51},[50],[47,1475,1477],{"className":1476},[55],[57,1478,1479],{"xmlns":59},[61,1480,1481,1485],{},[64,1482,1483],{},[107,1484,212],{},[71,1486,212],{"encoding":73},[47,1488,1490],{"className":1489,"ariaHidden":78},[77],[47,1491,1493,1496],{"className":1492},[82],[47,1494],{"className":1495,"style":121},[86],[47,1497,212],{"className":1498},[91],". We want to calculate the probability for the interval from ",[47,1501,1503,1516],{"className":1502,"translate":51},[50],[47,1504,1506],{"className":1505},[55],[57,1507,1508],{"xmlns":59},[61,1509,1510,1514],{},[64,1511,1512],{},[107,1513,1127],{},[71,1515,1127],{"encoding":73},[47,1517,1519],{"className":1518,"ariaHidden":78},[77],[47,1520,1522,1525],{"className":1521},[82],[47,1523],{"className":1524,"style":121},[86],[47,1526,1127],{"className":1527},[91],[47,1529,1531,1544],{"className":1530,"translate":51},[50],[47,1532,1534],{"className":1533},[55],[57,1535,1536],{"xmlns":59},[61,1537,1538,1542],{},[64,1539,1540],{},[107,1541,212],{},[71,1543,212],{"encoding":73},[47,1545,1547],{"className":1546,"ariaHidden":78},[77],[47,1548,1550,1553],{"className":1549},[82],[47,1551],{"className":1552,"style":121},[86],[47,1554,212],{"className":1555},[91],[47,1557,1559],{"className":1558,"translate":51},[331],[47,1560,1562,1618],{"className":1561,"translate":51},[50],[47,1563,1565],{"className":1564},[55],[57,1566,1567],{"xmlns":59,"display":340},[61,1568,1569,1615],{},[64,1570,1571,1573,1575,1577,1579,1581,1583,1585,1587,1589,1601,1603,1610,1612],{},[67,1572,350],{},[214,1574,387],{"stretchy":386},[107,1576,1127],{},[214,1578,1226],{},[67,1580,1229],{},[214,1582,1226],{},[107,1584,212],{},[214,1586,405],{"stretchy":386},[214,1588,375],{},[377,1590,1591,1599],{},[64,1592,1593,1595,1597],{},[107,1594,212],{},[214,1596,394],{},[107,1598,1127],{},[107,1600,212],{},[214,1602,375],{},[377,1604,1605,1608],{},[107,1606,1607],{},"200",[107,1609,212],{},[214,1611,375],{},[107,1613,1614],{},"0.2",[71,1616,1617],{"encoding":73},"P(800 \\leq X \\leq 1000) = \\frac{1000 - 800}{1000} = \\frac{200}{1000} = 0.2",[47,1619,1621,1645,1663,1684,1773,1850],{"className":1620,"ariaHidden":78},[77],[47,1622,1624,1627,1630,1633,1636,1639,1642],{"className":1623},[82],[47,1625],{"className":1626,"style":1264},[86],[47,1628,350],{"className":1629,"style":425},[91,92],[47,1631,387],{"className":1632},[523],[47,1634,1127],{"className":1635},[91],[47,1637],{"className":1638,"style":502},[262],[47,1640,1226],{"className":1641},[506],[47,1643],{"className":1644,"style":502},[262],[47,1646,1648,1651,1654,1657,1660],{"className":1647},[82],[47,1649],{"className":1650,"style":1289},[86],[47,1652,1229],{"className":1653,"style":1293},[91,92],[47,1655],{"className":1656,"style":502},[262],[47,1658,1226],{"className":1659},[506],[47,1661],{"className":1662,"style":502},[262],[47,1664,1666,1669,1672,1675,1678,1681],{"className":1665},[82],[47,1667],{"className":1668,"style":1264},[86],[47,1670,212],{"className":1671},[91],[47,1673,405],{"className":1674},[590],[47,1676],{"className":1677,"style":502},[262],[47,1679,375],{"className":1680},[506],[47,1682],{"className":1683,"style":502},[262],[47,1685,1687,1690,1764,1767,1770],{"className":1686},[82],[47,1688],{"className":1689,"style":951},[86],[47,1691,1693,1696,1761],{"className":1692},[91],[47,1694],{"className":1695},[523,524],[47,1697,1699],{"className":1698},[377],[47,1700,1702,1753],{"className":1701},[433,434],[47,1703,1705,1750],{"className":1704},[438],[47,1706,1708,1719,1727],{"className":1707,"style":537},[442],[47,1709,1710,1713],{"style":540},[47,1711],{"className":1712,"style":544},[450],[47,1714,1716],{"className":1715},[91],[47,1717,212],{"className":1718},[91],[47,1720,1721,1724],{"style":593},[47,1722],{"className":1723,"style":544},[450],[47,1725],{"className":1726,"style":601},[600],[47,1728,1729,1732],{"style":604},[47,1730],{"className":1731,"style":544},[450],[47,1733,1735,1738,1741,1744,1747],{"className":1734},[91],[47,1736,212],{"className":1737},[91],[47,1739],{"className":1740,"style":561},[262],[47,1742,394],{"className":1743},[565],[47,1745],{"className":1746,"style":561},[262],[47,1748,1127],{"className":1749},[91],[47,1751,489],{"className":1752},[488],[47,1754,1756],{"className":1755},[438],[47,1757,1759],{"className":1758,"style":1009},[442],[47,1760],{},[47,1762],{"className":1763},[590,524],[47,1765],{"className":1766,"style":502},[262],[47,1768,375],{"className":1769},[506],[47,1771],{"className":1772,"style":502},[262],[47,1774,1776,1779,1841,1844,1847],{"className":1775},[82],[47,1777],{"className":1778,"style":951},[86],[47,1780,1782,1785,1838],{"className":1781},[91],[47,1783],{"className":1784},[523,524],[47,1786,1788],{"className":1787},[377],[47,1789,1791,1830],{"className":1790},[433,434],[47,1792,1794,1827],{"className":1793},[438],[47,1795,1797,1808,1816],{"className":1796,"style":537},[442],[47,1798,1799,1802],{"style":540},[47,1800],{"className":1801,"style":544},[450],[47,1803,1805],{"className":1804},[91],[47,1806,212],{"className":1807},[91],[47,1809,1810,1813],{"style":593},[47,1811],{"className":1812,"style":544},[450],[47,1814],{"className":1815,"style":601},[600],[47,1817,1818,1821],{"style":604},[47,1819],{"className":1820,"style":544},[450],[47,1822,1824],{"className":1823},[91],[47,1825,1607],{"className":1826},[91],[47,1828,489],{"className":1829},[488],[47,1831,1833],{"className":1832},[438],[47,1834,1836],{"className":1835,"style":1009},[442],[47,1837],{},[47,1839],{"className":1840},[590,524],[47,1842],{"className":1843,"style":502},[262],[47,1845,375],{"className":1846},[506],[47,1848],{"className":1849,"style":502},[262],[47,1851,1853,1856],{"className":1852},[82],[47,1854],{"className":1855,"style":121},[86],[47,1857,1614],{"className":1858},[91],[15,1860,1861,1862,125,1890,1046,1918,230],{},"Thus, the probability of randomly selecting a treasure box with a prize between ",[47,1863,1865,1878],{"className":1864,"translate":51},[50],[47,1866,1868],{"className":1867},[55],[57,1869,1870],{"xmlns":59},[61,1871,1872,1876],{},[64,1873,1874],{},[107,1875,1127],{},[71,1877,1127],{"encoding":73},[47,1879,1881],{"className":1880,"ariaHidden":78},[77],[47,1882,1884,1887],{"className":1883},[82],[47,1885],{"className":1886,"style":121},[86],[47,1888,1127],{"className":1889},[91],[47,1891,1893,1906],{"className":1892,"translate":51},[50],[47,1894,1896],{"className":1895},[55],[57,1897,1898],{"xmlns":59},[61,1899,1900,1904],{},[64,1901,1902],{},[107,1903,212],{},[71,1905,212],{"encoding":73},[47,1907,1909],{"className":1908,"ariaHidden":78},[77],[47,1910,1912,1915],{"className":1911},[82],[47,1913],{"className":1914,"style":121},[86],[47,1916,212],{"className":1917},[91],[47,1919,1921,1938],{"className":1920,"translate":51},[50],[47,1922,1924],{"className":1923},[55],[57,1925,1926],{"xmlns":59},[61,1927,1928,1935],{},[64,1929,1930,1933],{},[107,1931,1932],{},"20",[67,1934,1064],{"mathvariant":229},[71,1936,1937],{"encoding":73},"20\\%",[47,1939,1941],{"className":1940,"ariaHidden":78},[77],[47,1942,1944,1947],{"className":1943},[82],[47,1945],{"className":1946,"style":1085},[86],[47,1948,1950],{"className":1949},[91],"20%",[15,1952,1953,1954,1984],{},"While the probability has increased significantly, ",[47,1955,1957,1972],{"className":1956,"translate":51},[50],[47,1958,1960],{"className":1959},[55],[57,1961,1962],{"xmlns":59},[61,1963,1964,1970],{},[64,1965,1966,1968],{},[107,1967,1932],{},[67,1969,1064],{"mathvariant":229},[71,1971,1937],{"encoding":73},[47,1973,1975],{"className":1974,"ariaHidden":78},[77],[47,1976,1978,1981],{"className":1977},[82],[47,1979],{"className":1980,"style":1085},[86],[47,1982,1950],{"className":1983},[91]," is still relatively low.",[15,1986,1987],{},"So, if you randomly pick a treasure box and decide to keep it, the chances of getting a high prize are still slim, which clearly isn’t the optimal strategy.",[10,1989,1991],{"id":1990},"optimization","Optimization",[15,1993,1994,1997],{},[36,1995,1996],{},"In most cases, we don’t know the maximum prize inside the treasure boxes."," Just like in the story of Socrates' students picking the largest ear of wheat, they didn’t know how big the largest ear was.",[15,1999,2000],{},"Let’s change our approach. Even though we don’t know the maximum prize, we do know that if we sample a portion of the treasure boxes, the probability of finding the jackpot in the sample is equal to its probability in the entire set. By sampling some treasure boxes, we can estimate the value of the jackpot based on the maximum prize in the sample.",[15,2002,2003,2004,2035,2036,2103,2104,2165,2166,230],{},"Let’s denote the jackpot in the entire set as ",[47,2005,2007,2021],{"className":2006,"translate":51},[50],[47,2008,2010],{"className":2009},[55],[57,2011,2012],{"xmlns":59},[61,2013,2014,2019],{},[64,2015,2016],{},[67,2017,2018],{},"M",[71,2020,2018],{"encoding":73},[47,2022,2024],{"className":2023,"ariaHidden":78},[77],[47,2025,2027,2031],{"className":2026},[82],[47,2028],{"className":2029,"style":2030},[86],"height:0.6833em;",[47,2032,2018],{"className":2033,"style":2034},[91,92],"margin-right:0.10903em;",". Since the jackpot is uniformly distributed among the treasure boxes, we can take a sample from the boxes and record the maximum prize in that sample, denoted as ",[47,2037,2039,2060],{"className":2038,"translate":51},[50],[47,2040,2042],{"className":2041},[55],[57,2043,2044],{"xmlns":59},[61,2045,2046,2057],{},[64,2047,2048],{},[2049,2050,2051,2053],"msup",{},[67,2052,2018],{},[214,2054,2056],{"mathvariant":229,"lspace":2055,"rspace":2055},"0em","′",[71,2058,2059],{"encoding":73},"M'",[47,2061,2063],{"className":2062,"ariaHidden":78},[77],[47,2064,2066,2070],{"className":2065},[82],[47,2067],{"className":2068,"style":2069},[86],"height:0.7519em;",[47,2071,2073,2076],{"className":2072},[91],[47,2074,2018],{"className":2075,"style":2034},[91,92],[47,2077,2079],{"className":2078},[429],[47,2080,2082],{"className":2081},[433],[47,2083,2085],{"className":2084},[438],[47,2086,2088],{"className":2087,"style":2069},[442],[47,2089,2091,2094],{"style":2090},"top:-3.063em;margin-right:0.05em;",[47,2092],{"className":2093,"style":451},[450],[47,2095,2097],{"className":2096},[455,456,457,458],[47,2098,2100],{"className":2099},[91,458],[47,2101,2056],{"className":2102},[91,458],". Clearly, this sample maximum ",[47,2105,2107,2124],{"className":2106,"translate":51},[50],[47,2108,2110],{"className":2109},[55],[57,2111,2112],{"xmlns":59},[61,2113,2114,2122],{},[64,2115,2116],{},[2049,2117,2118,2120],{},[67,2119,2018],{},[214,2121,2056],{"mathvariant":229,"lspace":2055,"rspace":2055},[71,2123,2059],{"encoding":73},[47,2125,2127],{"className":2126,"ariaHidden":78},[77],[47,2128,2130,2133],{"className":2129},[82],[47,2131],{"className":2132,"style":2069},[86],[47,2134,2136,2139],{"className":2135},[91],[47,2137,2018],{"className":2138,"style":2034},[91,92],[47,2140,2142],{"className":2141},[429],[47,2143,2145],{"className":2144},[433],[47,2146,2148],{"className":2147},[438],[47,2149,2151],{"className":2150,"style":2069},[442],[47,2152,2153,2156],{"style":2090},[47,2154],{"className":2155,"style":451},[450],[47,2157,2159],{"className":2158},[455,456,457,458],[47,2160,2162],{"className":2161},[91,458],[47,2163,2056],{"className":2164},[91,458]," will be close to the overall jackpot ",[47,2167,2169,2182],{"className":2168,"translate":51},[50],[47,2170,2172],{"className":2171},[55],[57,2173,2174],{"xmlns":59},[61,2175,2176,2180],{},[64,2177,2178],{},[67,2179,2018],{},[71,2181,2018],{"encoding":73},[47,2183,2185],{"className":2184,"ariaHidden":78},[77],[47,2186,2188,2191],{"className":2187},[82],[47,2189],{"className":2190,"style":2030},[86],[47,2192,2018],{"className":2193,"style":2034},[91,92],[15,2195,2196,2197,2225,2226,2287,2288,230],{},"Sampling a portion of the boxes to estimate the jackpot ",[47,2198,2200,2213],{"className":2199,"translate":51},[50],[47,2201,2203],{"className":2202},[55],[57,2204,2205],{"xmlns":59},[61,2206,2207,2211],{},[64,2208,2209],{},[67,2210,2018],{},[71,2212,2018],{"encoding":73},[47,2214,2216],{"className":2215,"ariaHidden":78},[77],[47,2217,2219,2222],{"className":2218},[82],[47,2220],{"className":2221,"style":2030},[86],[47,2223,2018],{"className":2224,"style":2034},[91,92]," is feasible, and the more samples we take, the closer ",[47,2227,2229,2246],{"className":2228,"translate":51},[50],[47,2230,2232],{"className":2231},[55],[57,2233,2234],{"xmlns":59},[61,2235,2236,2244],{},[64,2237,2238],{},[2049,2239,2240,2242],{},[67,2241,2018],{},[214,2243,2056],{"mathvariant":229,"lspace":2055,"rspace":2055},[71,2245,2059],{"encoding":73},[47,2247,2249],{"className":2248,"ariaHidden":78},[77],[47,2250,2252,2255],{"className":2251},[82],[47,2253],{"className":2254,"style":2069},[86],[47,2256,2258,2261],{"className":2257},[91],[47,2259,2018],{"className":2260,"style":2034},[91,92],[47,2262,2264],{"className":2263},[429],[47,2265,2267],{"className":2266},[433],[47,2268,2270],{"className":2269},[438],[47,2271,2273],{"className":2272,"style":2069},[442],[47,2274,2275,2278],{"style":2090},[47,2276],{"className":2277,"style":451},[450],[47,2279,2281],{"className":2280},[455,456,457,458],[47,2282,2284],{"className":2283},[91,458],[47,2285,2056],{"className":2286},[91,458]," will be to ",[47,2289,2291,2304],{"className":2290,"translate":51},[50],[47,2292,2294],{"className":2293},[55],[57,2295,2296],{"xmlns":59},[61,2297,2298,2302],{},[64,2299,2300],{},[67,2301,2018],{},[71,2303,2018],{"encoding":73},[47,2305,2307],{"className":2306,"ariaHidden":78},[77],[47,2308,2310,2313],{"className":2309},[82],[47,2311],{"className":2312,"style":2030},[86],[47,2314,2018],{"className":2315,"style":2034},[91,92],[15,2317,2318,2319,125,2380,2408,2409,2437,2438,2466,2467,2528,2529,2557],{},"However, the sample size cannot be too small, or the difference between ",[47,2320,2322,2339],{"className":2321,"translate":51},[50],[47,2323,2325],{"className":2324},[55],[57,2326,2327],{"xmlns":59},[61,2328,2329,2337],{},[64,2330,2331],{},[2049,2332,2333,2335],{},[67,2334,2018],{},[214,2336,2056],{"mathvariant":229,"lspace":2055,"rspace":2055},[71,2338,2059],{"encoding":73},[47,2340,2342],{"className":2341,"ariaHidden":78},[77],[47,2343,2345,2348],{"className":2344},[82],[47,2346],{"className":2347,"style":2069},[86],[47,2349,2351,2354],{"className":2350},[91],[47,2352,2018],{"className":2353,"style":2034},[91,92],[47,2355,2357],{"className":2356},[429],[47,2358,2360],{"className":2359},[433],[47,2361,2363],{"className":2362},[438],[47,2364,2366],{"className":2365,"style":2069},[442],[47,2367,2368,2371],{"style":2090},[47,2369],{"className":2370,"style":451},[450],[47,2372,2374],{"className":2373},[455,456,457,458],[47,2375,2377],{"className":2376},[91,458],[47,2378,2056],{"className":2379},[91,458],[47,2381,2383,2396],{"className":2382,"translate":51},[50],[47,2384,2386],{"className":2385},[55],[57,2387,2388],{"xmlns":59},[61,2389,2390,2394],{},[64,2391,2392],{},[67,2393,2018],{},[71,2395,2018],{"encoding":73},[47,2397,2399],{"className":2398,"ariaHidden":78},[77],[47,2400,2402,2405],{"className":2401},[82],[47,2403],{"className":2404,"style":2030},[86],[47,2406,2018],{"className":2407,"style":2034},[91,92]," will be large, making the estimate ineffective. On the other hand, the sample size cannot be too large either, because there will be fewer treasure boxes left to choose from, reducing the chance of finding the jackpot. Consider an extreme example: if we sample ",[47,2410,2412,2425],{"className":2411,"translate":51},[50],[47,2413,2415],{"className":2414},[55],[57,2416,2417],{"xmlns":59},[61,2418,2419,2423],{},[64,2420,2421],{},[107,2422,219],{},[71,2424,219],{"encoding":73},[47,2426,2428],{"className":2427,"ariaHidden":78},[77],[47,2429,2431,2434],{"className":2430},[82],[47,2432],{"className":2433,"style":121},[86],[47,2435,219],{"className":2436},[91]," treasure boxes out of ",[47,2439,2441,2454],{"className":2440,"translate":51},[50],[47,2442,2444],{"className":2443},[55],[57,2445,2446],{"xmlns":59},[61,2447,2448,2452],{},[64,2449,2450],{},[107,2451,212],{},[71,2453,212],{"encoding":73},[47,2455,2457],{"className":2456,"ariaHidden":78},[77],[47,2458,2460,2463],{"className":2459},[82],[47,2461],{"className":2462,"style":121},[86],[47,2464,212],{"className":2465},[91],", ",[47,2468,2470,2487],{"className":2469,"translate":51},[50],[47,2471,2473],{"className":2472},[55],[57,2474,2475],{"xmlns":59},[61,2476,2477,2485],{},[64,2478,2479],{},[2049,2480,2481,2483],{},[67,2482,2018],{},[214,2484,2056],{"mathvariant":229,"lspace":2055,"rspace":2055},[71,2486,2059],{"encoding":73},[47,2488,2490],{"className":2489,"ariaHidden":78},[77],[47,2491,2493,2496],{"className":2492},[82],[47,2494],{"className":2495,"style":2069},[86],[47,2497,2499,2502],{"className":2498},[91],[47,2500,2018],{"className":2501,"style":2034},[91,92],[47,2503,2505],{"className":2504},[429],[47,2506,2508],{"className":2507},[433],[47,2509,2511],{"className":2510},[438],[47,2512,2514],{"className":2513,"style":2069},[442],[47,2515,2516,2519],{"style":2090},[47,2517],{"className":2518,"style":451},[450],[47,2520,2522],{"className":2521},[455,456,457,458],[47,2523,2525],{"className":2524},[91,458],[47,2526,2056],{"className":2527},[91,458]," will likely be almost equal to ",[47,2530,2532,2545],{"className":2531,"translate":51},[50],[47,2533,2535],{"className":2534},[55],[57,2536,2537],{"xmlns":59},[61,2538,2539,2543],{},[64,2540,2541],{},[67,2542,2018],{},[71,2544,2018],{"encoding":73},[47,2546,2548],{"className":2547,"ariaHidden":78},[77],[47,2549,2551,2554],{"className":2550},[82],[47,2552],{"className":2553,"style":2030},[86],[47,2555,2018],{"className":2556,"style":2034},[91,92],", but with only one treasure box remaining, the chance of hitting the jackpot becomes negligible.",[15,2559,2560,2561,2622,2623,2651],{},"Thus, it is important to balance the sample size with the total number of treasure boxes. Is there an optimal sample size where ",[47,2562,2564,2581],{"className":2563,"translate":51},[50],[47,2565,2567],{"className":2566},[55],[57,2568,2569],{"xmlns":59},[61,2570,2571,2579],{},[64,2572,2573],{},[2049,2574,2575,2577],{},[67,2576,2018],{},[214,2578,2056],{"mathvariant":229,"lspace":2055,"rspace":2055},[71,2580,2059],{"encoding":73},[47,2582,2584],{"className":2583,"ariaHidden":78},[77],[47,2585,2587,2590],{"className":2586},[82],[47,2588],{"className":2589,"style":2069},[86],[47,2591,2593,2596],{"className":2592},[91],[47,2594,2018],{"className":2595,"style":2034},[91,92],[47,2597,2599],{"className":2598},[429],[47,2600,2602],{"className":2601},[433],[47,2603,2605],{"className":2604},[438],[47,2606,2608],{"className":2607,"style":2069},[442],[47,2609,2610,2613],{"style":2090},[47,2611],{"className":2612,"style":451},[450],[47,2614,2616],{"className":2615},[455,456,457,458],[47,2617,2619],{"className":2618},[91,458],[47,2620,2056],{"className":2621},[91,458]," is close to ",[47,2624,2626,2639],{"className":2625,"translate":51},[50],[47,2627,2629],{"className":2628},[55],[57,2630,2631],{"xmlns":59},[61,2632,2633,2637],{},[64,2634,2635],{},[67,2636,2018],{},[71,2638,2018],{"encoding":73},[47,2640,2642],{"className":2641,"ariaHidden":78},[77],[47,2643,2645,2648],{"className":2644},[82],[47,2646],{"className":2647,"style":2030},[86],[47,2649,2018],{"className":2650,"style":2034},[91,92]," and the probability of winning a high prize is maximized afterward?",[15,2653,2654,2655,2683,2684,2714,2715,230],{},"Next, we will use mathematical methods to derive this “balance point.” Suppose the total number of treasure boxes is ",[47,2656,2658,2671],{"className":2657,"translate":51},[50],[47,2659,2661],{"className":2660},[55],[57,2662,2663],{"xmlns":59},[61,2664,2665,2669],{},[64,2666,2667],{},[67,2668,69],{},[71,2670,69],{"encoding":73},[47,2672,2674],{"className":2673,"ariaHidden":78},[77],[47,2675,2677,2680],{"className":2676},[82],[47,2678],{"className":2679,"style":87},[86],[47,2681,69],{"className":2682},[91,92],", and the sample size is ",[47,2685,2687,2701],{"className":2686,"translate":51},[50],[47,2688,2690],{"className":2689},[55],[57,2691,2692],{"xmlns":59},[61,2693,2694,2699],{},[64,2695,2696],{},[67,2697,2698],{},"r",[71,2700,2698],{"encoding":73},[47,2702,2704],{"className":2703,"ariaHidden":78},[77],[47,2705,2707,2710],{"className":2706},[82],[47,2708],{"className":2709,"style":87},[86],[47,2711,2698],{"className":2712,"style":2713},[91,92],"margin-right:0.02778em;",". We will find the expression for the optimal ",[47,2716,2718,2731],{"className":2717,"translate":51},[50],[47,2719,2721],{"className":2720},[55],[57,2722,2723],{"xmlns":59},[61,2724,2725,2729],{},[64,2726,2727],{},[67,2728,2698],{},[71,2730,2698],{"encoding":73},[47,2732,2734],{"className":2733,"ariaHidden":78},[77],[47,2735,2737,2740],{"className":2736},[82],[47,2738],{"className":2739,"style":87},[86],[47,2741,2698],{"className":2742,"style":2713},[91,92],[10,2744,2746],{"id":2745},"mathematical-derivation","Mathematical Derivation",[15,2748,2749,2750,2779,2780,2808,2809,2837],{},"Assume we have a total of ",[47,2751,2753,2767],{"className":2752,"translate":51},[50],[47,2754,2756],{"className":2755},[55],[57,2757,2758],{"xmlns":59},[61,2759,2760,2765],{},[64,2761,2762],{},[67,2763,2764],{},"N",[71,2766,2764],{"encoding":73},[47,2768,2770],{"className":2769,"ariaHidden":78},[77],[47,2771,2773,2776],{"className":2772},[82],[47,2774],{"className":2775,"style":2030},[86],[47,2777,2764],{"className":2778,"style":2034},[91,92]," treasure boxes, and we select the first ",[47,2781,2783,2796],{"className":2782,"translate":51},[50],[47,2784,2786],{"className":2785},[55],[57,2787,2788],{"xmlns":59},[61,2789,2790,2794],{},[64,2791,2792],{},[67,2793,2698],{},[71,2795,2698],{"encoding":73},[47,2797,2799],{"className":2798,"ariaHidden":78},[77],[47,2800,2802,2805],{"className":2801},[82],[47,2803],{"className":2804,"style":87},[86],[47,2806,2698],{"className":2807,"style":2713},[91,92]," boxes as a sample. In this scenario, we aim to find the optimal sample size ",[47,2810,2812,2825],{"className":2811,"translate":51},[50],[47,2813,2815],{"className":2814},[55],[57,2816,2817],{"xmlns":59},[61,2818,2819,2823],{},[64,2820,2821],{},[67,2822,2698],{},[71,2824,2698],{"encoding":73},[47,2826,2828],{"className":2827,"ariaHidden":78},[77],[47,2829,2831,2834],{"className":2830},[82],[47,2832],{"className":2833,"style":87},[86],[47,2835,2698],{"className":2836,"style":2713},[91,92]," that maximizes the probability of selecting the highest prize after discarding the sample.",[2839,2840,2842],"h3",{"id":2841},"probability-derivation","Probability Derivation",[15,2844,2845,2846,2874,2875,2904,2905,2957,2958,2986,2987,230],{},"For a fixed ",[47,2847,2849,2862],{"className":2848,"translate":51},[50],[47,2850,2852],{"className":2851},[55],[57,2853,2854],{"xmlns":59},[61,2855,2856,2860],{},[64,2857,2858],{},[67,2859,2698],{},[71,2861,2698],{"encoding":73},[47,2863,2865],{"className":2864,"ariaHidden":78},[77],[47,2866,2868,2871],{"className":2867},[82],[47,2869],{"className":2870,"style":87},[86],[47,2872,2698],{"className":2873,"style":2713},[91,92],", suppose the highest prize appears in the ",[47,2876,2878,2891],{"className":2877,"translate":51},[50],[47,2879,2881],{"className":2880},[55],[57,2882,2883],{"xmlns":59},[61,2884,2885,2889],{},[64,2886,2887],{},[67,2888,364],{},[71,2890,364],{"encoding":73},[47,2892,2894],{"className":2893,"ariaHidden":78},[77],[47,2895,2897,2901],{"className":2896},[82],[47,2898],{"className":2899,"style":2900},[86],"height:0.6944em;",[47,2902,364],{"className":2903,"style":475},[91,92],"-th treasure box. To win the highest prize after discarding the sample, it is necessary that the second-highest prize appears in the first ",[47,2906,2908,2926],{"className":2907,"translate":51},[50],[47,2909,2911],{"className":2910},[55],[57,2912,2913],{"xmlns":59},[61,2914,2915,2923],{},[64,2916,2917,2919,2921],{},[67,2918,364],{},[214,2920,394],{},[107,2922,381],{},[71,2924,2925],{"encoding":73},"k-1",[47,2927,2929,2948],{"className":2928,"ariaHidden":78},[77],[47,2930,2932,2936,2939,2942,2945],{"className":2931},[82],[47,2933],{"className":2934,"style":2935},[86],"height:0.7778em;vertical-align:-0.0833em;",[47,2937,364],{"className":2938,"style":475},[91,92],[47,2940],{"className":2941,"style":561},[262],[47,2943,394],{"className":2944},[565],[47,2946],{"className":2947,"style":561},[262],[47,2949,2951,2954],{"className":2950},[82],[47,2952],{"className":2953,"style":121},[86],[47,2955,381],{"className":2956},[91]," boxes, and specifically, this second-highest prize must also appear within the sample (the first ",[47,2959,2961,2974],{"className":2960,"translate":51},[50],[47,2962,2964],{"className":2963},[55],[57,2965,2966],{"xmlns":59},[61,2967,2968,2972],{},[64,2969,2970],{},[67,2971,2698],{},[71,2973,2698],{"encoding":73},[47,2975,2977],{"className":2976,"ariaHidden":78},[77],[47,2978,2980,2983],{"className":2979},[82],[47,2981],{"className":2982,"style":87},[86],[47,2984,2698],{"className":2985,"style":2713},[91,92]," boxes). The probability of this happening is ",[47,2988,2990,3014],{"className":2989,"translate":51},[50],[47,2991,2993],{"className":2992},[55],[57,2994,2995],{"xmlns":59},[61,2996,2997,3011],{},[64,2998,2999],{},[377,3000,3001,3003],{},[67,3002,2698],{},[64,3004,3005,3007,3009],{},[67,3006,364],{},[214,3008,394],{},[107,3010,381],{},[71,3012,3013],{"encoding":73},"\\frac{r}{k-1}",[47,3015,3017],{"className":3016,"ariaHidden":78},[77],[47,3018,3020,3024],{"className":3019},[82],[47,3021],{"className":3022,"style":3023},[86],"height:1.0987em;vertical-align:-0.4033em;",[47,3025,3027,3030,3099],{"className":3026},[91],[47,3028],{"className":3029},[523,524],[47,3031,3033],{"className":3032},[377],[47,3034,3036,3090],{"className":3035},[433,434],[47,3037,3039,3087],{"className":3038},[438],[47,3040,3043,3064,3072],{"className":3041,"style":3042},[442],"height:0.6954em;",[47,3044,3046,3049],{"style":3045},"top:-2.655em;",[47,3047],{"className":3048,"style":544},[450],[47,3050,3052],{"className":3051},[455,456,457,458],[47,3053,3055,3058,3061],{"className":3054},[91,458],[47,3056,364],{"className":3057,"style":475},[91,92,458],[47,3059,394],{"className":3060},[565,458],[47,3062,381],{"className":3063},[91,458],[47,3065,3066,3069],{"style":593},[47,3067],{"className":3068,"style":544},[450],[47,3070],{"className":3071,"style":601},[600],[47,3073,3075,3078],{"style":3074},"top:-3.394em;",[47,3076],{"className":3077,"style":544},[450],[47,3079,3081],{"className":3080},[455,456,457,458],[47,3082,3084],{"className":3083},[91,458],[47,3085,2698],{"className":3086,"style":2713},[91,92,458],[47,3088,489],{"className":3089},[488],[47,3091,3093],{"className":3092},[438],[47,3094,3097],{"className":3095,"style":3096},[442],"height:0.4033em;",[47,3098],{},[47,3100],{"className":3101},[590,524],[15,3103,3104],{},"This can be visualized with a diagram, where the red arrow indicates the positions where the second-highest prize occurs:",[15,3106,3107],{},[3108,3109],"img",{"alt":3110,"src":3111},"","https:\u002F\u002Fimage-assets.dreams.plus\u002F20240922115506.webp",[15,3113,3114],{},"Thus, we derive the following equation:",[47,3116,3118],{"className":3117,"translate":51},[331],[47,3119,3121,3213],{"className":3120,"translate":51},[50],[47,3122,3124],{"className":3123},[55],[57,3125,3126],{"xmlns":59,"display":340},[61,3127,3128,3210],{},[64,3129,3130,3132,3134,3140,3160,3172,3174,3180,3198],{},[67,3131,350],{},[214,3133,375],{},[377,3135,3136,3138],{},[107,3137,381],{},[67,3139,2764],{},[3141,3142,3143,3146,3158],"munderover",{},[214,3144,3145],{},"∑",[64,3147,3148,3150,3152,3154,3156],{},[67,3149,364],{},[214,3151,375],{},[67,3153,2698],{},[214,3155,400],{},[107,3157,381],{},[67,3159,2764],{},[377,3161,3162,3164],{},[67,3163,2698],{},[64,3165,3166,3168,3170],{},[67,3167,364],{},[214,3169,394],{},[107,3171,381],{},[214,3173,375],{},[377,3175,3176,3178],{},[67,3177,2698],{},[67,3179,2764],{},[3141,3181,3182,3184,3196],{},[214,3183,3145],{},[64,3185,3186,3188,3190,3192,3194],{},[67,3187,364],{},[214,3189,375],{},[67,3191,2698],{},[214,3193,400],{},[107,3195,381],{},[67,3197,2764],{},[377,3199,3200,3202],{},[107,3201,381],{},[64,3203,3204,3206,3208],{},[67,3205,364],{},[214,3207,394],{},[107,3209,381],{},[71,3211,3212],{"encoding":73},"P = \\frac{1}{N} \\sum_{k=r+1}^{N} \\frac{r}{k-1} = \\frac{r}{N} \\sum_{k=r+1}^{N} \\frac{1}{k-1}",[47,3214,3216,3234,3477],{"className":3215,"ariaHidden":78},[77],[47,3217,3219,3222,3225,3228,3231],{"className":3218},[82],[47,3220],{"className":3221,"style":2030},[86],[47,3223,350],{"className":3224,"style":425},[91,92],[47,3226],{"className":3227,"style":502},[262],[47,3229,375],{"className":3230},[506],[47,3232],{"className":3233,"style":502},[262],[47,3235,3237,3241,3303,3306,3389,3392,3468,3471,3474],{"className":3236},[82],[47,3238],{"className":3239,"style":3240},[86],"height:3.1888em;vertical-align:-1.3604em;",[47,3242,3244,3247,3300],{"className":3243},[91],[47,3245],{"className":3246},[523,524],[47,3248,3250],{"className":3249},[377],[47,3251,3253,3292],{"className":3252},[433,434],[47,3254,3256,3289],{"className":3255},[438],[47,3257,3259,3270,3278],{"className":3258,"style":537},[442],[47,3260,3261,3264],{"style":540},[47,3262],{"className":3263,"style":544},[450],[47,3265,3267],{"className":3266},[91],[47,3268,2764],{"className":3269,"style":2034},[91,92],[47,3271,3272,3275],{"style":593},[47,3273],{"className":3274,"style":544},[450],[47,3276],{"className":3277,"style":601},[600],[47,3279,3280,3283],{"style":604},[47,3281],{"className":3282,"style":544},[450],[47,3284,3286],{"className":3285},[91],[47,3287,381],{"className":3288},[91],[47,3290,489],{"className":3291},[488],[47,3293,3295],{"className":3294},[438],[47,3296,3298],{"className":3297,"style":1009},[442],[47,3299],{},[47,3301],{"className":3302},[590,524],[47,3304],{"className":3305,"style":263},[262],[47,3307,3311],{"className":3308},[3309,3310],"mop","op-limits",[47,3312,3314,3380],{"className":3313},[433,434],[47,3315,3317,3377],{"className":3316},[438],[47,3318,3321,3349,3362],{"className":3319,"style":3320},[442],"height:1.8283em;",[47,3322,3324,3328],{"style":3323},"top:-1.8479em;margin-left:0em;",[47,3325],{"className":3326,"style":3327},[450],"height:3.05em;",[47,3329,3331],{"className":3330},[455,456,457,458],[47,3332,3334,3337,3340,3343,3346],{"className":3333},[91,458],[47,3335,364],{"className":3336,"style":475},[91,92,458],[47,3338,375],{"className":3339},[506,458],[47,3341,2698],{"className":3342,"style":2713},[91,92,458],[47,3344,400],{"className":3345},[565,458],[47,3347,381],{"className":3348},[91,458],[47,3350,3352,3355],{"style":3351},"top:-3.05em;",[47,3353],{"className":3354,"style":3327},[450],[47,3356,3357],{},[47,3358,3145],{"className":3359},[3309,3360,3361],"op-symbol","large-op",[47,3363,3365,3368],{"style":3364},"top:-4.3em;margin-left:0em;",[47,3366],{"className":3367,"style":3327},[450],[47,3369,3371],{"className":3370},[455,456,457,458],[47,3372,3374],{"className":3373},[91,458],[47,3375,2764],{"className":3376,"style":2034},[91,92,458],[47,3378,489],{"className":3379},[488],[47,3381,3383],{"className":3382},[438],[47,3384,3387],{"className":3385,"style":3386},[442],"height:1.3604em;",[47,3388],{},[47,3390],{"className":3391,"style":263},[262],[47,3393,3395,3398,3465],{"className":3394},[91],[47,3396],{"className":3397},[523,524],[47,3399,3401],{"className":3400},[377],[47,3402,3404,3456],{"className":3403},[433,434],[47,3405,3407,3453],{"className":3406},[438],[47,3408,3411,3434,3442],{"className":3409,"style":3410},[442],"height:1.1076em;",[47,3412,3413,3416],{"style":540},[47,3414],{"className":3415,"style":544},[450],[47,3417,3419,3422,3425,3428,3431],{"className":3418},[91],[47,3420,364],{"className":3421,"style":475},[91,92],[47,3423],{"className":3424,"style":561},[262],[47,3426,394],{"className":3427},[565],[47,3429],{"className":3430,"style":561},[262],[47,3432,381],{"className":3433},[91],[47,3435,3436,3439],{"style":593},[47,3437],{"className":3438,"style":544},[450],[47,3440],{"className":3441,"style":601},[600],[47,3443,3444,3447],{"style":604},[47,3445],{"className":3446,"style":544},[450],[47,3448,3450],{"className":3449},[91],[47,3451,2698],{"className":3452,"style":2713},[91,92],[47,3454,489],{"className":3455},[488],[47,3457,3459],{"className":3458},[438],[47,3460,3463],{"className":3461,"style":3462},[442],"height:0.7693em;",[47,3464],{},[47,3466],{"className":3467},[590,524],[47,3469],{"className":3470,"style":502},[262],[47,3472,375],{"className":3473},[506],[47,3475],{"className":3476,"style":502},[262],[47,3478,3480,3483,3545,3548,3621,3624],{"className":3479},[82],[47,3481],{"className":3482,"style":3240},[86],[47,3484,3486,3489,3542],{"className":3485},[91],[47,3487],{"className":3488},[523,524],[47,3490,3492],{"className":3491},[377],[47,3493,3495,3534],{"className":3494},[433,434],[47,3496,3498,3531],{"className":3497},[438],[47,3499,3501,3512,3520],{"className":3500,"style":3410},[442],[47,3502,3503,3506],{"style":540},[47,3504],{"className":3505,"style":544},[450],[47,3507,3509],{"className":3508},[91],[47,3510,2764],{"className":3511,"style":2034},[91,92],[47,3513,3514,3517],{"style":593},[47,3515],{"className":3516,"style":544},[450],[47,3518],{"className":3519,"style":601},[600],[47,3521,3522,3525],{"style":604},[47,3523],{"className":3524,"style":544},[450],[47,3526,3528],{"className":3527},[91],[47,3529,2698],{"className":3530,"style":2713},[91,92],[47,3532,489],{"className":3533},[488],[47,3535,3537],{"className":3536},[438],[47,3538,3540],{"className":3539,"style":1009},[442],[47,3541],{},[47,3543],{"className":3544},[590,524],[47,3546],{"className":3547,"style":263},[262],[47,3549,3551],{"className":3550},[3309,3310],[47,3552,3554,3613],{"className":3553},[433,434],[47,3555,3557,3610],{"className":3556},[438],[47,3558,3560,3586,3596],{"className":3559,"style":3320},[442],[47,3561,3562,3565],{"style":3323},[47,3563],{"className":3564,"style":3327},[450],[47,3566,3568],{"className":3567},[455,456,457,458],[47,3569,3571,3574,3577,3580,3583],{"className":3570},[91,458],[47,3572,364],{"className":3573,"style":475},[91,92,458],[47,3575,375],{"className":3576},[506,458],[47,3578,2698],{"className":3579,"style":2713},[91,92,458],[47,3581,400],{"className":3582},[565,458],[47,3584,381],{"className":3585},[91,458],[47,3587,3588,3591],{"style":3351},[47,3589],{"className":3590,"style":3327},[450],[47,3592,3593],{},[47,3594,3145],{"className":3595},[3309,3360,3361],[47,3597,3598,3601],{"style":3364},[47,3599],{"className":3600,"style":3327},[450],[47,3602,3604],{"className":3603},[455,456,457,458],[47,3605,3607],{"className":3606},[91,458],[47,3608,2764],{"className":3609,"style":2034},[91,92,458],[47,3611,489],{"className":3612},[488],[47,3614,3616],{"className":3615},[438],[47,3617,3619],{"className":3618,"style":3386},[442],[47,3620],{},[47,3622],{"className":3623,"style":263},[262],[47,3625,3627,3630,3695],{"className":3626},[91],[47,3628],{"className":3629},[523,524],[47,3631,3633],{"className":3632},[377],[47,3634,3636,3687],{"className":3635},[433,434],[47,3637,3639,3684],{"className":3638},[438],[47,3640,3642,3665,3673],{"className":3641,"style":537},[442],[47,3643,3644,3647],{"style":540},[47,3645],{"className":3646,"style":544},[450],[47,3648,3650,3653,3656,3659,3662],{"className":3649},[91],[47,3651,364],{"className":3652,"style":475},[91,92],[47,3654],{"className":3655,"style":561},[262],[47,3657,394],{"className":3658},[565],[47,3660],{"className":3661,"style":561},[262],[47,3663,381],{"className":3664},[91],[47,3666,3667,3670],{"style":593},[47,3668],{"className":3669,"style":544},[450],[47,3671],{"className":3672,"style":601},[600],[47,3674,3675,3678],{"style":604},[47,3676],{"className":3677,"style":544},[450],[47,3679,3681],{"className":3680},[91],[47,3682,381],{"className":3683},[91],[47,3685,489],{"className":3686},[488],[47,3688,3690],{"className":3689},[438],[47,3691,3693],{"className":3692,"style":3462},[442],[47,3694],{},[47,3696],{"className":3697},[590,524],[15,3699,3700,3701,3802,3803,3831,3832,3941,3942,3970,3971,3999],{},"In this equation, ",[47,3702,3704,3722],{"className":3703,"translate":51},[50],[47,3705,3707],{"className":3706},[55],[57,3708,3709],{"xmlns":59},[61,3710,3711,3719],{},[64,3712,3713],{},[377,3714,3715,3717],{},[107,3716,381],{},[67,3718,2764],{},[71,3720,3721],{"encoding":73},"\\frac{1}{N}",[47,3723,3725],{"className":3724,"ariaHidden":78},[77],[47,3726,3728,3732],{"className":3727},[82],[47,3729],{"className":3730,"style":3731},[86],"height:1.1901em;vertical-align:-0.345em;",[47,3733,3735,3738,3799],{"className":3734},[91],[47,3736],{"className":3737},[523,524],[47,3739,3741],{"className":3740},[377],[47,3742,3744,3790],{"className":3743},[433,434],[47,3745,3747,3787],{"className":3746},[438],[47,3748,3751,3765,3773],{"className":3749,"style":3750},[442],"height:0.8451em;",[47,3752,3753,3756],{"style":3045},[47,3754],{"className":3755,"style":544},[450],[47,3757,3759],{"className":3758},[455,456,457,458],[47,3760,3762],{"className":3761},[91,458],[47,3763,2764],{"className":3764,"style":2034},[91,92,458],[47,3766,3767,3770],{"style":593},[47,3768],{"className":3769,"style":544},[450],[47,3771],{"className":3772,"style":601},[600],[47,3774,3775,3778],{"style":3074},[47,3776],{"className":3777,"style":544},[450],[47,3779,3781],{"className":3780},[455,456,457,458],[47,3782,3784],{"className":3783},[91,458],[47,3785,381],{"className":3786},[91,458],[47,3788,489],{"className":3789},[488],[47,3791,3793],{"className":3792},[438],[47,3794,3797],{"className":3795,"style":3796},[442],"height:0.345em;",[47,3798],{},[47,3800],{"className":3801},[590,524]," represents the probability of the highest prize appearing in the ",[47,3804,3806,3819],{"className":3805,"translate":51},[50],[47,3807,3809],{"className":3808},[55],[57,3810,3811],{"xmlns":59},[61,3812,3813,3817],{},[64,3814,3815],{},[67,3816,364],{},[71,3818,364],{"encoding":73},[47,3820,3822],{"className":3821,"ariaHidden":78},[77],[47,3823,3825,3828],{"className":3824},[82],[47,3826],{"className":3827,"style":2900},[86],[47,3829,364],{"className":3830,"style":475},[91,92],"-th treasure box, and ",[47,3833,3835,3858],{"className":3834,"translate":51},[50],[47,3836,3838],{"className":3837},[55],[57,3839,3840],{"xmlns":59},[61,3841,3842,3856],{},[64,3843,3844],{},[377,3845,3846,3848],{},[67,3847,2698],{},[64,3849,3850,3852,3854],{},[67,3851,364],{},[214,3853,394],{},[107,3855,381],{},[71,3857,3013],{"encoding":73},[47,3859,3861],{"className":3860,"ariaHidden":78},[77],[47,3862,3864,3867],{"className":3863},[82],[47,3865],{"className":3866,"style":3023},[86],[47,3868,3870,3873,3938],{"className":3869},[91],[47,3871],{"className":3872},[523,524],[47,3874,3876],{"className":3875},[377],[47,3877,3879,3930],{"className":3878},[433,434],[47,3880,3882,3927],{"className":3881},[438],[47,3883,3885,3905,3913],{"className":3884,"style":3042},[442],[47,3886,3887,3890],{"style":3045},[47,3888],{"className":3889,"style":544},[450],[47,3891,3893],{"className":3892},[455,456,457,458],[47,3894,3896,3899,3902],{"className":3895},[91,458],[47,3897,364],{"className":3898,"style":475},[91,92,458],[47,3900,394],{"className":3901},[565,458],[47,3903,381],{"className":3904},[91,458],[47,3906,3907,3910],{"style":593},[47,3908],{"className":3909,"style":544},[450],[47,3911],{"className":3912,"style":601},[600],[47,3914,3915,3918],{"style":3074},[47,3916],{"className":3917,"style":544},[450],[47,3919,3921],{"className":3920},[455,456,457,458],[47,3922,3924],{"className":3923},[91,458],[47,3925,2698],{"className":3926,"style":2713},[91,92,458],[47,3928,489],{"className":3929},[488],[47,3931,3933],{"className":3932},[438],[47,3934,3936],{"className":3935,"style":3096},[442],[47,3937],{},[47,3939],{"className":3940},[590,524]," represents the probability that the second-highest prize is within the sample of the first ",[47,3943,3945,3958],{"className":3944,"translate":51},[50],[47,3946,3948],{"className":3947},[55],[57,3949,3950],{"xmlns":59},[61,3951,3952,3956],{},[64,3953,3954],{},[67,3955,2698],{},[71,3957,2698],{"encoding":73},[47,3959,3961],{"className":3960,"ariaHidden":78},[77],[47,3962,3964,3967],{"className":3963},[82],[47,3965],{"className":3966,"style":87},[86],[47,3968,2698],{"className":3969,"style":2713},[91,92]," boxes. ",[47,3972,3974,3987],{"className":3973,"translate":51},[50],[47,3975,3977],{"className":3976},[55],[57,3978,3979],{"xmlns":59},[61,3980,3981,3985],{},[64,3982,3983],{},[67,3984,350],{},[71,3986,350],{"encoding":73},[47,3988,3990],{"className":3989,"ariaHidden":78},[77],[47,3991,3993,3996],{"className":3992},[82],[47,3994],{"className":3995,"style":2030},[86],[47,3997,350],{"className":3998,"style":425},[91,92]," is the probability of winning the highest prize after discarding the sample.",[2839,4001,4003],{"id":4002},"summation-approximation","Summation Approximation",[15,4005,4006],{},"The above equation contains a harmonic series, which does not have a simple closed form but can be approximated using the following formula:",[47,4008,4010],{"className":4009,"translate":51},[331],[47,4011,4013,4064],{"className":4012,"translate":51},[50],[47,4014,4016],{"className":4015},[55],[57,4017,4018],{"xmlns":59,"display":340},[61,4019,4020,4061],{},[64,4021,4022,4036,4042,4044,4047,4050,4052,4054,4056,4058],{},[3141,4023,4024,4026,4034],{},[214,4025,3145],{},[64,4027,4028,4030,4032],{},[67,4029,364],{},[214,4031,375],{},[107,4033,381],{},[67,4035,2764],{},[377,4037,4038,4040],{},[107,4039,381],{},[67,4041,364],{},[214,4043,375],{},[67,4045,4046],{},"ln",[214,4048,4049],{},"⁡",[214,4051,387],{"stretchy":386},[67,4053,2764],{},[214,4055,405],{"stretchy":386},[214,4057,400],{},[67,4059,4060],{},"γ",[71,4062,4063],{"encoding":73},"\\sum_{k=1}^{N} \\frac{1}{k} = \\ln(N) + \\gamma",[47,4065,4067,4216,4243],{"className":4066,"ariaHidden":78},[77],[47,4068,4070,4074,4142,4145,4207,4210,4213],{"className":4069},[82],[47,4071],{"className":4072,"style":4073},[86],"height:3.1304em;vertical-align:-1.3021em;",[47,4075,4077],{"className":4076},[3309,3310],[47,4078,4080,4133],{"className":4079},[433,434],[47,4081,4083,4130],{"className":4082},[438],[47,4084,4086,4106,4116],{"className":4085,"style":3320},[442],[47,4087,4088,4091],{"style":3323},[47,4089],{"className":4090,"style":3327},[450],[47,4092,4094],{"className":4093},[455,456,457,458],[47,4095,4097,4100,4103],{"className":4096},[91,458],[47,4098,364],{"className":4099,"style":475},[91,92,458],[47,4101,375],{"className":4102},[506,458],[47,4104,381],{"className":4105},[91,458],[47,4107,4108,4111],{"style":3351},[47,4109],{"className":4110,"style":3327},[450],[47,4112,4113],{},[47,4114,3145],{"className":4115},[3309,3360,3361],[47,4117,4118,4121],{"style":3364},[47,4119],{"className":4120,"style":3327},[450],[47,4122,4124],{"className":4123},[455,456,457,458],[47,4125,4127],{"className":4126},[91,458],[47,4128,2764],{"className":4129,"style":2034},[91,92,458],[47,4131,489],{"className":4132},[488],[47,4134,4136],{"className":4135},[438],[47,4137,4140],{"className":4138,"style":4139},[442],"height:1.3021em;",[47,4141],{},[47,4143],{"className":4144,"style":263},[262],[47,4146,4148,4151,4204],{"className":4147},[91],[47,4149],{"className":4150},[523,524],[47,4152,4154],{"className":4153},[377],[47,4155,4157,4196],{"className":4156},[433,434],[47,4158,4160,4193],{"className":4159},[438],[47,4161,4163,4174,4182],{"className":4162,"style":537},[442],[47,4164,4165,4168],{"style":540},[47,4166],{"className":4167,"style":544},[450],[47,4169,4171],{"className":4170},[91],[47,4172,364],{"className":4173,"style":475},[91,92],[47,4175,4176,4179],{"style":593},[47,4177],{"className":4178,"style":544},[450],[47,4180],{"className":4181,"style":601},[600],[47,4183,4184,4187],{"style":604},[47,4185],{"className":4186,"style":544},[450],[47,4188,4190],{"className":4189},[91],[47,4191,381],{"className":4192},[91],[47,4194,489],{"className":4195},[488],[47,4197,4199],{"className":4198},[438],[47,4200,4202],{"className":4201,"style":1009},[442],[47,4203],{},[47,4205],{"className":4206},[590,524],[47,4208],{"className":4209,"style":502},[262],[47,4211,375],{"className":4212},[506],[47,4214],{"className":4215,"style":502},[262],[47,4217,4219,4222,4225,4228,4231,4234,4237,4240],{"className":4218},[82],[47,4220],{"className":4221,"style":1264},[86],[47,4223,4046],{"className":4224},[3309],[47,4226,387],{"className":4227},[523],[47,4229,2764],{"className":4230,"style":2034},[91,92],[47,4232,405],{"className":4233},[590],[47,4235],{"className":4236,"style":561},[262],[47,4238,400],{"className":4239},[565],[47,4241],{"className":4242,"style":561},[262],[47,4244,4246,4250],{"className":4245},[82],[47,4247],{"className":4248,"style":4249},[86],"height:0.625em;vertical-align:-0.1944em;",[47,4251,4060],{"className":4252,"style":4253},[91,92],"margin-right:0.05556em;",[15,4255,4256,4257,4303,4304,4333,4334,1441,4386,4414],{},"where ",[47,4258,4260,4282],{"className":4259,"translate":51},[50],[47,4261,4263],{"className":4262},[55],[57,4264,4265],{"xmlns":59},[61,4266,4267,4279],{},[64,4268,4269,4271,4273,4275,4277],{},[67,4270,4046],{},[214,4272,4049],{},[214,4274,387],{"stretchy":386},[67,4276,2764],{},[214,4278,405],{"stretchy":386},[71,4280,4281],{"encoding":73},"\\ln(N)",[47,4283,4285],{"className":4284,"ariaHidden":78},[77],[47,4286,4288,4291,4294,4297,4300],{"className":4287},[82],[47,4289],{"className":4290,"style":1264},[86],[47,4292,4046],{"className":4293},[3309],[47,4295,387],{"className":4296},[523],[47,4298,2764],{"className":4299,"style":2034},[91,92],[47,4301,405],{"className":4302},[590]," is the natural logarithm and ",[47,4305,4307,4321],{"className":4306,"translate":51},[50],[47,4308,4310],{"className":4309},[55],[57,4311,4312],{"xmlns":59},[61,4313,4314,4318],{},[64,4315,4316],{},[67,4317,4060],{},[71,4319,4320],{"encoding":73},"\\gamma",[47,4322,4324],{"className":4323,"ariaHidden":78},[77],[47,4325,4327,4330],{"className":4326},[82],[47,4328],{"className":4329,"style":4249},[86],[47,4331,4060],{"className":4332,"style":4253},[91,92]," is Euler's constant. Here, we want to approximate the summation from ",[47,4335,4337,4355],{"className":4336,"translate":51},[50],[47,4338,4340],{"className":4339},[55],[57,4341,4342],{"xmlns":59},[61,4343,4344,4352],{},[64,4345,4346,4348,4350],{},[67,4347,2698],{},[214,4349,400],{},[107,4351,381],{},[71,4353,4354],{"encoding":73},"r+1",[47,4356,4358,4377],{"className":4357,"ariaHidden":78},[77],[47,4359,4361,4365,4368,4371,4374],{"className":4360},[82],[47,4362],{"className":4363,"style":4364},[86],"height:0.6667em;vertical-align:-0.0833em;",[47,4366,2698],{"className":4367,"style":2713},[91,92],[47,4369],{"className":4370,"style":561},[262],[47,4372,400],{"className":4373},[565],[47,4375],{"className":4376,"style":561},[262],[47,4378,4380,4383],{"className":4379},[82],[47,4381],{"className":4382,"style":121},[86],[47,4384,381],{"className":4385},[91],[47,4387,4389,4402],{"className":4388,"translate":51},[50],[47,4390,4392],{"className":4391},[55],[57,4393,4394],{"xmlns":59},[61,4395,4396,4400],{},[64,4397,4398],{},[67,4399,2764],{},[71,4401,2764],{"encoding":73},[47,4403,4405],{"className":4404,"ariaHidden":78},[77],[47,4406,4408,4411],{"className":4407},[82],[47,4409],{"className":4410,"style":2030},[86],[47,4412,2764],{"className":4413,"style":2034},[91,92],". To do this, we rewrite the sum as:",[47,4416,4418],{"className":4417,"translate":51},[331],[47,4419,4421,4531],{"className":4420,"translate":51},[50],[47,4422,4424],{"className":4423},[55],[57,4425,4426],{"xmlns":59,"display":340},[61,4427,4428,4528],{},[64,4429,4430,4448,4460,4462,4494,4500,4502,4522],{},[3141,4431,4432,4434,4446],{},[214,4433,3145],{},[64,4435,4436,4438,4440,4442,4444],{},[67,4437,364],{},[214,4439,375],{},[67,4441,2698],{},[214,4443,400],{},[107,4445,381],{},[67,4447,2764],{},[377,4449,4450,4452],{},[107,4451,381],{},[64,4453,4454,4456,4458],{},[67,4455,364],{},[214,4457,394],{},[107,4459,381],{},[214,4461,375],{},[3141,4463,4464,4466,4474],{},[214,4465,3145],{},[64,4467,4468,4470,4472],{},[67,4469,364],{},[214,4471,375],{},[107,4473,381],{},[64,4475,4476,4478,4480,4482,4484,4486,4488,4490,4492],{},[67,4477,2764],{},[214,4479,394],{},[214,4481,387],{"stretchy":386},[67,4483,2698],{},[214,4485,400],{},[107,4487,381],{},[214,4489,405],{"stretchy":386},[214,4491,400],{},[107,4493,381],{},[377,4495,4496,4498],{},[107,4497,381],{},[67,4499,364],{},[214,4501,375],{},[3141,4503,4504,4506,4514],{},[214,4505,3145],{},[64,4507,4508,4510,4512],{},[67,4509,364],{},[214,4511,375],{},[107,4513,381],{},[64,4515,4516,4518,4520],{},[67,4517,2764],{},[214,4519,394],{},[67,4521,2698],{},[377,4523,4524,4526],{},[107,4525,381],{},[67,4527,364],{},[71,4529,4530],{"encoding":73},"\\sum_{k=r+1}^{N} \\frac{1}{k-1} = \\sum_{k=1}^{N-(r+1)+1} \\frac{1}{k} = \\sum_{k=1}^{N-r} \\frac{1}{k}",[47,4532,4534,4699,4873],{"className":4533,"ariaHidden":78},[77],[47,4535,4537,4540,4613,4616,4690,4693,4696],{"className":4536},[82],[47,4538],{"className":4539,"style":3240},[86],[47,4541,4543],{"className":4542},[3309,3310],[47,4544,4546,4605],{"className":4545},[433,434],[47,4547,4549,4602],{"className":4548},[438],[47,4550,4552,4578,4588],{"className":4551,"style":3320},[442],[47,4553,4554,4557],{"style":3323},[47,4555],{"className":4556,"style":3327},[450],[47,4558,4560],{"className":4559},[455,456,457,458],[47,4561,4563,4566,4569,4572,4575],{"className":4562},[91,458],[47,4564,364],{"className":4565,"style":475},[91,92,458],[47,4567,375],{"className":4568},[506,458],[47,4570,2698],{"className":4571,"style":2713},[91,92,458],[47,4573,400],{"className":4574},[565,458],[47,4576,381],{"className":4577},[91,458],[47,4579,4580,4583],{"style":3351},[47,4581],{"className":4582,"style":3327},[450],[47,4584,4585],{},[47,4586,3145],{"className":4587},[3309,3360,3361],[47,4589,4590,4593],{"style":3364},[47,4591],{"className":4592,"style":3327},[450],[47,4594,4596],{"className":4595},[455,456,457,458],[47,4597,4599],{"className":4598},[91,458],[47,4600,2764],{"className":4601,"style":2034},[91,92,458],[47,4603,489],{"className":4604},[488],[47,4606,4608],{"className":4607},[438],[47,4609,4611],{"className":4610,"style":3386},[442],[47,4612],{},[47,4614],{"className":4615,"style":263},[262],[47,4617,4619,4622,4687],{"className":4618},[91],[47,4620],{"className":4621},[523,524],[47,4623,4625],{"className":4624},[377],[47,4626,4628,4679],{"className":4627},[433,434],[47,4629,4631,4676],{"className":4630},[438],[47,4632,4634,4657,4665],{"className":4633,"style":537},[442],[47,4635,4636,4639],{"style":540},[47,4637],{"className":4638,"style":544},[450],[47,4640,4642,4645,4648,4651,4654],{"className":4641},[91],[47,4643,364],{"className":4644,"style":475},[91,92],[47,4646],{"className":4647,"style":561},[262],[47,4649,394],{"className":4650},[565],[47,4652],{"className":4653,"style":561},[262],[47,4655,381],{"className":4656},[91],[47,4658,4659,4662],{"style":593},[47,4660],{"className":4661,"style":544},[450],[47,4663],{"className":4664,"style":601},[600],[47,4666,4667,4670],{"style":604},[47,4668],{"className":4669,"style":544},[450],[47,4671,4673],{"className":4672},[91],[47,4674,381],{"className":4675},[91],[47,4677,489],{"className":4678},[488],[47,4680,4682],{"className":4681},[438],[47,4683,4685],{"className":4684,"style":3462},[442],[47,4686],{},[47,4688],{"className":4689},[590,524],[47,4691],{"className":4692,"style":502},[262],[47,4694,375],{"className":4695},[506],[47,4697],{"className":4698,"style":502},[262],[47,4700,4702,4706,4799,4802,4864,4867,4870],{"className":4701},[82],[47,4703],{"className":4704,"style":4705},[86],"height:3.2631em;vertical-align:-1.3021em;",[47,4707,4709],{"className":4708},[3309,3310],[47,4710,4712,4791],{"className":4711},[433,434],[47,4713,4715,4788],{"className":4714},[438],[47,4716,4719,4739,4749],{"className":4717,"style":4718},[442],"height:1.961em;",[47,4720,4721,4724],{"style":3323},[47,4722],{"className":4723,"style":3327},[450],[47,4725,4727],{"className":4726},[455,456,457,458],[47,4728,4730,4733,4736],{"className":4729},[91,458],[47,4731,364],{"className":4732,"style":475},[91,92,458],[47,4734,375],{"className":4735},[506,458],[47,4737,381],{"className":4738},[91,458],[47,4740,4741,4744],{"style":3351},[47,4742],{"className":4743,"style":3327},[450],[47,4745,4746],{},[47,4747,3145],{"className":4748},[3309,3360,3361],[47,4750,4752,4755],{"style":4751},"top:-4.386em;margin-left:0em;",[47,4753],{"className":4754,"style":3327},[450],[47,4756,4758],{"className":4757},[455,456,457,458],[47,4759,4761,4764,4767,4770,4773,4776,4779,4782,4785],{"className":4760},[91,458],[47,4762,2764],{"className":4763,"style":2034},[91,92,458],[47,4765,394],{"className":4766},[565,458],[47,4768,387],{"className":4769},[523,458],[47,4771,2698],{"className":4772,"style":2713},[91,92,458],[47,4774,400],{"className":4775},[565,458],[47,4777,381],{"className":4778},[91,458],[47,4780,405],{"className":4781},[590,458],[47,4783,400],{"className":4784},[565,458],[47,4786,381],{"className":4787},[91,458],[47,4789,489],{"className":4790},[488],[47,4792,4794],{"className":4793},[438],[47,4795,4797],{"className":4796,"style":4139},[442],[47,4798],{},[47,4800],{"className":4801,"style":263},[262],[47,4803,4805,4808,4861],{"className":4804},[91],[47,4806],{"className":4807},[523,524],[47,4809,4811],{"className":4810},[377],[47,4812,4814,4853],{"className":4813},[433,434],[47,4815,4817,4850],{"className":4816},[438],[47,4818,4820,4831,4839],{"className":4819,"style":537},[442],[47,4821,4822,4825],{"style":540},[47,4823],{"className":4824,"style":544},[450],[47,4826,4828],{"className":4827},[91],[47,4829,364],{"className":4830,"style":475},[91,92],[47,4832,4833,4836],{"style":593},[47,4834],{"className":4835,"style":544},[450],[47,4837],{"className":4838,"style":601},[600],[47,4840,4841,4844],{"style":604},[47,4842],{"className":4843,"style":544},[450],[47,4845,4847],{"className":4846},[91],[47,4848,381],{"className":4849},[91],[47,4851,489],{"className":4852},[488],[47,4854,4856],{"className":4855},[438],[47,4857,4859],{"className":4858,"style":1009},[442],[47,4860],{},[47,4862],{"className":4863},[590,524],[47,4865],{"className":4866,"style":502},[262],[47,4868,375],{"className":4869},[506],[47,4871],{"className":4872,"style":502},[262],[47,4874,4876,4879,4952,4955],{"className":4875},[82],[47,4877],{"className":4878,"style":4073},[86],[47,4880,4882],{"className":4881},[3309,3310],[47,4883,4885,4944],{"className":4884},[433,434],[47,4886,4888,4941],{"className":4887},[438],[47,4889,4891,4911,4921],{"className":4890,"style":3320},[442],[47,4892,4893,4896],{"style":3323},[47,4894],{"className":4895,"style":3327},[450],[47,4897,4899],{"className":4898},[455,456,457,458],[47,4900,4902,4905,4908],{"className":4901},[91,458],[47,4903,364],{"className":4904,"style":475},[91,92,458],[47,4906,375],{"className":4907},[506,458],[47,4909,381],{"className":4910},[91,458],[47,4912,4913,4916],{"style":3351},[47,4914],{"className":4915,"style":3327},[450],[47,4917,4918],{},[47,4919,3145],{"className":4920},[3309,3360,3361],[47,4922,4923,4926],{"style":3364},[47,4924],{"className":4925,"style":3327},[450],[47,4927,4929],{"className":4928},[455,456,457,458],[47,4930,4932,4935,4938],{"className":4931},[91,458],[47,4933,2764],{"className":4934,"style":2034},[91,92,458],[47,4936,394],{"className":4937},[565,458],[47,4939,2698],{"className":4940,"style":2713},[91,92,458],[47,4942,489],{"className":4943},[488],[47,4945,4947],{"className":4946},[438],[47,4948,4950],{"className":4949,"style":4139},[442],[47,4951],{},[47,4953],{"className":4954,"style":263},[262],[47,4956,4958,4961,5014],{"className":4957},[91],[47,4959],{"className":4960},[523,524],[47,4962,4964],{"className":4963},[377],[47,4965,4967,5006],{"className":4966},[433,434],[47,4968,4970,5003],{"className":4969},[438],[47,4971,4973,4984,4992],{"className":4972,"style":537},[442],[47,4974,4975,4978],{"style":540},[47,4976],{"className":4977,"style":544},[450],[47,4979,4981],{"className":4980},[91],[47,4982,364],{"className":4983,"style":475},[91,92],[47,4985,4986,4989],{"style":593},[47,4987],{"className":4988,"style":544},[450],[47,4990],{"className":4991,"style":601},[600],[47,4993,4994,4997],{"style":604},[47,4995],{"className":4996,"style":544},[450],[47,4998,5000],{"className":4999},[91],[47,5001,381],{"className":5002},[91],[47,5004,489],{"className":5005},[488],[47,5007,5009],{"className":5008},[438],[47,5010,5012],{"className":5011,"style":1009},[442],[47,5013],{},[47,5015],{"className":5016},[590,524],[15,5018,5019,5020,5071],{},"Thus, the sum starts from ",[47,5021,5023,5041],{"className":5022,"translate":51},[50],[47,5024,5026],{"className":5025},[55],[57,5027,5028],{"xmlns":59},[61,5029,5030,5038],{},[64,5031,5032,5034,5036],{},[67,5033,364],{},[214,5035,375],{},[107,5037,381],{},[71,5039,5040],{"encoding":73},"k=1",[47,5042,5044,5062],{"className":5043,"ariaHidden":78},[77],[47,5045,5047,5050,5053,5056,5059],{"className":5046},[82],[47,5048],{"className":5049,"style":2900},[86],[47,5051,364],{"className":5052,"style":475},[91,92],[47,5054],{"className":5055,"style":502},[262],[47,5057,375],{"className":5058},[506],[47,5060],{"className":5061,"style":502},[262],[47,5063,5065,5068],{"className":5064},[82],[47,5066],{"className":5067,"style":121},[86],[47,5069,381],{"className":5070},[91]," and follows the same harmonic series approximation:",[47,5073,5075],{"className":5074,"translate":51},[331],[47,5076,5078,5188],{"className":5077,"translate":51},[50],[47,5079,5081],{"className":5080},[55],[57,5082,5083],{"xmlns":59,"display":340},[61,5084,5085,5185],{},[64,5086,5087,5105,5117,5119,5139,5141,5161,5163,5165,5167,5169,5171,5173,5175,5177,5179,5181,5183],{},[3141,5088,5089,5091,5103],{},[214,5090,3145],{},[64,5092,5093,5095,5097,5099,5101],{},[67,5094,364],{},[214,5096,375],{},[67,5098,2698],{},[214,5100,400],{},[107,5102,381],{},[67,5104,2764],{},[377,5106,5107,5109],{},[107,5108,381],{},[64,5110,5111,5113,5115],{},[67,5112,364],{},[214,5114,394],{},[107,5116,381],{},[214,5118,375],{},[64,5120,5121,5123,5125,5127,5129,5131,5133,5135,5137],{},[214,5122,387],{"fence":78},[67,5124,4046],{},[214,5126,4049],{},[214,5128,387],{"stretchy":386},[67,5130,2764],{},[214,5132,405],{"stretchy":386},[214,5134,400],{},[67,5136,4060],{},[214,5138,405],{"fence":78},[214,5140,394],{},[64,5142,5143,5145,5147,5149,5151,5153,5155,5157,5159],{},[214,5144,387],{"fence":78},[67,5146,4046],{},[214,5148,4049],{},[214,5150,387],{"stretchy":386},[67,5152,2698],{},[214,5154,405],{"stretchy":386},[214,5156,400],{},[67,5158,4060],{},[214,5160,405],{"fence":78},[214,5162,375],{},[67,5164,4046],{},[214,5166,4049],{},[214,5168,387],{"stretchy":386},[67,5170,2764],{},[214,5172,405],{"stretchy":386},[214,5174,394],{},[67,5176,4046],{},[214,5178,4049],{},[214,5180,387],{"stretchy":386},[67,5182,2698],{},[214,5184,405],{"stretchy":386},[71,5186,5187],{"encoding":73},"\\sum_{k=r+1}^{N} \\frac{1}{k-1} = \\left( \\ln(N) + \\gamma \\right) - \\left( \\ln(r) + \\gamma \\right) = \\ln(N) - \\ln(r)",[47,5189,5191,5356,5407,5455,5482],{"className":5190,"ariaHidden":78},[77],[47,5192,5194,5197,5270,5273,5347,5350,5353],{"className":5193},[82],[47,5195],{"className":5196,"style":3240},[86],[47,5198,5200],{"className":5199},[3309,3310],[47,5201,5203,5262],{"className":5202},[433,434],[47,5204,5206,5259],{"className":5205},[438],[47,5207,5209,5235,5245],{"className":5208,"style":3320},[442],[47,5210,5211,5214],{"style":3323},[47,5212],{"className":5213,"style":3327},[450],[47,5215,5217],{"className":5216},[455,456,457,458],[47,5218,5220,5223,5226,5229,5232],{"className":5219},[91,458],[47,5221,364],{"className":5222,"style":475},[91,92,458],[47,5224,375],{"className":5225},[506,458],[47,5227,2698],{"className":5228,"style":2713},[91,92,458],[47,5230,400],{"className":5231},[565,458],[47,5233,381],{"className":5234},[91,458],[47,5236,5237,5240],{"style":3351},[47,5238],{"className":5239,"style":3327},[450],[47,5241,5242],{},[47,5243,3145],{"className":5244},[3309,3360,3361],[47,5246,5247,5250],{"style":3364},[47,5248],{"className":5249,"style":3327},[450],[47,5251,5253],{"className":5252},[455,456,457,458],[47,5254,5256],{"className":5255},[91,458],[47,5257,2764],{"className":5258,"style":2034},[91,92,458],[47,5260,489],{"className":5261},[488],[47,5263,5265],{"className":5264},[438],[47,5266,5268],{"className":5267,"style":3386},[442],[47,5269],{},[47,5271],{"className":5272,"style":263},[262],[47,5274,5276,5279,5344],{"className":5275},[91],[47,5277],{"className":5278},[523,524],[47,5280,5282],{"className":5281},[377],[47,5283,5285,5336],{"className":5284},[433,434],[47,5286,5288,5333],{"className":5287},[438],[47,5289,5291,5314,5322],{"className":5290,"style":537},[442],[47,5292,5293,5296],{"style":540},[47,5294],{"className":5295,"style":544},[450],[47,5297,5299,5302,5305,5308,5311],{"className":5298},[91],[47,5300,364],{"className":5301,"style":475},[91,92],[47,5303],{"className":5304,"style":561},[262],[47,5306,394],{"className":5307},[565],[47,5309],{"className":5310,"style":561},[262],[47,5312,381],{"className":5313},[91],[47,5315,5316,5319],{"style":593},[47,5317],{"className":5318,"style":544},[450],[47,5320],{"className":5321,"style":601},[600],[47,5323,5324,5327],{"style":604},[47,5325],{"className":5326,"style":544},[450],[47,5328,5330],{"className":5329},[91],[47,5331,381],{"className":5332},[91],[47,5334,489],{"className":5335},[488],[47,5337,5339],{"className":5338},[438],[47,5340,5342],{"className":5341,"style":3462},[442],[47,5343],{},[47,5345],{"className":5346},[590,524],[47,5348],{"className":5349,"style":502},[262],[47,5351,375],{"className":5352},[506],[47,5354],{"className":5355,"style":502},[262],[47,5357,5359,5362,5398,5401,5404],{"className":5358},[82],[47,5360],{"className":5361,"style":1264},[86],[47,5363,5366,5371,5374,5377,5380,5383,5386,5389,5392,5395],{"className":5364},[5365],"minner",[47,5367,387],{"className":5368,"style":5370},[523,5369],"delimcenter","top:0em;",[47,5372,4046],{"className":5373},[3309],[47,5375,387],{"className":5376},[523],[47,5378,2764],{"className":5379,"style":2034},[91,92],[47,5381,405],{"className":5382},[590],[47,5384],{"className":5385,"style":561},[262],[47,5387,400],{"className":5388},[565],[47,5390],{"className":5391,"style":561},[262],[47,5393,4060],{"className":5394,"style":4253},[91,92],[47,5396,405],{"className":5397,"style":5370},[590,5369],[47,5399],{"className":5400,"style":561},[262],[47,5402,394],{"className":5403},[565],[47,5405],{"className":5406,"style":561},[262],[47,5408,5410,5413,5446,5449,5452],{"className":5409},[82],[47,5411],{"className":5412,"style":1264},[86],[47,5414,5416,5419,5422,5425,5428,5431,5434,5437,5440,5443],{"className":5415},[5365],[47,5417,387],{"className":5418,"style":5370},[523,5369],[47,5420,4046],{"className":5421},[3309],[47,5423,387],{"className":5424},[523],[47,5426,2698],{"className":5427,"style":2713},[91,92],[47,5429,405],{"className":5430},[590],[47,5432],{"className":5433,"style":561},[262],[47,5435,400],{"className":5436},[565],[47,5438],{"className":5439,"style":561},[262],[47,5441,4060],{"className":5442,"style":4253},[91,92],[47,5444,405],{"className":5445,"style":5370},[590,5369],[47,5447],{"className":5448,"style":502},[262],[47,5450,375],{"className":5451},[506],[47,5453],{"className":5454,"style":502},[262],[47,5456,5458,5461,5464,5467,5470,5473,5476,5479],{"className":5457},[82],[47,5459],{"className":5460,"style":1264},[86],[47,5462,4046],{"className":5463},[3309],[47,5465,387],{"className":5466},[523],[47,5468,2764],{"className":5469,"style":2034},[91,92],[47,5471,405],{"className":5472},[590],[47,5474],{"className":5475,"style":561},[262],[47,5477,394],{"className":5478},[565],[47,5480],{"className":5481,"style":561},[262],[47,5483,5485,5488,5491,5494,5497],{"className":5484},[82],[47,5486],{"className":5487,"style":1264},[86],[47,5489,4046],{"className":5490},[3309],[47,5492,387],{"className":5493},[523],[47,5495,2698],{"className":5496,"style":2713},[91,92],[47,5498,405],{"className":5499},[590],[15,5501,5502],{},"Finally, using logarithmic properties, we simplify this to:",[47,5504,5506],{"className":5505,"translate":51},[331],[47,5507,5509,5561],{"className":5508,"translate":51},[50],[47,5510,5512],{"className":5511},[55],[57,5513,5514],{"xmlns":59,"display":340},[61,5515,5516,5558],{},[64,5517,5518,5520,5522,5524,5526,5528,5530,5532,5534,5536,5538,5540,5542,5544,5546],{},[67,5519,4046],{},[214,5521,4049],{},[214,5523,387],{"stretchy":386},[67,5525,2764],{},[214,5527,405],{"stretchy":386},[214,5529,394],{},[67,5531,4046],{},[214,5533,4049],{},[214,5535,387],{"stretchy":386},[67,5537,2698],{},[214,5539,405],{"stretchy":386},[214,5541,375],{},[67,5543,4046],{},[214,5545,4049],{},[64,5547,5548,5550,5556],{},[214,5549,387],{"fence":78},[377,5551,5552,5554],{},[67,5553,2764],{},[67,5555,2698],{},[214,5557,405],{"fence":78},[71,5559,5560],{"encoding":73},"\\ln(N) - \\ln(r) = \\ln\\left(\\frac{N}{r}\\right)",[47,5562,5564,5591,5618],{"className":5563,"ariaHidden":78},[77],[47,5565,5567,5570,5573,5576,5579,5582,5585,5588],{"className":5566},[82],[47,5568],{"className":5569,"style":1264},[86],[47,5571,4046],{"className":5572},[3309],[47,5574,387],{"className":5575},[523],[47,5577,2764],{"className":5578,"style":2034},[91,92],[47,5580,405],{"className":5581},[590],[47,5583],{"className":5584,"style":561},[262],[47,5586,394],{"className":5587},[565],[47,5589],{"className":5590,"style":561},[262],[47,5592,5594,5597,5600,5603,5606,5609,5612,5615],{"className":5593},[82],[47,5595],{"className":5596,"style":1264},[86],[47,5598,4046],{"className":5599},[3309],[47,5601,387],{"className":5602},[523],[47,5604,2698],{"className":5605,"style":2713},[91,92],[47,5607,405],{"className":5608},[590],[47,5610],{"className":5611,"style":502},[262],[47,5613,375],{"className":5614},[506],[47,5616],{"className":5617,"style":502},[262],[47,5619,5621,5625,5628,5631],{"className":5620},[82],[47,5622],{"className":5623,"style":5624},[86],"height:2.4em;vertical-align:-0.95em;",[47,5626,4046],{"className":5627},[3309],[47,5629],{"className":5630,"style":263},[262],[47,5632,5634,5641,5704],{"className":5633},[5365],[47,5635,5637],{"className":5636,"style":5370},[523,5369],[47,5638,387],{"className":5639},[5640,457],"delimsizing",[47,5642,5644,5647,5701],{"className":5643},[91],[47,5645],{"className":5646},[523,524],[47,5648,5650],{"className":5649},[377],[47,5651,5653,5693],{"className":5652},[433,434],[47,5654,5656,5690],{"className":5655},[438],[47,5657,5660,5671,5679],{"className":5658,"style":5659},[442],"height:1.3603em;",[47,5661,5662,5665],{"style":540},[47,5663],{"className":5664,"style":544},[450],[47,5666,5668],{"className":5667},[91],[47,5669,2698],{"className":5670,"style":2713},[91,92],[47,5672,5673,5676],{"style":593},[47,5674],{"className":5675,"style":544},[450],[47,5677],{"className":5678,"style":601},[600],[47,5680,5681,5684],{"style":604},[47,5682],{"className":5683,"style":544},[450],[47,5685,5687],{"className":5686},[91],[47,5688,2764],{"className":5689,"style":2034},[91,92],[47,5691,489],{"className":5692},[488],[47,5694,5696],{"className":5695},[438],[47,5697,5699],{"className":5698,"style":1009},[442],[47,5700],{},[47,5702],{"className":5703},[590,524],[47,5705,5707],{"className":5706,"style":5370},[590,5369],[47,5708,405],{"className":5709},[5640,457],[2839,5711,5713],{"id":5712},"substituting-into-the-original-equation","Substituting into the Original Equation",[15,5715,5716],{},"Now, we substitute the simplified sum into the original probability equation:",[47,5718,5720],{"className":5719,"translate":51},[331],[47,5721,5723,5761],{"className":5722,"translate":51},[50],[47,5724,5726],{"className":5725},[55],[57,5727,5728],{"xmlns":59,"display":340},[61,5729,5730,5758],{},[64,5731,5732,5734,5736,5742,5744,5746],{},[67,5733,350],{},[214,5735,375],{},[377,5737,5738,5740],{},[67,5739,2698],{},[67,5741,2764],{},[67,5743,4046],{},[214,5745,4049],{},[64,5747,5748,5750,5756],{},[214,5749,387],{"fence":78},[377,5751,5752,5754],{},[67,5753,2764],{},[67,5755,2698],{},[214,5757,405],{"fence":78},[71,5759,5760],{"encoding":73},"P = \\frac{r}{N} \\ln\\left(\\frac{N}{r}\\right)",[47,5762,5764,5782],{"className":5763,"ariaHidden":78},[77],[47,5765,5767,5770,5773,5776,5779],{"className":5766},[82],[47,5768],{"className":5769,"style":2030},[86],[47,5771,350],{"className":5772,"style":425},[91,92],[47,5774],{"className":5775,"style":502},[262],[47,5777,375],{"className":5778},[506],[47,5780],{"className":5781,"style":502},[262],[47,5783,5785,5788,5850,5853,5856,5859],{"className":5784},[82],[47,5786],{"className":5787,"style":5624},[86],[47,5789,5791,5794,5847],{"className":5790},[91],[47,5792],{"className":5793},[523,524],[47,5795,5797],{"className":5796},[377],[47,5798,5800,5839],{"className":5799},[433,434],[47,5801,5803,5836],{"className":5802},[438],[47,5804,5806,5817,5825],{"className":5805,"style":3410},[442],[47,5807,5808,5811],{"style":540},[47,5809],{"className":5810,"style":544},[450],[47,5812,5814],{"className":5813},[91],[47,5815,2764],{"className":5816,"style":2034},[91,92],[47,5818,5819,5822],{"style":593},[47,5820],{"className":5821,"style":544},[450],[47,5823],{"className":5824,"style":601},[600],[47,5826,5827,5830],{"style":604},[47,5828],{"className":5829,"style":544},[450],[47,5831,5833],{"className":5832},[91],[47,5834,2698],{"className":5835,"style":2713},[91,92],[47,5837,489],{"className":5838},[488],[47,5840,5842],{"className":5841},[438],[47,5843,5845],{"className":5844,"style":1009},[442],[47,5846],{},[47,5848],{"className":5849},[590,524],[47,5851],{"className":5852,"style":263},[262],[47,5854,4046],{"className":5855},[3309],[47,5857],{"className":5858,"style":263},[262],[47,5860,5862,5868,5930],{"className":5861},[5365],[47,5863,5865],{"className":5864,"style":5370},[523,5369],[47,5866,387],{"className":5867},[5640,457],[47,5869,5871,5874,5927],{"className":5870},[91],[47,5872],{"className":5873},[523,524],[47,5875,5877],{"className":5876},[377],[47,5878,5880,5919],{"className":5879},[433,434],[47,5881,5883,5916],{"className":5882},[438],[47,5884,5886,5897,5905],{"className":5885,"style":5659},[442],[47,5887,5888,5891],{"style":540},[47,5889],{"className":5890,"style":544},[450],[47,5892,5894],{"className":5893},[91],[47,5895,2698],{"className":5896,"style":2713},[91,92],[47,5898,5899,5902],{"style":593},[47,5900],{"className":5901,"style":544},[450],[47,5903],{"className":5904,"style":601},[600],[47,5906,5907,5910],{"style":604},[47,5908],{"className":5909,"style":544},[450],[47,5911,5913],{"className":5912},[91],[47,5914,2764],{"className":5915,"style":2034},[91,92],[47,5917,489],{"className":5918},[488],[47,5920,5922],{"className":5921},[438],[47,5923,5925],{"className":5924,"style":1009},[442],[47,5926],{},[47,5928],{"className":5929},[590,524],[47,5931,5933],{"className":5932,"style":5370},[590,5369],[47,5934,405],{"className":5935},[5640,457],[15,5937,5938,5939,6060],{},"Next, we introduce a new variable ",[47,5940,5942,5964],{"className":5941,"translate":51},[50],[47,5943,5945],{"className":5944},[55],[57,5946,5947],{"xmlns":59},[61,5948,5949,5961],{},[64,5950,5951,5953,5955],{},[67,5952,140],{},[214,5954,375],{},[377,5956,5957,5959],{},[67,5958,2698],{},[67,5960,2764],{},[71,5962,5963],{"encoding":73},"x = \\frac{r}{N}",[47,5965,5967,5985],{"className":5966,"ariaHidden":78},[77],[47,5968,5970,5973,5976,5979,5982],{"className":5969},[82],[47,5971],{"className":5972,"style":87},[86],[47,5974,140],{"className":5975},[91,92],[47,5977],{"className":5978,"style":502},[262],[47,5980,375],{"className":5981},[506],[47,5983],{"className":5984,"style":502},[262],[47,5986,5988,5992],{"className":5987},[82],[47,5989],{"className":5990,"style":5991},[86],"height:1.0404em;vertical-align:-0.345em;",[47,5993,5995,5998,6057],{"className":5994},[91],[47,5996],{"className":5997},[523,524],[47,5999,6001],{"className":6000},[377],[47,6002,6004,6049],{"className":6003},[433,434],[47,6005,6007,6046],{"className":6006},[438],[47,6008,6010,6024,6032],{"className":6009,"style":3042},[442],[47,6011,6012,6015],{"style":3045},[47,6013],{"className":6014,"style":544},[450],[47,6016,6018],{"className":6017},[455,456,457,458],[47,6019,6021],{"className":6020},[91,458],[47,6022,2764],{"className":6023,"style":2034},[91,92,458],[47,6025,6026,6029],{"style":593},[47,6027],{"className":6028,"style":544},[450],[47,6030],{"className":6031,"style":601},[600],[47,6033,6034,6037],{"style":3074},[47,6035],{"className":6036,"style":544},[450],[47,6038,6040],{"className":6039},[455,456,457,458],[47,6041,6043],{"className":6042},[91,458],[47,6044,2698],{"className":6045,"style":2713},[91,92,458],[47,6047,489],{"className":6048},[488],[47,6050,6052],{"className":6051},[438],[47,6053,6055],{"className":6054,"style":3796},[442],[47,6056],{},[47,6058],{"className":6059},[590,524]," to represent the proportion of the total boxes chosen as the sample, yielding:",[47,6062,6064],{"className":6063,"translate":51},[331],[47,6065,6067,6101],{"className":6066,"translate":51},[50],[47,6068,6070],{"className":6069},[55],[57,6071,6072],{"xmlns":59,"display":340},[61,6073,6074,6098],{},[64,6075,6076,6078,6080,6082,6084,6086],{},[67,6077,350],{},[214,6079,375],{},[67,6081,140],{},[67,6083,4046],{},[214,6085,4049],{},[64,6087,6088,6090,6096],{},[214,6089,387],{"fence":78},[377,6091,6092,6094],{},[107,6093,381],{},[67,6095,140],{},[214,6097,405],{"fence":78},[71,6099,6100],{"encoding":73},"P = x \\ln\\left(\\frac{1}{x}\\right)",[47,6102,6104,6122],{"className":6103,"ariaHidden":78},[77],[47,6105,6107,6110,6113,6116,6119],{"className":6106},[82],[47,6108],{"className":6109,"style":2030},[86],[47,6111,350],{"className":6112,"style":425},[91,92],[47,6114],{"className":6115,"style":502},[262],[47,6117,375],{"className":6118},[506],[47,6120],{"className":6121,"style":502},[262],[47,6123,6125,6128,6131,6134,6137,6140],{"className":6124},[82],[47,6126],{"className":6127,"style":5624},[86],[47,6129,140],{"className":6130},[91,92],[47,6132],{"className":6133,"style":263},[262],[47,6135,4046],{"className":6136},[3309],[47,6138],{"className":6139,"style":263},[262],[47,6141,6143,6149,6211],{"className":6142},[5365],[47,6144,6146],{"className":6145,"style":5370},[523,5369],[47,6147,387],{"className":6148},[5640,457],[47,6150,6152,6155,6208],{"className":6151},[91],[47,6153],{"className":6154},[523,524],[47,6156,6158],{"className":6157},[377],[47,6159,6161,6200],{"className":6160},[433,434],[47,6162,6164,6197],{"className":6163},[438],[47,6165,6167,6178,6186],{"className":6166,"style":537},[442],[47,6168,6169,6172],{"style":540},[47,6170],{"className":6171,"style":544},[450],[47,6173,6175],{"className":6174},[91],[47,6176,140],{"className":6177},[91,92],[47,6179,6180,6183],{"style":593},[47,6181],{"className":6182,"style":544},[450],[47,6184],{"className":6185,"style":601},[600],[47,6187,6188,6191],{"style":604},[47,6189],{"className":6190,"style":544},[450],[47,6192,6194],{"className":6193},[91],[47,6195,381],{"className":6196},[91],[47,6198,489],{"className":6199},[488],[47,6201,6203],{"className":6202},[438],[47,6204,6206],{"className":6205,"style":1009},[442],[47,6207],{},[47,6209],{"className":6210},[590,524],[47,6212,6214],{"className":6213,"style":5370},[590,5369],[47,6215,405],{"className":6216},[5640,457],[2839,6218,6220],{"id":6219},"maximizing-the-probability","Maximizing the Probability",[15,6222,6223,6224,6252,6253,6413,6414,6442],{},"To maximize ",[47,6225,6227,6240],{"className":6226,"translate":51},[50],[47,6228,6230],{"className":6229},[55],[57,6231,6232],{"xmlns":59},[61,6233,6234,6238],{},[64,6235,6236],{},[67,6237,350],{},[71,6239,350],{"encoding":73},[47,6241,6243],{"className":6242,"ariaHidden":78},[77],[47,6244,6246,6249],{"className":6245},[82],[47,6247],{"className":6248,"style":2030},[86],[47,6250,350],{"className":6251,"style":425},[91,92],", we take the derivative of ",[47,6254,6256,6289],{"className":6255,"translate":51},[50],[47,6257,6259],{"className":6258},[55],[57,6260,6261],{"xmlns":59},[61,6262,6263,6287],{},[64,6264,6265,6267,6269,6271,6273,6275],{},[67,6266,350],{},[214,6268,375],{},[67,6270,140],{},[67,6272,4046],{},[214,6274,4049],{},[64,6276,6277,6279,6285],{},[214,6278,387],{"fence":78},[377,6280,6281,6283],{},[107,6282,381],{},[67,6284,140],{},[214,6286,405],{"fence":78},[71,6288,6100],{"encoding":73},[47,6290,6292,6310],{"className":6291,"ariaHidden":78},[77],[47,6293,6295,6298,6301,6304,6307],{"className":6294},[82],[47,6296],{"className":6297,"style":2030},[86],[47,6299,350],{"className":6300,"style":425},[91,92],[47,6302],{"className":6303,"style":502},[262],[47,6305,375],{"className":6306},[506],[47,6308],{"className":6309,"style":502},[262],[47,6311,6313,6317,6320,6323,6326,6329],{"className":6312},[82],[47,6314],{"className":6315,"style":6316},[86],"height:1.2em;vertical-align:-0.35em;",[47,6318,140],{"className":6319},[91,92],[47,6321],{"className":6322,"style":263},[262],[47,6324,4046],{"className":6325},[3309],[47,6327],{"className":6328,"style":263},[262],[47,6330,6332,6339,6407],{"className":6331},[5365],[47,6333,6335],{"className":6334,"style":5370},[523,5369],[47,6336,387],{"className":6337},[5640,6338],"size1",[47,6340,6342,6345,6404],{"className":6341},[91],[47,6343],{"className":6344},[523,524],[47,6346,6348],{"className":6347},[377],[47,6349,6351,6396],{"className":6350},[433,434],[47,6352,6354,6393],{"className":6353},[438],[47,6355,6357,6371,6379],{"className":6356,"style":3750},[442],[47,6358,6359,6362],{"style":3045},[47,6360],{"className":6361,"style":544},[450],[47,6363,6365],{"className":6364},[455,456,457,458],[47,6366,6368],{"className":6367},[91,458],[47,6369,140],{"className":6370},[91,92,458],[47,6372,6373,6376],{"style":593},[47,6374],{"className":6375,"style":544},[450],[47,6377],{"className":6378,"style":601},[600],[47,6380,6381,6384],{"style":3074},[47,6382],{"className":6383,"style":544},[450],[47,6385,6387],{"className":6386},[455,456,457,458],[47,6388,6390],{"className":6389},[91,458],[47,6391,381],{"className":6392},[91,458],[47,6394,489],{"className":6395},[488],[47,6397,6399],{"className":6398},[438],[47,6400,6402],{"className":6401,"style":3796},[442],[47,6403],{},[47,6405],{"className":6406},[590,524],[47,6408,6410],{"className":6409,"style":5370},[590,5369],[47,6411,405],{"className":6412},[5640,6338]," with respect to ",[47,6415,6417,6430],{"className":6416,"translate":51},[50],[47,6418,6420],{"className":6419},[55],[57,6421,6422],{"xmlns":59},[61,6423,6424,6428],{},[64,6425,6426],{},[67,6427,140],{},[71,6429,140],{"encoding":73},[47,6431,6433],{"className":6432,"ariaHidden":78},[77],[47,6434,6436,6439],{"className":6435},[82],[47,6437],{"className":6438,"style":87},[86],[47,6440,140],{"className":6441},[91,92],":",[47,6444,6446],{"className":6445,"translate":51},[331],[47,6447,6449,6512],{"className":6448,"translate":51},[50],[47,6450,6452],{"className":6451},[55],[57,6453,6454],{"xmlns":59,"display":340},[61,6455,6456,6509],{},[64,6457,6458,6473,6475,6485],{},[377,6459,6460,6467],{},[64,6461,6462,6465],{},[67,6463,6464],{},"d",[67,6466,350],{},[64,6468,6469,6471],{},[67,6470,6464],{},[67,6472,140],{},[214,6474,375],{},[377,6476,6477,6479],{},[67,6478,6464],{},[64,6480,6481,6483],{},[67,6482,6464],{},[67,6484,140],{},[64,6486,6487,6489,6491,6493,6495,6507],{},[214,6488,387],{"fence":78},[67,6490,140],{},[67,6492,4046],{},[214,6494,4049],{},[64,6496,6497,6499,6505],{},[214,6498,387],{"fence":78},[377,6500,6501,6503],{},[107,6502,381],{},[67,6504,140],{},[214,6506,405],{"fence":78},[214,6508,405],{"fence":78},[71,6510,6511],{"encoding":73},"\\frac{dP}{dx} = \\frac{d}{dx} \\left(x \\ln\\left(\\frac{1}{x}\\right)\\right)",[47,6513,6515,6599],{"className":6514,"ariaHidden":78},[77],[47,6516,6518,6522,6590,6593,6596],{"className":6517},[82],[47,6519],{"className":6520,"style":6521},[86],"height:2.0574em;vertical-align:-0.686em;",[47,6523,6525,6528,6587],{"className":6524},[91],[47,6526],{"className":6527},[523,524],[47,6529,6531],{"className":6530},[377],[47,6532,6534,6579],{"className":6533},[433,434],[47,6535,6537,6576],{"className":6536},[438],[47,6538,6540,6554,6562],{"className":6539,"style":1349},[442],[47,6541,6542,6545],{"style":540},[47,6543],{"className":6544,"style":544},[450],[47,6546,6548,6551],{"className":6547},[91],[47,6549,6464],{"className":6550},[91,92],[47,6552,140],{"className":6553},[91,92],[47,6555,6556,6559],{"style":593},[47,6557],{"className":6558,"style":544},[450],[47,6560],{"className":6561,"style":601},[600],[47,6563,6564,6567],{"style":604},[47,6565],{"className":6566,"style":544},[450],[47,6568,6570,6573],{"className":6569},[91],[47,6571,6464],{"className":6572},[91,92],[47,6574,350],{"className":6575,"style":425},[91,92],[47,6577,489],{"className":6578},[488],[47,6580,6582],{"className":6581},[438],[47,6583,6585],{"className":6584,"style":1009},[442],[47,6586],{},[47,6588],{"className":6589},[590,524],[47,6591],{"className":6592,"style":502},[262],[47,6594,375],{"className":6595},[506],[47,6597],{"className":6598,"style":502},[262],[47,6600,6602,6605,6670,6673],{"className":6601},[82],[47,6603],{"className":6604,"style":5624},[86],[47,6606,6608,6611,6667],{"className":6607},[91],[47,6609],{"className":6610},[523,524],[47,6612,6614],{"className":6613},[377],[47,6615,6617,6659],{"className":6616},[433,434],[47,6618,6620,6656],{"className":6619},[438],[47,6621,6623,6637,6645],{"className":6622,"style":1349},[442],[47,6624,6625,6628],{"style":540},[47,6626],{"className":6627,"style":544},[450],[47,6629,6631,6634],{"className":6630},[91],[47,6632,6464],{"className":6633},[91,92],[47,6635,140],{"className":6636},[91,92],[47,6638,6639,6642],{"style":593},[47,6640],{"className":6641,"style":544},[450],[47,6643],{"className":6644,"style":601},[600],[47,6646,6647,6650],{"style":604},[47,6648],{"className":6649,"style":544},[450],[47,6651,6653],{"className":6652},[91],[47,6654,6464],{"className":6655},[91,92],[47,6657,489],{"className":6658},[488],[47,6660,6662],{"className":6661},[438],[47,6663,6665],{"className":6664,"style":1009},[442],[47,6666],{},[47,6668],{"className":6669},[590,524],[47,6671],{"className":6672,"style":263},[262],[47,6674,6676,6682,6685,6688,6691,6694,6771],{"className":6675},[5365],[47,6677,6679],{"className":6678,"style":5370},[523,5369],[47,6680,387],{"className":6681},[5640,457],[47,6683,140],{"className":6684},[91,92],[47,6686],{"className":6687,"style":263},[262],[47,6689,4046],{"className":6690},[3309],[47,6692],{"className":6693,"style":263},[262],[47,6695,6697,6703,6765],{"className":6696},[5365],[47,6698,6700],{"className":6699,"style":5370},[523,5369],[47,6701,387],{"className":6702},[5640,457],[47,6704,6706,6709,6762],{"className":6705},[91],[47,6707],{"className":6708},[523,524],[47,6710,6712],{"className":6711},[377],[47,6713,6715,6754],{"className":6714},[433,434],[47,6716,6718,6751],{"className":6717},[438],[47,6719,6721,6732,6740],{"className":6720,"style":537},[442],[47,6722,6723,6726],{"style":540},[47,6724],{"className":6725,"style":544},[450],[47,6727,6729],{"className":6728},[91],[47,6730,140],{"className":6731},[91,92],[47,6733,6734,6737],{"style":593},[47,6735],{"className":6736,"style":544},[450],[47,6738],{"className":6739,"style":601},[600],[47,6741,6742,6745],{"style":604},[47,6743],{"className":6744,"style":544},[450],[47,6746,6748],{"className":6747},[91],[47,6749,381],{"className":6750},[91],[47,6752,489],{"className":6753},[488],[47,6755,6757],{"className":6756},[438],[47,6758,6760],{"className":6759,"style":1009},[442],[47,6761],{},[47,6763],{"className":6764},[590,524],[47,6766,6768],{"className":6767,"style":5370},[590,5369],[47,6769,405],{"className":6770},[5640,457],[47,6772,6774],{"className":6773,"style":5370},[590,5369],[47,6775,405],{"className":6776},[5640,457],[15,6778,6779],{},"Using the product rule:",[47,6781,6783],{"className":6782,"translate":51},[331],[47,6784,6786,6854],{"className":6785,"translate":51},[50],[47,6787,6789],{"className":6788},[55],[57,6790,6791],{"xmlns":59,"display":340},[61,6792,6793,6851],{},[64,6794,6795,6805,6807,6810,6813,6815,6817,6819,6833,6835,6837],{},[377,6796,6797,6799],{},[67,6798,6464],{},[64,6800,6801,6803],{},[67,6802,6464],{},[67,6804,140],{},[214,6806,387],{"stretchy":386},[67,6808,6809],{},"u",[67,6811,6812],{},"v",[214,6814,405],{"stretchy":386},[214,6816,375],{},[67,6818,6809],{},[377,6820,6821,6827],{},[64,6822,6823,6825],{},[67,6824,6464],{},[67,6826,6812],{},[64,6828,6829,6831],{},[67,6830,6464],{},[67,6832,140],{},[214,6834,400],{},[67,6836,6812],{},[377,6838,6839,6845],{},[64,6840,6841,6843],{},[67,6842,6464],{},[67,6844,6809],{},[64,6846,6847,6849],{},[67,6848,6464],{},[67,6850,140],{},[71,6852,6853],{"encoding":73},"\\frac{d}{dx}(uv) = u \\frac{dv}{dx} + v \\frac{du}{dx}",[47,6855,6857,6948,7034],{"className":6856,"ariaHidden":78},[77],[47,6858,6860,6863,6928,6931,6936,6939,6942,6945],{"className":6859},[82],[47,6861],{"className":6862,"style":6521},[86],[47,6864,6866,6869,6925],{"className":6865},[91],[47,6867],{"className":6868},[523,524],[47,6870,6872],{"className":6871},[377],[47,6873,6875,6917],{"className":6874},[433,434],[47,6876,6878,6914],{"className":6877},[438],[47,6879,6881,6895,6903],{"className":6880,"style":1349},[442],[47,6882,6883,6886],{"style":540},[47,6884],{"className":6885,"style":544},[450],[47,6887,6889,6892],{"className":6888},[91],[47,6890,6464],{"className":6891},[91,92],[47,6893,140],{"className":6894},[91,92],[47,6896,6897,6900],{"style":593},[47,6898],{"className":6899,"style":544},[450],[47,6901],{"className":6902,"style":601},[600],[47,6904,6905,6908],{"style":604},[47,6906],{"className":6907,"style":544},[450],[47,6909,6911],{"className":6910},[91],[47,6912,6464],{"className":6913},[91,92],[47,6915,489],{"className":6916},[488],[47,6918,6920],{"className":6919},[438],[47,6921,6923],{"className":6922,"style":1009},[442],[47,6924],{},[47,6926],{"className":6927},[590,524],[47,6929,387],{"className":6930},[523],[47,6932,6935],{"className":6933,"style":6934},[91,92],"margin-right:0.03588em;","uv",[47,6937,405],{"className":6938},[590],[47,6940],{"className":6941,"style":502},[262],[47,6943,375],{"className":6944},[506],[47,6946],{"className":6947,"style":502},[262],[47,6949,6951,6954,6957,7025,7028,7031],{"className":6950},[82],[47,6952],{"className":6953,"style":6521},[86],[47,6955,6809],{"className":6956},[91,92],[47,6958,6960,6963,7022],{"className":6959},[91],[47,6961],{"className":6962},[523,524],[47,6964,6966],{"className":6965},[377],[47,6967,6969,7014],{"className":6968},[433,434],[47,6970,6972,7011],{"className":6971},[438],[47,6973,6975,6989,6997],{"className":6974,"style":1349},[442],[47,6976,6977,6980],{"style":540},[47,6978],{"className":6979,"style":544},[450],[47,6981,6983,6986],{"className":6982},[91],[47,6984,6464],{"className":6985},[91,92],[47,6987,140],{"className":6988},[91,92],[47,6990,6991,6994],{"style":593},[47,6992],{"className":6993,"style":544},[450],[47,6995],{"className":6996,"style":601},[600],[47,6998,6999,7002],{"style":604},[47,7000],{"className":7001,"style":544},[450],[47,7003,7005,7008],{"className":7004},[91],[47,7006,6464],{"className":7007},[91,92],[47,7009,6812],{"className":7010,"style":6934},[91,92],[47,7012,489],{"className":7013},[488],[47,7015,7017],{"className":7016},[438],[47,7018,7020],{"className":7019,"style":1009},[442],[47,7021],{},[47,7023],{"className":7024},[590,524],[47,7026],{"className":7027,"style":561},[262],[47,7029,400],{"className":7030},[565],[47,7032],{"className":7033,"style":561},[262],[47,7035,7037,7040,7043],{"className":7036},[82],[47,7038],{"className":7039,"style":6521},[86],[47,7041,6812],{"className":7042,"style":6934},[91,92],[47,7044,7046,7049,7108],{"className":7045},[91],[47,7047],{"className":7048},[523,524],[47,7050,7052],{"className":7051},[377],[47,7053,7055,7100],{"className":7054},[433,434],[47,7056,7058,7097],{"className":7057},[438],[47,7059,7061,7075,7083],{"className":7060,"style":1349},[442],[47,7062,7063,7066],{"style":540},[47,7064],{"className":7065,"style":544},[450],[47,7067,7069,7072],{"className":7068},[91],[47,7070,6464],{"className":7071},[91,92],[47,7073,140],{"className":7074},[91,92],[47,7076,7077,7080],{"style":593},[47,7078],{"className":7079,"style":544},[450],[47,7081],{"className":7082,"style":601},[600],[47,7084,7085,7088],{"style":604},[47,7086],{"className":7087,"style":544},[450],[47,7089,7091,7094],{"className":7090},[91],[47,7092,6464],{"className":7093},[91,92],[47,7095,6809],{"className":7096},[91,92],[47,7098,489],{"className":7099},[488],[47,7101,7103],{"className":7102},[438],[47,7104,7106],{"className":7105,"style":1009},[442],[47,7107],{},[47,7109],{"className":7110},[590,524],[15,7112,4256,7113,125,7164,7315],{},[47,7114,7116,7134],{"className":7115,"translate":51},[50],[47,7117,7119],{"className":7118},[55],[57,7120,7121],{"xmlns":59},[61,7122,7123,7131],{},[64,7124,7125,7127,7129],{},[67,7126,6809],{},[214,7128,375],{},[67,7130,140],{},[71,7132,7133],{"encoding":73},"u = x",[47,7135,7137,7155],{"className":7136,"ariaHidden":78},[77],[47,7138,7140,7143,7146,7149,7152],{"className":7139},[82],[47,7141],{"className":7142,"style":87},[86],[47,7144,6809],{"className":7145},[91,92],[47,7147],{"className":7148,"style":502},[262],[47,7150,375],{"className":7151},[506],[47,7153],{"className":7154,"style":502},[262],[47,7156,7158,7161],{"className":7157},[82],[47,7159],{"className":7160,"style":87},[86],[47,7162,140],{"className":7163},[91,92],[47,7165,7167,7199],{"className":7166,"translate":51},[50],[47,7168,7170],{"className":7169},[55],[57,7171,7172],{"xmlns":59},[61,7173,7174,7196],{},[64,7175,7176,7178,7180,7182,7184],{},[67,7177,6812],{},[214,7179,375],{},[67,7181,4046],{},[214,7183,4049],{},[64,7185,7186,7188,7194],{},[214,7187,387],{"fence":78},[377,7189,7190,7192],{},[107,7191,381],{},[67,7193,140],{},[214,7195,405],{"fence":78},[71,7197,7198],{"encoding":73},"v = \\ln\\left(\\frac{1}{x}\\right)",[47,7200,7202,7220],{"className":7201,"ariaHidden":78},[77],[47,7203,7205,7208,7211,7214,7217],{"className":7204},[82],[47,7206],{"className":7207,"style":87},[86],[47,7209,6812],{"className":7210,"style":6934},[91,92],[47,7212],{"className":7213,"style":502},[262],[47,7215,375],{"className":7216},[506],[47,7218],{"className":7219,"style":502},[262],[47,7221,7223,7226,7229,7232],{"className":7222},[82],[47,7224],{"className":7225,"style":6316},[86],[47,7227,4046],{"className":7228},[3309],[47,7230],{"className":7231,"style":263},[262],[47,7233,7235,7241,7309],{"className":7234},[5365],[47,7236,7238],{"className":7237,"style":5370},[523,5369],[47,7239,387],{"className":7240},[5640,6338],[47,7242,7244,7247,7306],{"className":7243},[91],[47,7245],{"className":7246},[523,524],[47,7248,7250],{"className":7249},[377],[47,7251,7253,7298],{"className":7252},[433,434],[47,7254,7256,7295],{"className":7255},[438],[47,7257,7259,7273,7281],{"className":7258,"style":3750},[442],[47,7260,7261,7264],{"style":3045},[47,7262],{"className":7263,"style":544},[450],[47,7265,7267],{"className":7266},[455,456,457,458],[47,7268,7270],{"className":7269},[91,458],[47,7271,140],{"className":7272},[91,92,458],[47,7274,7275,7278],{"style":593},[47,7276],{"className":7277,"style":544},[450],[47,7279],{"className":7280,"style":601},[600],[47,7282,7283,7286],{"style":3074},[47,7284],{"className":7285,"style":544},[450],[47,7287,7289],{"className":7288},[455,456,457,458],[47,7290,7292],{"className":7291},[91,458],[47,7293,381],{"className":7294},[91,458],[47,7296,489],{"className":7297},[488],[47,7299,7301],{"className":7300},[438],[47,7302,7304],{"className":7303,"style":3796},[442],[47,7305],{},[47,7307],{"className":7308},[590,524],[47,7310,7312],{"className":7311,"style":5370},[590,5369],[47,7313,405],{"className":7314},[5640,6338],", we calculate the derivatives:",[47,7317,7319],{"className":7318,"translate":51},[331],[47,7320,7322,7352],{"className":7321,"translate":51},[50],[47,7323,7325],{"className":7324},[55],[57,7326,7327],{"xmlns":59,"display":340},[61,7328,7329,7349],{},[64,7330,7331,7345,7347],{},[377,7332,7333,7339],{},[64,7334,7335,7337],{},[67,7336,6464],{},[67,7338,6809],{},[64,7340,7341,7343],{},[67,7342,6464],{},[67,7344,140],{},[214,7346,375],{},[107,7348,381],{},[71,7350,7351],{"encoding":73},"\\frac{du}{dx} = 1",[47,7353,7355,7438],{"className":7354,"ariaHidden":78},[77],[47,7356,7358,7361,7429,7432,7435],{"className":7357},[82],[47,7359],{"className":7360,"style":6521},[86],[47,7362,7364,7367,7426],{"className":7363},[91],[47,7365],{"className":7366},[523,524],[47,7368,7370],{"className":7369},[377],[47,7371,7373,7418],{"className":7372},[433,434],[47,7374,7376,7415],{"className":7375},[438],[47,7377,7379,7393,7401],{"className":7378,"style":1349},[442],[47,7380,7381,7384],{"style":540},[47,7382],{"className":7383,"style":544},[450],[47,7385,7387,7390],{"className":7386},[91],[47,7388,6464],{"className":7389},[91,92],[47,7391,140],{"className":7392},[91,92],[47,7394,7395,7398],{"style":593},[47,7396],{"className":7397,"style":544},[450],[47,7399],{"className":7400,"style":601},[600],[47,7402,7403,7406],{"style":604},[47,7404],{"className":7405,"style":544},[450],[47,7407,7409,7412],{"className":7408},[91],[47,7410,6464],{"className":7411},[91,92],[47,7413,6809],{"className":7414},[91,92],[47,7416,489],{"className":7417},[488],[47,7419,7421],{"className":7420},[438],[47,7422,7424],{"className":7423,"style":1009},[442],[47,7425],{},[47,7427],{"className":7428},[590,524],[47,7430],{"className":7431,"style":502},[262],[47,7433,375],{"className":7434},[506],[47,7436],{"className":7437,"style":502},[262],[47,7439,7441,7444],{"className":7440},[82],[47,7442],{"className":7443,"style":121},[86],[47,7445,381],{"className":7446},[91],[15,7448,7449],{},"and",[47,7451,7453],{"className":7452,"translate":51},[331],[47,7454,7456,7550],{"className":7455,"translate":51},[50],[47,7457,7459],{"className":7458},[55],[57,7460,7461],{"xmlns":59,"display":340},[61,7462,7463,7547],{},[64,7464,7465,7479,7481,7491,7493,7495,7507,7509,7519,7537,7539,7541],{},[377,7466,7467,7473],{},[64,7468,7469,7471],{},[67,7470,6464],{},[67,7472,6812],{},[64,7474,7475,7477],{},[67,7476,6464],{},[67,7478,140],{},[214,7480,375],{},[377,7482,7483,7485],{},[67,7484,6464],{},[64,7486,7487,7489],{},[67,7488,6464],{},[67,7490,140],{},[67,7492,4046],{},[214,7494,4049],{},[64,7496,7497,7499,7505],{},[214,7498,387],{"fence":78},[377,7500,7501,7503],{},[107,7502,381],{},[67,7504,140],{},[214,7506,405],{"fence":78},[214,7508,375],{},[377,7510,7511,7513],{},[67,7512,6464],{},[64,7514,7515,7517],{},[67,7516,6464],{},[67,7518,140],{},[64,7520,7521,7523,7525,7527,7529,7531,7533,7535],{},[214,7522,387],{"fence":78},[214,7524,394],{},[67,7526,4046],{},[214,7528,4049],{},[214,7530,387],{"stretchy":386},[67,7532,140],{},[214,7534,405],{"stretchy":386},[214,7536,405],{"fence":78},[214,7538,375],{},[214,7540,394],{},[377,7542,7543,7545],{},[107,7544,381],{},[67,7546,140],{},[71,7548,7549],{"encoding":73},"\\frac{dv}{dx} = \\frac{d}{dx} \\ln\\left(\\frac{1}{x}\\right) = \\frac{d}{dx} \\left(-\\ln(x)\\right) = -\\frac{1}{x}",[47,7551,7553,7636,7802,7912],{"className":7552,"ariaHidden":78},[77],[47,7554,7556,7559,7627,7630,7633],{"className":7555},[82],[47,7557],{"className":7558,"style":6521},[86],[47,7560,7562,7565,7624],{"className":7561},[91],[47,7563],{"className":7564},[523,524],[47,7566,7568],{"className":7567},[377],[47,7569,7571,7616],{"className":7570},[433,434],[47,7572,7574,7613],{"className":7573},[438],[47,7575,7577,7591,7599],{"className":7576,"style":1349},[442],[47,7578,7579,7582],{"style":540},[47,7580],{"className":7581,"style":544},[450],[47,7583,7585,7588],{"className":7584},[91],[47,7586,6464],{"className":7587},[91,92],[47,7589,140],{"className":7590},[91,92],[47,7592,7593,7596],{"style":593},[47,7594],{"className":7595,"style":544},[450],[47,7597],{"className":7598,"style":601},[600],[47,7600,7601,7604],{"style":604},[47,7602],{"className":7603,"style":544},[450],[47,7605,7607,7610],{"className":7606},[91],[47,7608,6464],{"className":7609},[91,92],[47,7611,6812],{"className":7612,"style":6934},[91,92],[47,7614,489],{"className":7615},[488],[47,7617,7619],{"className":7618},[438],[47,7620,7622],{"className":7621,"style":1009},[442],[47,7623],{},[47,7625],{"className":7626},[590,524],[47,7628],{"className":7629,"style":502},[262],[47,7631,375],{"className":7632},[506],[47,7634],{"className":7635,"style":502},[262],[47,7637,7639,7642,7707,7710,7713,7716,7793,7796,7799],{"className":7638},[82],[47,7640],{"className":7641,"style":5624},[86],[47,7643,7645,7648,7704],{"className":7644},[91],[47,7646],{"className":7647},[523,524],[47,7649,7651],{"className":7650},[377],[47,7652,7654,7696],{"className":7653},[433,434],[47,7655,7657,7693],{"className":7656},[438],[47,7658,7660,7674,7682],{"className":7659,"style":1349},[442],[47,7661,7662,7665],{"style":540},[47,7663],{"className":7664,"style":544},[450],[47,7666,7668,7671],{"className":7667},[91],[47,7669,6464],{"className":7670},[91,92],[47,7672,140],{"className":7673},[91,92],[47,7675,7676,7679],{"style":593},[47,7677],{"className":7678,"style":544},[450],[47,7680],{"className":7681,"style":601},[600],[47,7683,7684,7687],{"style":604},[47,7685],{"className":7686,"style":544},[450],[47,7688,7690],{"className":7689},[91],[47,7691,6464],{"className":7692},[91,92],[47,7694,489],{"className":7695},[488],[47,7697,7699],{"className":7698},[438],[47,7700,7702],{"className":7701,"style":1009},[442],[47,7703],{},[47,7705],{"className":7706},[590,524],[47,7708],{"className":7709,"style":263},[262],[47,7711,4046],{"className":7712},[3309],[47,7714],{"className":7715,"style":263},[262],[47,7717,7719,7725,7787],{"className":7718},[5365],[47,7720,7722],{"className":7721,"style":5370},[523,5369],[47,7723,387],{"className":7724},[5640,457],[47,7726,7728,7731,7784],{"className":7727},[91],[47,7729],{"className":7730},[523,524],[47,7732,7734],{"className":7733},[377],[47,7735,7737,7776],{"className":7736},[433,434],[47,7738,7740,7773],{"className":7739},[438],[47,7741,7743,7754,7762],{"className":7742,"style":537},[442],[47,7744,7745,7748],{"style":540},[47,7746],{"className":7747,"style":544},[450],[47,7749,7751],{"className":7750},[91],[47,7752,140],{"className":7753},[91,92],[47,7755,7756,7759],{"style":593},[47,7757],{"className":7758,"style":544},[450],[47,7760],{"className":7761,"style":601},[600],[47,7763,7764,7767],{"style":604},[47,7765],{"className":7766,"style":544},[450],[47,7768,7770],{"className":7769},[91],[47,7771,381],{"className":7772},[91],[47,7774,489],{"className":7775},[488],[47,7777,7779],{"className":7778},[438],[47,7780,7782],{"className":7781,"style":1009},[442],[47,7783],{},[47,7785],{"className":7786},[590,524],[47,7788,7790],{"className":7789,"style":5370},[590,5369],[47,7791,405],{"className":7792},[5640,457],[47,7794],{"className":7795,"style":502},[262],[47,7797,375],{"className":7798},[506],[47,7800],{"className":7801,"style":502},[262],[47,7803,7805,7808,7873,7876,7903,7906,7909],{"className":7804},[82],[47,7806],{"className":7807,"style":6521},[86],[47,7809,7811,7814,7870],{"className":7810},[91],[47,7812],{"className":7813},[523,524],[47,7815,7817],{"className":7816},[377],[47,7818,7820,7862],{"className":7819},[433,434],[47,7821,7823,7859],{"className":7822},[438],[47,7824,7826,7840,7848],{"className":7825,"style":1349},[442],[47,7827,7828,7831],{"style":540},[47,7829],{"className":7830,"style":544},[450],[47,7832,7834,7837],{"className":7833},[91],[47,7835,6464],{"className":7836},[91,92],[47,7838,140],{"className":7839},[91,92],[47,7841,7842,7845],{"style":593},[47,7843],{"className":7844,"style":544},[450],[47,7846],{"className":7847,"style":601},[600],[47,7849,7850,7853],{"style":604},[47,7851],{"className":7852,"style":544},[450],[47,7854,7856],{"className":7855},[91],[47,7857,6464],{"className":7858},[91,92],[47,7860,489],{"className":7861},[488],[47,7863,7865],{"className":7864},[438],[47,7866,7868],{"className":7867,"style":1009},[442],[47,7869],{},[47,7871],{"className":7872},[590,524],[47,7874],{"className":7875,"style":263},[262],[47,7877,7879,7882,7885,7888,7891,7894,7897,7900],{"className":7878},[5365],[47,7880,387],{"className":7881,"style":5370},[523,5369],[47,7883,394],{"className":7884},[91],[47,7886],{"className":7887,"style":263},[262],[47,7889,4046],{"className":7890},[3309],[47,7892,387],{"className":7893},[523],[47,7895,140],{"className":7896},[91,92],[47,7898,405],{"className":7899},[590],[47,7901,405],{"className":7902,"style":5370},[590,5369],[47,7904],{"className":7905,"style":502},[262],[47,7907,375],{"className":7908},[506],[47,7910],{"className":7911,"style":502},[262],[47,7913,7915,7918,7921],{"className":7914},[82],[47,7916],{"className":7917,"style":951},[86],[47,7919,394],{"className":7920},[91],[47,7922,7924,7927,7980],{"className":7923},[91],[47,7925],{"className":7926},[523,524],[47,7928,7930],{"className":7929},[377],[47,7931,7933,7972],{"className":7932},[433,434],[47,7934,7936,7969],{"className":7935},[438],[47,7937,7939,7950,7958],{"className":7938,"style":537},[442],[47,7940,7941,7944],{"style":540},[47,7942],{"className":7943,"style":544},[450],[47,7945,7947],{"className":7946},[91],[47,7948,140],{"className":7949},[91,92],[47,7951,7952,7955],{"style":593},[47,7953],{"className":7954,"style":544},[450],[47,7956],{"className":7957,"style":601},[600],[47,7959,7960,7963],{"style":604},[47,7961],{"className":7962,"style":544},[450],[47,7964,7966],{"className":7965},[91],[47,7967,381],{"className":7968},[91],[47,7970,489],{"className":7971},[488],[47,7973,7975],{"className":7974},[438],[47,7976,7978],{"className":7977,"style":1009},[442],[47,7979],{},[47,7981],{"className":7982},[590,524],[15,7984,7985],{},"Substituting these into the product rule:",[47,7987,7989],{"className":7988,"translate":51},[331],[47,7990,7992,8083],{"className":7991,"translate":51},[50],[47,7993,7995],{"className":7994},[55],[57,7996,7997],{"xmlns":59,"display":340},[61,7998,7999,8080],{},[64,8000,8001,8015,8017,8019,8033,8035,8037,8039,8051,8054,8056,8058,8060,8062,8064,8066,8068],{},[377,8002,8003,8009],{},[64,8004,8005,8007],{},[67,8006,6464],{},[67,8008,350],{},[64,8010,8011,8013],{},[67,8012,6464],{},[67,8014,140],{},[214,8016,375],{},[67,8018,140],{},[64,8020,8021,8023,8025,8031],{},[214,8022,387],{"fence":78},[214,8024,394],{},[377,8026,8027,8029],{},[107,8028,381],{},[67,8030,140],{},[214,8032,405],{"fence":78},[214,8034,400],{},[67,8036,4046],{},[214,8038,4049],{},[64,8040,8041,8043,8049],{},[214,8042,387],{"fence":78},[377,8044,8045,8047],{},[107,8046,381],{},[67,8048,140],{},[214,8050,405],{"fence":78},[214,8052,8053],{},"⋅",[107,8055,381],{},[214,8057,375],{},[214,8059,394],{},[107,8061,381],{},[214,8063,400],{},[67,8065,4046],{},[214,8067,4049],{},[64,8069,8070,8072,8078],{},[214,8071,387],{"fence":78},[377,8073,8074,8076],{},[107,8075,381],{},[67,8077,140],{},[214,8079,405],{"fence":78},[71,8081,8082],{"encoding":73},"\\frac{dP}{dx} = x \\left(-\\frac{1}{x}\\right) + \\ln\\left(\\frac{1}{x}\\right) \\cdot 1 = -1 + \\ln\\left(\\frac{1}{x}\\right)",[47,8084,8086,8169,8270,8368,8386,8408],{"className":8085,"ariaHidden":78},[77],[47,8087,8089,8092,8160,8163,8166],{"className":8088},[82],[47,8090],{"className":8091,"style":6521},[86],[47,8093,8095,8098,8157],{"className":8094},[91],[47,8096],{"className":8097},[523,524],[47,8099,8101],{"className":8100},[377],[47,8102,8104,8149],{"className":8103},[433,434],[47,8105,8107,8146],{"className":8106},[438],[47,8108,8110,8124,8132],{"className":8109,"style":1349},[442],[47,8111,8112,8115],{"style":540},[47,8113],{"className":8114,"style":544},[450],[47,8116,8118,8121],{"className":8117},[91],[47,8119,6464],{"className":8120},[91,92],[47,8122,140],{"className":8123},[91,92],[47,8125,8126,8129],{"style":593},[47,8127],{"className":8128,"style":544},[450],[47,8130],{"className":8131,"style":601},[600],[47,8133,8134,8137],{"style":604},[47,8135],{"className":8136,"style":544},[450],[47,8138,8140,8143],{"className":8139},[91],[47,8141,6464],{"className":8142},[91,92],[47,8144,350],{"className":8145,"style":425},[91,92],[47,8147,489],{"className":8148},[488],[47,8150,8152],{"className":8151},[438],[47,8153,8155],{"className":8154,"style":1009},[442],[47,8156],{},[47,8158],{"className":8159},[590,524],[47,8161],{"className":8162,"style":502},[262],[47,8164,375],{"className":8165},[506],[47,8167],{"className":8168,"style":502},[262],[47,8170,8172,8175,8178,8181,8261,8264,8267],{"className":8171},[82],[47,8173],{"className":8174,"style":5624},[86],[47,8176,140],{"className":8177},[91,92],[47,8179],{"className":8180,"style":263},[262],[47,8182,8184,8190,8193,8255],{"className":8183},[5365],[47,8185,8187],{"className":8186,"style":5370},[523,5369],[47,8188,387],{"className":8189},[5640,457],[47,8191,394],{"className":8192},[91],[47,8194,8196,8199,8252],{"className":8195},[91],[47,8197],{"className":8198},[523,524],[47,8200,8202],{"className":8201},[377],[47,8203,8205,8244],{"className":8204},[433,434],[47,8206,8208,8241],{"className":8207},[438],[47,8209,8211,8222,8230],{"className":8210,"style":537},[442],[47,8212,8213,8216],{"style":540},[47,8214],{"className":8215,"style":544},[450],[47,8217,8219],{"className":8218},[91],[47,8220,140],{"className":8221},[91,92],[47,8223,8224,8227],{"style":593},[47,8225],{"className":8226,"style":544},[450],[47,8228],{"className":8229,"style":601},[600],[47,8231,8232,8235],{"style":604},[47,8233],{"className":8234,"style":544},[450],[47,8236,8238],{"className":8237},[91],[47,8239,381],{"className":8240},[91],[47,8242,489],{"className":8243},[488],[47,8245,8247],{"className":8246},[438],[47,8248,8250],{"className":8249,"style":1009},[442],[47,8251],{},[47,8253],{"className":8254},[590,524],[47,8256,8258],{"className":8257,"style":5370},[590,5369],[47,8259,405],{"className":8260},[5640,457],[47,8262],{"className":8263,"style":561},[262],[47,8265,400],{"className":8266},[565],[47,8268],{"className":8269,"style":561},[262],[47,8271,8273,8276,8279,8282,8359,8362,8365],{"className":8272},[82],[47,8274],{"className":8275,"style":5624},[86],[47,8277,4046],{"className":8278},[3309],[47,8280],{"className":8281,"style":263},[262],[47,8283,8285,8291,8353],{"className":8284},[5365],[47,8286,8288],{"className":8287,"style":5370},[523,5369],[47,8289,387],{"className":8290},[5640,457],[47,8292,8294,8297,8350],{"className":8293},[91],[47,8295],{"className":8296},[523,524],[47,8298,8300],{"className":8299},[377],[47,8301,8303,8342],{"className":8302},[433,434],[47,8304,8306,8339],{"className":8305},[438],[47,8307,8309,8320,8328],{"className":8308,"style":537},[442],[47,8310,8311,8314],{"style":540},[47,8312],{"className":8313,"style":544},[450],[47,8315,8317],{"className":8316},[91],[47,8318,140],{"className":8319},[91,92],[47,8321,8322,8325],{"style":593},[47,8323],{"className":8324,"style":544},[450],[47,8326],{"className":8327,"style":601},[600],[47,8329,8330,8333],{"style":604},[47,8331],{"className":8332,"style":544},[450],[47,8334,8336],{"className":8335},[91],[47,8337,381],{"className":8338},[91],[47,8340,489],{"className":8341},[488],[47,8343,8345],{"className":8344},[438],[47,8346,8348],{"className":8347,"style":1009},[442],[47,8349],{},[47,8351],{"className":8352},[590,524],[47,8354,8356],{"className":8355,"style":5370},[590,5369],[47,8357,405],{"className":8358},[5640,457],[47,8360],{"className":8361,"style":561},[262],[47,8363,8053],{"className":8364},[565],[47,8366],{"className":8367,"style":561},[262],[47,8369,8371,8374,8377,8380,8383],{"className":8370},[82],[47,8372],{"className":8373,"style":121},[86],[47,8375,381],{"className":8376},[91],[47,8378],{"className":8379,"style":502},[262],[47,8381,375],{"className":8382},[506],[47,8384],{"className":8385,"style":502},[262],[47,8387,8389,8393,8396,8399,8402,8405],{"className":8388},[82],[47,8390],{"className":8391,"style":8392},[86],"height:0.7278em;vertical-align:-0.0833em;",[47,8394,394],{"className":8395},[91],[47,8397,381],{"className":8398},[91],[47,8400],{"className":8401,"style":561},[262],[47,8403,400],{"className":8404},[565],[47,8406],{"className":8407,"style":561},[262],[47,8409,8411,8414,8417,8420],{"className":8410},[82],[47,8412],{"className":8413,"style":5624},[86],[47,8415,4046],{"className":8416},[3309],[47,8418],{"className":8419,"style":263},[262],[47,8421,8423,8429,8491],{"className":8422},[5365],[47,8424,8426],{"className":8425,"style":5370},[523,5369],[47,8427,387],{"className":8428},[5640,457],[47,8430,8432,8435,8488],{"className":8431},[91],[47,8433],{"className":8434},[523,524],[47,8436,8438],{"className":8437},[377],[47,8439,8441,8480],{"className":8440},[433,434],[47,8442,8444,8477],{"className":8443},[438],[47,8445,8447,8458,8466],{"className":8446,"style":537},[442],[47,8448,8449,8452],{"style":540},[47,8450],{"className":8451,"style":544},[450],[47,8453,8455],{"className":8454},[91],[47,8456,140],{"className":8457},[91,92],[47,8459,8460,8463],{"style":593},[47,8461],{"className":8462,"style":544},[450],[47,8464],{"className":8465,"style":601},[600],[47,8467,8468,8471],{"style":604},[47,8469],{"className":8470,"style":544},[450],[47,8472,8474],{"className":8473},[91],[47,8475,381],{"className":8476},[91],[47,8478,489],{"className":8479},[488],[47,8481,8483],{"className":8482},[438],[47,8484,8486],{"className":8485,"style":1009},[442],[47,8487],{},[47,8489],{"className":8490},[590,524],[47,8492,8494],{"className":8493,"style":5370},[590,5369],[47,8495,405],{"className":8496},[5640,457],[15,8498,8499,8500,8528],{},"To find the maximum probability ",[47,8501,8503,8516],{"className":8502,"translate":51},[50],[47,8504,8506],{"className":8505},[55],[57,8507,8508],{"xmlns":59},[61,8509,8510,8514],{},[64,8511,8512],{},[67,8513,350],{},[71,8515,350],{"encoding":73},[47,8517,8519],{"className":8518,"ariaHidden":78},[77],[47,8520,8522,8525],{"className":8521},[82],[47,8523],{"className":8524,"style":2030},[86],[47,8526,350],{"className":8527,"style":425},[91,92],", we set the derivative equal to zero:",[47,8530,8532],{"className":8531,"translate":51},[331],[47,8533,8535,8573],{"className":8534,"translate":51},[50],[47,8536,8538],{"className":8537},[55],[57,8539,8540],{"xmlns":59,"display":340},[61,8541,8542,8570],{},[64,8543,8544,8546,8548,8550,8552,8554,8566,8568],{},[214,8545,394],{},[107,8547,381],{},[214,8549,400],{},[67,8551,4046],{},[214,8553,4049],{},[64,8555,8556,8558,8564],{},[214,8557,387],{"fence":78},[377,8559,8560,8562],{},[107,8561,381],{},[67,8563,140],{},[214,8565,405],{"fence":78},[214,8567,375],{},[107,8569,109],{},[71,8571,8572],{"encoding":73},"-1 + \\ln\\left(\\frac{1}{x}\\right) = 0",[47,8574,8576,8597,8695],{"className":8575,"ariaHidden":78},[77],[47,8577,8579,8582,8585,8588,8591,8594],{"className":8578},[82],[47,8580],{"className":8581,"style":8392},[86],[47,8583,394],{"className":8584},[91],[47,8586,381],{"className":8587},[91],[47,8589],{"className":8590,"style":561},[262],[47,8592,400],{"className":8593},[565],[47,8595],{"className":8596,"style":561},[262],[47,8598,8600,8603,8606,8609,8686,8689,8692],{"className":8599},[82],[47,8601],{"className":8602,"style":5624},[86],[47,8604,4046],{"className":8605},[3309],[47,8607],{"className":8608,"style":263},[262],[47,8610,8612,8618,8680],{"className":8611},[5365],[47,8613,8615],{"className":8614,"style":5370},[523,5369],[47,8616,387],{"className":8617},[5640,457],[47,8619,8621,8624,8677],{"className":8620},[91],[47,8622],{"className":8623},[523,524],[47,8625,8627],{"className":8626},[377],[47,8628,8630,8669],{"className":8629},[433,434],[47,8631,8633,8666],{"className":8632},[438],[47,8634,8636,8647,8655],{"className":8635,"style":537},[442],[47,8637,8638,8641],{"style":540},[47,8639],{"className":8640,"style":544},[450],[47,8642,8644],{"className":8643},[91],[47,8645,140],{"className":8646},[91,92],[47,8648,8649,8652],{"style":593},[47,8650],{"className":8651,"style":544},[450],[47,8653],{"className":8654,"style":601},[600],[47,8656,8657,8660],{"style":604},[47,8658],{"className":8659,"style":544},[450],[47,8661,8663],{"className":8662},[91],[47,8664,381],{"className":8665},[91],[47,8667,489],{"className":8668},[488],[47,8670,8672],{"className":8671},[438],[47,8673,8675],{"className":8674,"style":1009},[442],[47,8676],{},[47,8678],{"className":8679},[590,524],[47,8681,8683],{"className":8682,"style":5370},[590,5369],[47,8684,405],{"className":8685},[5640,457],[47,8687],{"className":8688,"style":502},[262],[47,8690,375],{"className":8691},[506],[47,8693],{"className":8694,"style":502},[262],[47,8696,8698,8701],{"className":8697},[82],[47,8699],{"className":8700,"style":121},[86],[47,8702,109],{"className":8703},[91],[15,8705,8706],{},"Thus, we have:",[47,8708,8710],{"className":8709,"translate":51},[331],[47,8711,8713,8745],{"className":8712,"translate":51},[50],[47,8714,8716],{"className":8715},[55],[57,8717,8718],{"xmlns":59,"display":340},[61,8719,8720,8742],{},[64,8721,8722,8724,8726,8738,8740],{},[67,8723,4046],{},[214,8725,4049],{},[64,8727,8728,8730,8736],{},[214,8729,387],{"fence":78},[377,8731,8732,8734],{},[107,8733,381],{},[67,8735,140],{},[214,8737,405],{"fence":78},[214,8739,375],{},[107,8741,381],{},[71,8743,8744],{"encoding":73},"\\ln\\left(\\frac{1}{x}\\right) = 1",[47,8746,8748,8846],{"className":8747,"ariaHidden":78},[77],[47,8749,8751,8754,8757,8760,8837,8840,8843],{"className":8750},[82],[47,8752],{"className":8753,"style":5624},[86],[47,8755,4046],{"className":8756},[3309],[47,8758],{"className":8759,"style":263},[262],[47,8761,8763,8769,8831],{"className":8762},[5365],[47,8764,8766],{"className":8765,"style":5370},[523,5369],[47,8767,387],{"className":8768},[5640,457],[47,8770,8772,8775,8828],{"className":8771},[91],[47,8773],{"className":8774},[523,524],[47,8776,8778],{"className":8777},[377],[47,8779,8781,8820],{"className":8780},[433,434],[47,8782,8784,8817],{"className":8783},[438],[47,8785,8787,8798,8806],{"className":8786,"style":537},[442],[47,8788,8789,8792],{"style":540},[47,8790],{"className":8791,"style":544},[450],[47,8793,8795],{"className":8794},[91],[47,8796,140],{"className":8797},[91,92],[47,8799,8800,8803],{"style":593},[47,8801],{"className":8802,"style":544},[450],[47,8804],{"className":8805,"style":601},[600],[47,8807,8808,8811],{"style":604},[47,8809],{"className":8810,"style":544},[450],[47,8812,8814],{"className":8813},[91],[47,8815,381],{"className":8816},[91],[47,8818,489],{"className":8819},[488],[47,8821,8823],{"className":8822},[438],[47,8824,8826],{"className":8825,"style":1009},[442],[47,8827],{},[47,8829],{"className":8830},[590,524],[47,8832,8834],{"className":8833,"style":5370},[590,5369],[47,8835,405],{"className":8836},[5640,457],[47,8838],{"className":8839,"style":502},[262],[47,8841,375],{"className":8842},[506],[47,8844],{"className":8845,"style":502},[262],[47,8847,8849,8852],{"className":8848},[82],[47,8850],{"className":8851,"style":121},[86],[47,8853,381],{"className":8854},[91],[15,8856,8857],{},"Exponentiating both sides, we get:",[47,8859,8861],{"className":8860,"translate":51},[331],[47,8862,8864,8887],{"className":8863,"translate":51},[50],[47,8865,8867],{"className":8866},[55],[57,8868,8869],{"xmlns":59,"display":340},[61,8870,8871,8884],{},[64,8872,8873,8879,8881],{},[377,8874,8875,8877],{},[107,8876,381],{},[67,8878,140],{},[214,8880,375],{},[67,8882,8883],{},"e",[71,8885,8886],{"encoding":73},"\\frac{1}{x} = e",[47,8888,8890,8967],{"className":8889,"ariaHidden":78},[77],[47,8891,8893,8896,8958,8961,8964],{"className":8892},[82],[47,8894],{"className":8895,"style":951},[86],[47,8897,8899,8902,8955],{"className":8898},[91],[47,8900],{"className":8901},[523,524],[47,8903,8905],{"className":8904},[377],[47,8906,8908,8947],{"className":8907},[433,434],[47,8909,8911,8944],{"className":8910},[438],[47,8912,8914,8925,8933],{"className":8913,"style":537},[442],[47,8915,8916,8919],{"style":540},[47,8917],{"className":8918,"style":544},[450],[47,8920,8922],{"className":8921},[91],[47,8923,140],{"className":8924},[91,92],[47,8926,8927,8930],{"style":593},[47,8928],{"className":8929,"style":544},[450],[47,8931],{"className":8932,"style":601},[600],[47,8934,8935,8938],{"style":604},[47,8936],{"className":8937,"style":544},[450],[47,8939,8941],{"className":8940},[91],[47,8942,381],{"className":8943},[91],[47,8945,489],{"className":8946},[488],[47,8948,8950],{"className":8949},[438],[47,8951,8953],{"className":8952,"style":1009},[442],[47,8954],{},[47,8956],{"className":8957},[590,524],[47,8959],{"className":8960,"style":502},[262],[47,8962,375],{"className":8963},[506],[47,8965],{"className":8966,"style":502},[262],[47,8968,8970,8973],{"className":8969},[82],[47,8971],{"className":8972,"style":87},[86],[47,8974,8883],{"className":8975},[91,92],[15,8977,8978,8979,9099,9100,230],{},"Therefore, ",[47,8980,8982,9004],{"className":8981,"translate":51},[50],[47,8983,8985],{"className":8984},[55],[57,8986,8987],{"xmlns":59},[61,8988,8989,9001],{},[64,8990,8991,8993,8995],{},[67,8992,140],{},[214,8994,375],{},[377,8996,8997,8999],{},[107,8998,381],{},[67,9000,8883],{},[71,9002,9003],{"encoding":73},"x = \\frac{1}{e}",[47,9005,9007,9025],{"className":9006,"ariaHidden":78},[77],[47,9008,9010,9013,9016,9019,9022],{"className":9009},[82],[47,9011],{"className":9012,"style":87},[86],[47,9014,140],{"className":9015},[91,92],[47,9017],{"className":9018,"style":502},[262],[47,9020,375],{"className":9021},[506],[47,9023],{"className":9024,"style":502},[262],[47,9026,9028,9031],{"className":9027},[82],[47,9029],{"className":9030,"style":3731},[86],[47,9032,9034,9037,9096],{"className":9033},[91],[47,9035],{"className":9036},[523,524],[47,9038,9040],{"className":9039},[377],[47,9041,9043,9088],{"className":9042},[433,434],[47,9044,9046,9085],{"className":9045},[438],[47,9047,9049,9063,9071],{"className":9048,"style":3750},[442],[47,9050,9051,9054],{"style":3045},[47,9052],{"className":9053,"style":544},[450],[47,9055,9057],{"className":9056},[455,456,457,458],[47,9058,9060],{"className":9059},[91,458],[47,9061,8883],{"className":9062},[91,92,458],[47,9064,9065,9068],{"style":593},[47,9066],{"className":9067,"style":544},[450],[47,9069],{"className":9070,"style":601},[600],[47,9072,9073,9076],{"style":3074},[47,9074],{"className":9075,"style":544},[450],[47,9077,9079],{"className":9078},[455,456,457,458],[47,9080,9082],{"className":9081},[91,458],[47,9083,381],{"className":9084},[91,458],[47,9086,489],{"className":9087},[488],[47,9089,9091],{"className":9090},[438],[47,9092,9094],{"className":9093,"style":3796},[442],[47,9095],{},[47,9097],{"className":9098},[590,524],", which means the probability is maximized when ",[47,9101,9103,9129],{"className":9102,"translate":51},[50],[47,9104,9106],{"className":9105},[55],[57,9107,9108],{"xmlns":59},[61,9109,9110,9126],{},[64,9111,9112,9118,9120],{},[377,9113,9114,9116],{},[67,9115,2698],{},[67,9117,2764],{},[214,9119,375],{},[377,9121,9122,9124],{},[107,9123,381],{},[67,9125,8883],{},[71,9127,9128],{"encoding":73},"\\frac{r}{N} = \\frac{1}{e}",[47,9130,9132,9215],{"className":9131,"ariaHidden":78},[77],[47,9133,9135,9138,9206,9209,9212],{"className":9134},[82],[47,9136],{"className":9137,"style":5991},[86],[47,9139,9141,9144,9203],{"className":9140},[91],[47,9142],{"className":9143},[523,524],[47,9145,9147],{"className":9146},[377],[47,9148,9150,9195],{"className":9149},[433,434],[47,9151,9153,9192],{"className":9152},[438],[47,9154,9156,9170,9178],{"className":9155,"style":3042},[442],[47,9157,9158,9161],{"style":3045},[47,9159],{"className":9160,"style":544},[450],[47,9162,9164],{"className":9163},[455,456,457,458],[47,9165,9167],{"className":9166},[91,458],[47,9168,2764],{"className":9169,"style":2034},[91,92,458],[47,9171,9172,9175],{"style":593},[47,9173],{"className":9174,"style":544},[450],[47,9176],{"className":9177,"style":601},[600],[47,9179,9180,9183],{"style":3074},[47,9181],{"className":9182,"style":544},[450],[47,9184,9186],{"className":9185},[455,456,457,458],[47,9187,9189],{"className":9188},[91,458],[47,9190,2698],{"className":9191,"style":2713},[91,92,458],[47,9193,489],{"className":9194},[488],[47,9196,9198],{"className":9197},[438],[47,9199,9201],{"className":9200,"style":3796},[442],[47,9202],{},[47,9204],{"className":9205},[590,524],[47,9207],{"className":9208,"style":502},[262],[47,9210,375],{"className":9211},[506],[47,9213],{"className":9214,"style":502},[262],[47,9216,9218,9221],{"className":9217},[82],[47,9219],{"className":9220,"style":3731},[86],[47,9222,9224,9227,9286],{"className":9223},[91],[47,9225],{"className":9226},[523,524],[47,9228,9230],{"className":9229},[377],[47,9231,9233,9278],{"className":9232},[433,434],[47,9234,9236,9275],{"className":9235},[438],[47,9237,9239,9253,9261],{"className":9238,"style":3750},[442],[47,9240,9241,9244],{"style":3045},[47,9242],{"className":9243,"style":544},[450],[47,9245,9247],{"className":9246},[455,456,457,458],[47,9248,9250],{"className":9249},[91,458],[47,9251,8883],{"className":9252},[91,92,458],[47,9254,9255,9258],{"style":593},[47,9256],{"className":9257,"style":544},[450],[47,9259],{"className":9260,"style":601},[600],[47,9262,9263,9266],{"style":3074},[47,9264],{"className":9265,"style":544},[450],[47,9267,9269],{"className":9268},[455,456,457,458],[47,9270,9272],{"className":9271},[91,458],[47,9273,381],{"className":9274},[91,458],[47,9276,489],{"className":9277},[488],[47,9279,9281],{"className":9280},[438],[47,9282,9284],{"className":9283,"style":3796},[442],[47,9285],{},[47,9287],{"className":9288},[590,524],[2839,9290,9292],{"id":9291},"maximum-probability","Maximum Probability",[15,9294,9295],{},"At this optimal point, the maximum probability is:",[47,9297,9299],{"className":9298,"translate":51},[331],[47,9300,9302,9343],{"className":9301,"translate":51},[50],[47,9303,9305],{"className":9304},[55],[57,9306,9307],{"xmlns":59,"display":340},[61,9308,9309,9340],{},[64,9310,9311,9313,9315,9327,9329,9335,9337],{},[67,9312,350],{},[214,9314,375],{},[346,9316,9317,9319],{},[67,9318,350],{},[64,9320,9321,9323,9325],{},[67,9322,69],{},[67,9324,358],{},[67,9326,140],{},[214,9328,375],{},[377,9330,9331,9333],{},[107,9332,381],{},[67,9334,8883],{},[214,9336,1067],{},[107,9338,9339],{},"0.368",[71,9341,9342],{"encoding":73},"P = P_{max} = \\frac{1}{e} \\approx 0.368",[47,9344,9346,9364,9429,9506],{"className":9345,"ariaHidden":78},[77],[47,9347,9349,9352,9355,9358,9361],{"className":9348},[82],[47,9350],{"className":9351,"style":2030},[86],[47,9353,350],{"className":9354,"style":425},[91,92],[47,9356],{"className":9357,"style":502},[262],[47,9359,375],{"className":9360},[506],[47,9362],{"className":9363,"style":502},[262],[47,9365,9367,9371,9420,9423,9426],{"className":9366},[82],[47,9368],{"className":9369,"style":9370},[86],"height:0.8333em;vertical-align:-0.15em;",[47,9372,9374,9377],{"className":9373},[91],[47,9375,350],{"className":9376,"style":425},[91,92],[47,9378,9380],{"className":9379},[429],[47,9381,9383,9411],{"className":9382},[433,434],[47,9384,9386,9408],{"className":9385},[438],[47,9387,9390],{"className":9388,"style":9389},[442],"height:0.1514em;",[47,9391,9392,9395],{"style":446},[47,9393],{"className":9394,"style":451},[450],[47,9396,9398],{"className":9397},[455,456,457,458],[47,9399,9401,9405],{"className":9400},[91,458],[47,9402,9404],{"className":9403},[91,92,458],"ma",[47,9406,140],{"className":9407},[91,92,458],[47,9409,489],{"className":9410},[488],[47,9412,9414],{"className":9413},[438],[47,9415,9418],{"className":9416,"style":9417},[442],"height:0.15em;",[47,9419],{},[47,9421],{"className":9422,"style":502},[262],[47,9424,375],{"className":9425},[506],[47,9427],{"className":9428,"style":502},[262],[47,9430,9432,9435,9497,9500,9503],{"className":9431},[82],[47,9433],{"className":9434,"style":951},[86],[47,9436,9438,9441,9494],{"className":9437},[91],[47,9439],{"className":9440},[523,524],[47,9442,9444],{"className":9443},[377],[47,9445,9447,9486],{"className":9446},[433,434],[47,9448,9450,9483],{"className":9449},[438],[47,9451,9453,9464,9472],{"className":9452,"style":537},[442],[47,9454,9455,9458],{"style":540},[47,9456],{"className":9457,"style":544},[450],[47,9459,9461],{"className":9460},[91],[47,9462,8883],{"className":9463},[91,92],[47,9465,9466,9469],{"style":593},[47,9467],{"className":9468,"style":544},[450],[47,9470],{"className":9471,"style":601},[600],[47,9473,9474,9477],{"style":604},[47,9475],{"className":9476,"style":544},[450],[47,9478,9480],{"className":9479},[91],[47,9481,381],{"className":9482},[91],[47,9484,489],{"className":9485},[488],[47,9487,9489],{"className":9488},[438],[47,9490,9492],{"className":9491,"style":1009},[442],[47,9493],{},[47,9495],{"className":9496},[590,524],[47,9498],{"className":9499,"style":502},[262],[47,9501,1067],{"className":9502},[506],[47,9504],{"className":9505,"style":502},[262],[47,9507,9509,9512],{"className":9508},[82],[47,9510],{"className":9511,"style":121},[86],[47,9513,9339],{"className":9514},[91],[15,9516,9517,9518,9616],{},"This shows that the optimal strategy is to sample approximately ",[47,9519,9521,9539],{"className":9520,"translate":51},[50],[47,9522,9524],{"className":9523},[55],[57,9525,9526],{"xmlns":59},[61,9527,9528,9536],{},[64,9529,9530],{},[377,9531,9532,9534],{},[107,9533,381],{},[67,9535,8883],{},[71,9537,9538],{"encoding":73},"\\frac{1}{e}",[47,9540,9542],{"className":9541,"ariaHidden":78},[77],[47,9543,9545,9548],{"className":9544},[82],[47,9546],{"className":9547,"style":3731},[86],[47,9549,9551,9554,9613],{"className":9550},[91],[47,9552],{"className":9553},[523,524],[47,9555,9557],{"className":9556},[377],[47,9558,9560,9605],{"className":9559},[433,434],[47,9561,9563,9602],{"className":9562},[438],[47,9564,9566,9580,9588],{"className":9565,"style":3750},[442],[47,9567,9568,9571],{"style":3045},[47,9569],{"className":9570,"style":544},[450],[47,9572,9574],{"className":9573},[455,456,457,458],[47,9575,9577],{"className":9576},[91,458],[47,9578,8883],{"className":9579},[91,92,458],[47,9581,9582,9585],{"style":593},[47,9583],{"className":9584,"style":544},[450],[47,9586],{"className":9587,"style":601},[600],[47,9589,9590,9593],{"style":3074},[47,9591],{"className":9592,"style":544},[450],[47,9594,9596],{"className":9595},[455,456,457,458],[47,9597,9599],{"className":9598},[91,458],[47,9600,381],{"className":9601},[91,458],[47,9603,489],{"className":9604},[488],[47,9606,9608],{"className":9607},[438],[47,9609,9611],{"className":9610,"style":3796},[442],[47,9612],{},[47,9614],{"className":9615},[590,524]," (about 36.8%) of the total treasure boxes and then select the next box that is greater than all the ones in the sample. This strategy maximizes the chance of selecting the highest prize.",[10,9618,9620],{"id":9619},"program-simulation","Program Simulation",[15,9622,9623],{},"We can use TypeScript to simulate the process described above.",[15,9625,9626],{},"To verify the probability of winning the highest prize using this strategy, we can calculate the ratio of successful trials where the highest prize is obtained to the total number of trials. Below is a TypeScript program that simulates this process:",[9628,9629,9633],"pre",{"className":9630,"code":9631,"language":9632,"meta":3110,"style":3110},"language-typescript shiki shiki-themes github-dark","function simulateGamblingGame(m: number, x: number, trials: number): number {\n  let successCount = 0;\n\n  for (let trial = 0; trial \u003C trials; trial++) {\n    const boxes = Array.from({ length: m }, () => Math.random() * x);\n    const firstPart = boxes.slice(0, Math.floor(m * 0.37));\n\n    \u002F\u002F Remember the highest prize in the first 37% of boxes\n    const p = Math.max(...firstPart);\n\n    \u002F\u002F Find the maximum prize in all boxes\n    const maxPrize = Math.max(...boxes);\n\n    \u002F\u002F Check from the 37% onward if a prize higher than p is found\n    for (const prize of boxes.slice(Math.floor(m * 0.37))) {\n      if (prize > p) {\n        if (prize === maxPrize) {\n          successCount++;\n        }\n        break; \u002F\u002F Stop once a higher prize is found\n      }\n    }\n  }\n\n  return successCount \u002F trials; \u002F\u002F Return the success probability\n}\n\n\u002F\u002F Set parameters\nconst m = 100; \u002F\u002F Total number of boxes\nconst x = 1000; \u002F\u002F Maximum prize in each box\nconst trials = 100000; \u002F\u002F Number of trials\n\nconst successProbability = simulateGamblingGame(m, x, trials);\nconsole.log(`Probability of winning the highest prize: ${successProbability}`);\n","typescript",[9634,9635,9636,9686,9703,9710,9744,9783,9820,9825,9832,9854,9859,9865,9886,9891,9897,9932,9947,9961,9971,9977,9989,9995,10001,10007,10012,10029,10035,10040,10046,10064,10082,10100,10105,10120],"code",{"__ignoreMap":3110},[47,9637,9640,9644,9648,9651,9654,9656,9660,9662,9664,9666,9668,9670,9673,9675,9677,9679,9681,9683],{"class":9638,"line":9639},"line",1,[47,9641,9643],{"class":9642},"snl16","function",[47,9645,9647],{"class":9646},"svObZ"," simulateGamblingGame",[47,9649,387],{"class":9650},"s95oV",[47,9652,69],{"class":9653},"s9osk",[47,9655,6442],{"class":9642},[47,9657,9659],{"class":9658},"sDLfK"," number",[47,9661,2466],{"class":9650},[47,9663,140],{"class":9653},[47,9665,6442],{"class":9642},[47,9667,9659],{"class":9658},[47,9669,2466],{"class":9650},[47,9671,9672],{"class":9653},"trials",[47,9674,6442],{"class":9642},[47,9676,9659],{"class":9658},[47,9678,405],{"class":9650},[47,9680,6442],{"class":9642},[47,9682,9659],{"class":9658},[47,9684,9685],{"class":9650}," {\n",[47,9687,9689,9692,9695,9697,9700],{"class":9638,"line":9688},2,[47,9690,9691],{"class":9642},"  let",[47,9693,9694],{"class":9650}," successCount ",[47,9696,375],{"class":9642},[47,9698,9699],{"class":9658}," 0",[47,9701,9702],{"class":9650},";\n",[47,9704,9706],{"class":9638,"line":9705},3,[47,9707,9709],{"emptyLinePlaceholder":9708},true,"\n",[47,9711,9713,9716,9719,9722,9725,9727,9729,9732,9735,9738,9741],{"class":9638,"line":9712},4,[47,9714,9715],{"class":9642},"  for",[47,9717,9718],{"class":9650}," (",[47,9720,9721],{"class":9642},"let",[47,9723,9724],{"class":9650}," trial ",[47,9726,375],{"class":9642},[47,9728,9699],{"class":9658},[47,9730,9731],{"class":9650},"; trial ",[47,9733,9734],{"class":9642},"\u003C",[47,9736,9737],{"class":9650}," trials; trial",[47,9739,9740],{"class":9642},"++",[47,9742,9743],{"class":9650},") {\n",[47,9745,9747,9750,9753,9756,9759,9762,9765,9768,9771,9774,9777,9780],{"class":9638,"line":9746},5,[47,9748,9749],{"class":9642},"    const",[47,9751,9752],{"class":9658}," boxes",[47,9754,9755],{"class":9642}," =",[47,9757,9758],{"class":9650}," Array.",[47,9760,9761],{"class":9646},"from",[47,9763,9764],{"class":9650},"({ length: m }, () ",[47,9766,9767],{"class":9642},"=>",[47,9769,9770],{"class":9650}," Math.",[47,9772,9773],{"class":9646},"random",[47,9775,9776],{"class":9650},"() ",[47,9778,9779],{"class":9642},"*",[47,9781,9782],{"class":9650}," x);\n",[47,9784,9786,9788,9791,9793,9796,9799,9801,9803,9806,9809,9812,9814,9817],{"class":9638,"line":9785},6,[47,9787,9749],{"class":9642},[47,9789,9790],{"class":9658}," firstPart",[47,9792,9755],{"class":9642},[47,9794,9795],{"class":9650}," boxes.",[47,9797,9798],{"class":9646},"slice",[47,9800,387],{"class":9650},[47,9802,109],{"class":9658},[47,9804,9805],{"class":9650},", Math.",[47,9807,9808],{"class":9646},"floor",[47,9810,9811],{"class":9650},"(m ",[47,9813,9779],{"class":9642},[47,9815,9816],{"class":9658}," 0.37",[47,9818,9819],{"class":9650},"));\n",[47,9821,9823],{"class":9638,"line":9822},7,[47,9824,9709],{"emptyLinePlaceholder":9708},[47,9826,9828],{"class":9638,"line":9827},8,[47,9829,9831],{"class":9830},"sAwPA","    \u002F\u002F Remember the highest prize in the first 37% of boxes\n",[47,9833,9835,9837,9840,9842,9844,9847,9849,9851],{"class":9638,"line":9834},9,[47,9836,9749],{"class":9642},[47,9838,9839],{"class":9658}," p",[47,9841,9755],{"class":9642},[47,9843,9770],{"class":9650},[47,9845,9846],{"class":9646},"max",[47,9848,387],{"class":9650},[47,9850,285],{"class":9642},[47,9852,9853],{"class":9650},"firstPart);\n",[47,9855,9857],{"class":9638,"line":9856},10,[47,9858,9709],{"emptyLinePlaceholder":9708},[47,9860,9862],{"class":9638,"line":9861},11,[47,9863,9864],{"class":9830},"    \u002F\u002F Find the maximum prize in all boxes\n",[47,9866,9868,9870,9873,9875,9877,9879,9881,9883],{"class":9638,"line":9867},12,[47,9869,9749],{"class":9642},[47,9871,9872],{"class":9658}," maxPrize",[47,9874,9755],{"class":9642},[47,9876,9770],{"class":9650},[47,9878,9846],{"class":9646},[47,9880,387],{"class":9650},[47,9882,285],{"class":9642},[47,9884,9885],{"class":9650},"boxes);\n",[47,9887,9889],{"class":9638,"line":9888},13,[47,9890,9709],{"emptyLinePlaceholder":9708},[47,9892,9894],{"class":9638,"line":9893},14,[47,9895,9896],{"class":9830},"    \u002F\u002F Check from the 37% onward if a prize higher than p is found\n",[47,9898,9900,9903,9905,9908,9911,9914,9916,9918,9921,9923,9925,9927,9929],{"class":9638,"line":9899},15,[47,9901,9902],{"class":9642},"    for",[47,9904,9718],{"class":9650},[47,9906,9907],{"class":9642},"const",[47,9909,9910],{"class":9658}," prize",[47,9912,9913],{"class":9642}," of",[47,9915,9795],{"class":9650},[47,9917,9798],{"class":9646},[47,9919,9920],{"class":9650},"(Math.",[47,9922,9808],{"class":9646},[47,9924,9811],{"class":9650},[47,9926,9779],{"class":9642},[47,9928,9816],{"class":9658},[47,9930,9931],{"class":9650},"))) {\n",[47,9933,9935,9938,9941,9944],{"class":9638,"line":9934},16,[47,9936,9937],{"class":9642},"      if",[47,9939,9940],{"class":9650}," (prize ",[47,9942,9943],{"class":9642},">",[47,9945,9946],{"class":9650}," p) {\n",[47,9948,9950,9953,9955,9958],{"class":9638,"line":9949},17,[47,9951,9952],{"class":9642},"        if",[47,9954,9940],{"class":9650},[47,9956,9957],{"class":9642},"===",[47,9959,9960],{"class":9650}," maxPrize) {\n",[47,9962,9964,9967,9969],{"class":9638,"line":9963},18,[47,9965,9966],{"class":9650},"          successCount",[47,9968,9740],{"class":9642},[47,9970,9702],{"class":9650},[47,9972,9974],{"class":9638,"line":9973},19,[47,9975,9976],{"class":9650},"        }\n",[47,9978,9980,9983,9986],{"class":9638,"line":9979},20,[47,9981,9982],{"class":9642},"        break",[47,9984,9985],{"class":9650},"; ",[47,9987,9988],{"class":9830},"\u002F\u002F Stop once a higher prize is found\n",[47,9990,9992],{"class":9638,"line":9991},21,[47,9993,9994],{"class":9650},"      }\n",[47,9996,9998],{"class":9638,"line":9997},22,[47,9999,10000],{"class":9650},"    }\n",[47,10002,10004],{"class":9638,"line":10003},23,[47,10005,10006],{"class":9650},"  }\n",[47,10008,10010],{"class":9638,"line":10009},24,[47,10011,9709],{"emptyLinePlaceholder":9708},[47,10013,10015,10018,10020,10023,10026],{"class":9638,"line":10014},25,[47,10016,10017],{"class":9642},"  return",[47,10019,9694],{"class":9650},[47,10021,10022],{"class":9642},"\u002F",[47,10024,10025],{"class":9650}," trials; ",[47,10027,10028],{"class":9830},"\u002F\u002F Return the success probability\n",[47,10030,10032],{"class":9638,"line":10031},26,[47,10033,10034],{"class":9650},"}\n",[47,10036,10038],{"class":9638,"line":10037},27,[47,10039,9709],{"emptyLinePlaceholder":9708},[47,10041,10043],{"class":9638,"line":10042},28,[47,10044,10045],{"class":9830},"\u002F\u002F Set parameters\n",[47,10047,10049,10051,10054,10056,10059,10061],{"class":9638,"line":10048},29,[47,10050,9907],{"class":9642},[47,10052,10053],{"class":9658}," m",[47,10055,9755],{"class":9642},[47,10057,10058],{"class":9658}," 100",[47,10060,9985],{"class":9650},[47,10062,10063],{"class":9830},"\u002F\u002F Total number of boxes\n",[47,10065,10067,10069,10072,10074,10077,10079],{"class":9638,"line":10066},30,[47,10068,9907],{"class":9642},[47,10070,10071],{"class":9658}," x",[47,10073,9755],{"class":9642},[47,10075,10076],{"class":9658}," 1000",[47,10078,9985],{"class":9650},[47,10080,10081],{"class":9830},"\u002F\u002F Maximum prize in each box\n",[47,10083,10085,10087,10090,10092,10095,10097],{"class":9638,"line":10084},31,[47,10086,9907],{"class":9642},[47,10088,10089],{"class":9658}," trials",[47,10091,9755],{"class":9642},[47,10093,10094],{"class":9658}," 100000",[47,10096,9985],{"class":9650},[47,10098,10099],{"class":9830},"\u002F\u002F Number of trials\n",[47,10101,10103],{"class":9638,"line":10102},32,[47,10104,9709],{"emptyLinePlaceholder":9708},[47,10106,10108,10110,10113,10115,10117],{"class":9638,"line":10107},33,[47,10109,9907],{"class":9642},[47,10111,10112],{"class":9658}," successProbability",[47,10114,9755],{"class":9642},[47,10116,9647],{"class":9646},[47,10118,10119],{"class":9650},"(m, x, trials);\n",[47,10121,10123,10126,10129,10131,10135,10138,10141],{"class":9638,"line":10122},34,[47,10124,10125],{"class":9650},"console.",[47,10127,10128],{"class":9646},"log",[47,10130,387],{"class":9650},[47,10132,10134],{"class":10133},"sU2Wk","`Probability of winning the highest prize: ${",[47,10136,10137],{"class":9650},"successProbability",[47,10139,10140],{"class":10133},"}`",[47,10142,10143],{"class":9650},");\n",[2839,10145,10147],{"id":10146},"explanation","Explanation",[10149,10150,10151,10175,10206],"ol",{},[10152,10153,10154,6442,10157],"li",{},[36,10155,10156],{},"Parameters",[10158,10159,10160,10165,10170],"ul",{},[10152,10161,10162,10164],{},[9634,10163,69],{},": Total number of treasure boxes.",[10152,10166,10167,10169],{},[9634,10168,140],{},": Maximum prize in each box.",[10152,10171,10172,10174],{},[9634,10173,9672],{},": Number of simulation trials. More trials increase accuracy.",[10152,10176,10177,6442,10180],{},[36,10178,10179],{},"Simulation Process",[10158,10181,10182,10185,10191,10198],{},[10152,10183,10184],{},"Randomly generate the prize for each box.",[10152,10186,10187,10188,10190],{},"Calculate the maximum prize in the first 37% of the boxes (",[9634,10189,15],{},").",[10152,10192,10193,10194,10197],{},"Find the highest prize (",[9634,10195,10196],{},"maxPrize",") in all boxes.",[10152,10199,10200,10201,10203,10204,230],{},"Starting from the 38% box, check if there is a prize higher than ",[9634,10202,15],{}," and if it equals ",[9634,10205,10196],{},[10152,10207,10208,6442,10211],{},[36,10209,10210],{},"Result",[10158,10212,10213],{},[10152,10214,10215],{},"Return the probability of winning the highest prize after many trials.",[15,10217,10218],{},"Running this simulation produces a result like this:",[9628,10220,10224],{"className":10221,"code":10222,"language":10223,"meta":3110,"style":3110},"language-plaintext shiki shiki-themes github-dark","Probability of winning the highest prize: 0.367999\n","plaintext",[9634,10225,10226],{"__ignoreMap":3110},[47,10227,10228],{"class":9638,"line":9639},[47,10229,10222],{},[15,10231,10232],{},"As shown, the probability of winning the highest prize increases to around 37%, which is a significant improvement from 0.1%.",[10234,10235],"hr",{},[15,10237,10238,10239,10273,10274,10307],{},"To further verify, let's simulate the probability of winning the top ",[47,10240,10242,10258],{"className":10241,"translate":51},[50],[47,10243,10245],{"className":10244},[55],[57,10246,10247],{"xmlns":59},[61,10248,10249,10255],{},[64,10250,10251,10253],{},[67,10252,372],{},[67,10254,1064],{"mathvariant":229},[71,10256,10257],{"encoding":73},"t\\%",[47,10259,10261],{"className":10260,"ariaHidden":78},[77],[47,10262,10264,10267,10270],{"className":10263},[82],[47,10265],{"className":10266,"style":1085},[86],[47,10268,372],{"className":10269},[91,92],[47,10271,1064],{"className":10272},[91]," prizes by dividing the prize ranges into ",[47,10275,10277,10294],{"className":10276,"translate":51},[50],[47,10278,10280],{"className":10279},[55],[57,10281,10282],{"xmlns":59},[61,10283,10284,10291],{},[64,10285,10286,10289],{},[107,10287,10288],{},"10",[67,10290,1064],{"mathvariant":229},[71,10292,10293],{"encoding":73},"10\\%",[47,10295,10297],{"className":10296,"ariaHidden":78},[77],[47,10298,10300,10303],{"className":10299},[82],[47,10301],{"className":10302,"style":1085},[86],[47,10304,10306],{"className":10305},[91],"10%"," increments (e.g., top 10%, top 20%, etc.).",[2839,10309,10311],{"id":10310},"modified-typescript-code-for-different-prize-ranges","Modified TypeScript Code for Different Prize Ranges",[9628,10313,10315],{"className":9630,"code":10314,"language":9632,"meta":3110,"style":3110},"function simulateGamblingGameWithRanges(\n  m: number,\n  x: number,\n  trials: number\n): number[] {\n  const successCounts = Array(10).fill(0);\n  \u002F\u002F Array to store the number of successes in each range\n\n  for (let trial = 0; trial \u003C trials; trial++) {\n    const boxes = Array.from({ length: m }, () => Math.random() * x);\n\n    const k = Math.floor(m * 0.37);\n    const firstPart = boxes.slice(0, k);\n\n    const p = Math.max(...firstPart);\n\n    let chosenPrize = 0;\n\n    \u002F\u002F Find the first box greater than p after the first 37% boxes\n    for (const prize of boxes.slice(k)) {\n      if (prize > p) {\n        chosenPrize = prize;\n        break;\n      }\n    }\n\n    \u002F\u002F If no box greater than p is found, select the last box\n    if (chosenPrize === 0) {\n      chosenPrize = boxes[boxes.length - 1];\n    }\n\n    const sortedBoxes = [...boxes].sort((a, b) => b - a);\n\n    \u002F\u002F Determine the prize range that the chosen prize falls into\n    for (let i = 1; i \u003C= 9; i++) {\n      const thresholdPrize = sortedBoxes[Math.floor((m * i * 10) \u002F 100)];\n      if (chosenPrize >= thresholdPrize) {\n        successCounts[i - 1]++;\n      }\n    }\n\n    \u002F\u002F The top 100% prize is always chosen, so increment this count\n    successCounts[9]++;\n  }\n\n  \u002F\u002F Convert counts to probabilities\n  const successProbabilities = successCounts.map((count) => count \u002F trials);\n\n  return successProbabilities;\n}\n\n\u002F\u002F Set parameters\nconst m = 100; \u002F\u002F Number of boxes\nconst x = 1000; \u002F\u002F Maximum prize\nconst trials = 100000; \u002F\u002F Number of trials\n\nconst successProbabilities = simulateGamblingGameWithRanges(m, x, trials);\n\n\u002F\u002F Output the probabilities for each prize range\nfor (let i = 1; i \u003C= 10; i++) {\n  console.log(\n    `Probability of winning a prize in the top ${i * 10}%: ${(\n      successProbabilities[i - 1] * 100\n    ).toFixed(2)}%`\n  );\n}\n",[9634,10316,10317,10327,10339,10350,10360,10371,10399,10404,10408,10432,10458,10462,10483,10502,10506,10524,10528,10542,10546,10551,10570,10580,10590,10596,10600,10604,10608,10613,10627,10649,10653,10657,10700,10704,10709,10741,10778,10791,10808,10813,10818,10823,10829,10844,10849,10854,10860,10893,10898,10906,10911,10916,10921,10937,10953,10968,10973,10986,10991,10997,11025,11035,11054,11077,11096,11102],{"__ignoreMap":3110},[47,10318,10319,10321,10324],{"class":9638,"line":9639},[47,10320,9643],{"class":9642},[47,10322,10323],{"class":9646}," simulateGamblingGameWithRanges",[47,10325,10326],{"class":9650},"(\n",[47,10328,10329,10332,10334,10336],{"class":9638,"line":9688},[47,10330,10331],{"class":9653},"  m",[47,10333,6442],{"class":9642},[47,10335,9659],{"class":9658},[47,10337,10338],{"class":9650},",\n",[47,10340,10341,10344,10346,10348],{"class":9638,"line":9705},[47,10342,10343],{"class":9653},"  x",[47,10345,6442],{"class":9642},[47,10347,9659],{"class":9658},[47,10349,10338],{"class":9650},[47,10351,10352,10355,10357],{"class":9638,"line":9712},[47,10353,10354],{"class":9653},"  trials",[47,10356,6442],{"class":9642},[47,10358,10359],{"class":9658}," number\n",[47,10361,10362,10364,10366,10368],{"class":9638,"line":9746},[47,10363,405],{"class":9650},[47,10365,6442],{"class":9642},[47,10367,9659],{"class":9658},[47,10369,10370],{"class":9650},"[] {\n",[47,10372,10373,10376,10379,10381,10384,10386,10388,10390,10393,10395,10397],{"class":9638,"line":9785},[47,10374,10375],{"class":9642},"  const",[47,10377,10378],{"class":9658}," successCounts",[47,10380,9755],{"class":9642},[47,10382,10383],{"class":9646}," Array",[47,10385,387],{"class":9650},[47,10387,10288],{"class":9658},[47,10389,10190],{"class":9650},[47,10391,10392],{"class":9646},"fill",[47,10394,387],{"class":9650},[47,10396,109],{"class":9658},[47,10398,10143],{"class":9650},[47,10400,10401],{"class":9638,"line":9822},[47,10402,10403],{"class":9830},"  \u002F\u002F Array to store the number of successes in each range\n",[47,10405,10406],{"class":9638,"line":9827},[47,10407,9709],{"emptyLinePlaceholder":9708},[47,10409,10410,10412,10414,10416,10418,10420,10422,10424,10426,10428,10430],{"class":9638,"line":9834},[47,10411,9715],{"class":9642},[47,10413,9718],{"class":9650},[47,10415,9721],{"class":9642},[47,10417,9724],{"class":9650},[47,10419,375],{"class":9642},[47,10421,9699],{"class":9658},[47,10423,9731],{"class":9650},[47,10425,9734],{"class":9642},[47,10427,9737],{"class":9650},[47,10429,9740],{"class":9642},[47,10431,9743],{"class":9650},[47,10433,10434,10436,10438,10440,10442,10444,10446,10448,10450,10452,10454,10456],{"class":9638,"line":9856},[47,10435,9749],{"class":9642},[47,10437,9752],{"class":9658},[47,10439,9755],{"class":9642},[47,10441,9758],{"class":9650},[47,10443,9761],{"class":9646},[47,10445,9764],{"class":9650},[47,10447,9767],{"class":9642},[47,10449,9770],{"class":9650},[47,10451,9773],{"class":9646},[47,10453,9776],{"class":9650},[47,10455,9779],{"class":9642},[47,10457,9782],{"class":9650},[47,10459,10460],{"class":9638,"line":9861},[47,10461,9709],{"emptyLinePlaceholder":9708},[47,10463,10464,10466,10469,10471,10473,10475,10477,10479,10481],{"class":9638,"line":9867},[47,10465,9749],{"class":9642},[47,10467,10468],{"class":9658}," k",[47,10470,9755],{"class":9642},[47,10472,9770],{"class":9650},[47,10474,9808],{"class":9646},[47,10476,9811],{"class":9650},[47,10478,9779],{"class":9642},[47,10480,9816],{"class":9658},[47,10482,10143],{"class":9650},[47,10484,10485,10487,10489,10491,10493,10495,10497,10499],{"class":9638,"line":9888},[47,10486,9749],{"class":9642},[47,10488,9790],{"class":9658},[47,10490,9755],{"class":9642},[47,10492,9795],{"class":9650},[47,10494,9798],{"class":9646},[47,10496,387],{"class":9650},[47,10498,109],{"class":9658},[47,10500,10501],{"class":9650},", k);\n",[47,10503,10504],{"class":9638,"line":9893},[47,10505,9709],{"emptyLinePlaceholder":9708},[47,10507,10508,10510,10512,10514,10516,10518,10520,10522],{"class":9638,"line":9899},[47,10509,9749],{"class":9642},[47,10511,9839],{"class":9658},[47,10513,9755],{"class":9642},[47,10515,9770],{"class":9650},[47,10517,9846],{"class":9646},[47,10519,387],{"class":9650},[47,10521,285],{"class":9642},[47,10523,9853],{"class":9650},[47,10525,10526],{"class":9638,"line":9934},[47,10527,9709],{"emptyLinePlaceholder":9708},[47,10529,10530,10533,10536,10538,10540],{"class":9638,"line":9949},[47,10531,10532],{"class":9642},"    let",[47,10534,10535],{"class":9650}," chosenPrize ",[47,10537,375],{"class":9642},[47,10539,9699],{"class":9658},[47,10541,9702],{"class":9650},[47,10543,10544],{"class":9638,"line":9963},[47,10545,9709],{"emptyLinePlaceholder":9708},[47,10547,10548],{"class":9638,"line":9973},[47,10549,10550],{"class":9830},"    \u002F\u002F Find the first box greater than p after the first 37% boxes\n",[47,10552,10553,10555,10557,10559,10561,10563,10565,10567],{"class":9638,"line":9979},[47,10554,9902],{"class":9642},[47,10556,9718],{"class":9650},[47,10558,9907],{"class":9642},[47,10560,9910],{"class":9658},[47,10562,9913],{"class":9642},[47,10564,9795],{"class":9650},[47,10566,9798],{"class":9646},[47,10568,10569],{"class":9650},"(k)) {\n",[47,10571,10572,10574,10576,10578],{"class":9638,"line":9991},[47,10573,9937],{"class":9642},[47,10575,9940],{"class":9650},[47,10577,9943],{"class":9642},[47,10579,9946],{"class":9650},[47,10581,10582,10585,10587],{"class":9638,"line":9997},[47,10583,10584],{"class":9650},"        chosenPrize ",[47,10586,375],{"class":9642},[47,10588,10589],{"class":9650}," prize;\n",[47,10591,10592,10594],{"class":9638,"line":10003},[47,10593,9982],{"class":9642},[47,10595,9702],{"class":9650},[47,10597,10598],{"class":9638,"line":10009},[47,10599,9994],{"class":9650},[47,10601,10602],{"class":9638,"line":10014},[47,10603,10000],{"class":9650},[47,10605,10606],{"class":9638,"line":10031},[47,10607,9709],{"emptyLinePlaceholder":9708},[47,10609,10610],{"class":9638,"line":10037},[47,10611,10612],{"class":9830},"    \u002F\u002F If no box greater than p is found, select the last box\n",[47,10614,10615,10618,10621,10623,10625],{"class":9638,"line":10042},[47,10616,10617],{"class":9642},"    if",[47,10619,10620],{"class":9650}," (chosenPrize ",[47,10622,9957],{"class":9642},[47,10624,9699],{"class":9658},[47,10626,9743],{"class":9650},[47,10628,10629,10632,10634,10637,10640,10643,10646],{"class":9638,"line":10048},[47,10630,10631],{"class":9650},"      chosenPrize ",[47,10633,375],{"class":9642},[47,10635,10636],{"class":9650}," boxes[boxes.",[47,10638,10639],{"class":9658},"length",[47,10641,10642],{"class":9642}," -",[47,10644,10645],{"class":9658}," 1",[47,10647,10648],{"class":9650},"];\n",[47,10650,10651],{"class":9638,"line":10066},[47,10652,10000],{"class":9650},[47,10654,10655],{"class":9638,"line":10084},[47,10656,9709],{"emptyLinePlaceholder":9708},[47,10658,10659,10661,10664,10666,10669,10671,10674,10677,10680,10682,10684,10686,10689,10691,10694,10697],{"class":9638,"line":10102},[47,10660,9749],{"class":9642},[47,10662,10663],{"class":9658}," sortedBoxes",[47,10665,9755],{"class":9642},[47,10667,10668],{"class":9650}," [",[47,10670,285],{"class":9642},[47,10672,10673],{"class":9650},"boxes].",[47,10675,10676],{"class":9646},"sort",[47,10678,10679],{"class":9650},"((",[47,10681,358],{"class":9653},[47,10683,2466],{"class":9650},[47,10685,1234],{"class":9653},[47,10687,10688],{"class":9650},") ",[47,10690,9767],{"class":9642},[47,10692,10693],{"class":9650}," b ",[47,10695,10696],{"class":9642},"-",[47,10698,10699],{"class":9650}," a);\n",[47,10701,10702],{"class":9638,"line":10107},[47,10703,9709],{"emptyLinePlaceholder":9708},[47,10705,10706],{"class":9638,"line":10122},[47,10707,10708],{"class":9830},"    \u002F\u002F Determine the prize range that the chosen prize falls into\n",[47,10710,10712,10714,10716,10718,10721,10723,10725,10728,10731,10734,10737,10739],{"class":9638,"line":10711},35,[47,10713,9902],{"class":9642},[47,10715,9718],{"class":9650},[47,10717,9721],{"class":9642},[47,10719,10720],{"class":9650}," i ",[47,10722,375],{"class":9642},[47,10724,10645],{"class":9658},[47,10726,10727],{"class":9650},"; i ",[47,10729,10730],{"class":9642},"\u003C=",[47,10732,10733],{"class":9658}," 9",[47,10735,10736],{"class":9650},"; i",[47,10738,9740],{"class":9642},[47,10740,9743],{"class":9650},[47,10742,10744,10747,10750,10752,10755,10757,10760,10762,10764,10766,10769,10771,10773,10775],{"class":9638,"line":10743},36,[47,10745,10746],{"class":9642},"      const",[47,10748,10749],{"class":9658}," thresholdPrize",[47,10751,9755],{"class":9642},[47,10753,10754],{"class":9650}," sortedBoxes[Math.",[47,10756,9808],{"class":9646},[47,10758,10759],{"class":9650},"((m ",[47,10761,9779],{"class":9642},[47,10763,10720],{"class":9650},[47,10765,9779],{"class":9642},[47,10767,10768],{"class":9658}," 10",[47,10770,10688],{"class":9650},[47,10772,10022],{"class":9642},[47,10774,10058],{"class":9658},[47,10776,10777],{"class":9650},")];\n",[47,10779,10781,10783,10785,10788],{"class":9638,"line":10780},37,[47,10782,9937],{"class":9642},[47,10784,10620],{"class":9650},[47,10786,10787],{"class":9642},">=",[47,10789,10790],{"class":9650}," thresholdPrize) {\n",[47,10792,10794,10797,10799,10801,10804,10806],{"class":9638,"line":10793},38,[47,10795,10796],{"class":9650},"        successCounts[i ",[47,10798,10696],{"class":9642},[47,10800,10645],{"class":9658},[47,10802,10803],{"class":9650},"]",[47,10805,9740],{"class":9642},[47,10807,9702],{"class":9650},[47,10809,10811],{"class":9638,"line":10810},39,[47,10812,9994],{"class":9650},[47,10814,10816],{"class":9638,"line":10815},40,[47,10817,10000],{"class":9650},[47,10819,10821],{"class":9638,"line":10820},41,[47,10822,9709],{"emptyLinePlaceholder":9708},[47,10824,10826],{"class":9638,"line":10825},42,[47,10827,10828],{"class":9830},"    \u002F\u002F The top 100% prize is always chosen, so increment this count\n",[47,10830,10832,10835,10838,10840,10842],{"class":9638,"line":10831},43,[47,10833,10834],{"class":9650},"    successCounts[",[47,10836,10837],{"class":9658},"9",[47,10839,10803],{"class":9650},[47,10841,9740],{"class":9642},[47,10843,9702],{"class":9650},[47,10845,10847],{"class":9638,"line":10846},44,[47,10848,10006],{"class":9650},[47,10850,10852],{"class":9638,"line":10851},45,[47,10853,9709],{"emptyLinePlaceholder":9708},[47,10855,10857],{"class":9638,"line":10856},46,[47,10858,10859],{"class":9830},"  \u002F\u002F Convert counts to probabilities\n",[47,10861,10863,10865,10868,10870,10873,10876,10878,10881,10883,10885,10888,10890],{"class":9638,"line":10862},47,[47,10864,10375],{"class":9642},[47,10866,10867],{"class":9658}," successProbabilities",[47,10869,9755],{"class":9642},[47,10871,10872],{"class":9650}," successCounts.",[47,10874,10875],{"class":9646},"map",[47,10877,10679],{"class":9650},[47,10879,10880],{"class":9653},"count",[47,10882,10688],{"class":9650},[47,10884,9767],{"class":9642},[47,10886,10887],{"class":9650}," count ",[47,10889,10022],{"class":9642},[47,10891,10892],{"class":9650}," trials);\n",[47,10894,10896],{"class":9638,"line":10895},48,[47,10897,9709],{"emptyLinePlaceholder":9708},[47,10899,10901,10903],{"class":9638,"line":10900},49,[47,10902,10017],{"class":9642},[47,10904,10905],{"class":9650}," successProbabilities;\n",[47,10907,10909],{"class":9638,"line":10908},50,[47,10910,10034],{"class":9650},[47,10912,10914],{"class":9638,"line":10913},51,[47,10915,9709],{"emptyLinePlaceholder":9708},[47,10917,10919],{"class":9638,"line":10918},52,[47,10920,10045],{"class":9830},[47,10922,10924,10926,10928,10930,10932,10934],{"class":9638,"line":10923},53,[47,10925,9907],{"class":9642},[47,10927,10053],{"class":9658},[47,10929,9755],{"class":9642},[47,10931,10058],{"class":9658},[47,10933,9985],{"class":9650},[47,10935,10936],{"class":9830},"\u002F\u002F Number of boxes\n",[47,10938,10940,10942,10944,10946,10948,10950],{"class":9638,"line":10939},54,[47,10941,9907],{"class":9642},[47,10943,10071],{"class":9658},[47,10945,9755],{"class":9642},[47,10947,10076],{"class":9658},[47,10949,9985],{"class":9650},[47,10951,10952],{"class":9830},"\u002F\u002F Maximum prize\n",[47,10954,10956,10958,10960,10962,10964,10966],{"class":9638,"line":10955},55,[47,10957,9907],{"class":9642},[47,10959,10089],{"class":9658},[47,10961,9755],{"class":9642},[47,10963,10094],{"class":9658},[47,10965,9985],{"class":9650},[47,10967,10099],{"class":9830},[47,10969,10971],{"class":9638,"line":10970},56,[47,10972,9709],{"emptyLinePlaceholder":9708},[47,10974,10976,10978,10980,10982,10984],{"class":9638,"line":10975},57,[47,10977,9907],{"class":9642},[47,10979,10867],{"class":9658},[47,10981,9755],{"class":9642},[47,10983,10323],{"class":9646},[47,10985,10119],{"class":9650},[47,10987,10989],{"class":9638,"line":10988},58,[47,10990,9709],{"emptyLinePlaceholder":9708},[47,10992,10994],{"class":9638,"line":10993},59,[47,10995,10996],{"class":9830},"\u002F\u002F Output the probabilities for each prize range\n",[47,10998,11000,11003,11005,11007,11009,11011,11013,11015,11017,11019,11021,11023],{"class":9638,"line":10999},60,[47,11001,11002],{"class":9642},"for",[47,11004,9718],{"class":9650},[47,11006,9721],{"class":9642},[47,11008,10720],{"class":9650},[47,11010,375],{"class":9642},[47,11012,10645],{"class":9658},[47,11014,10727],{"class":9650},[47,11016,10730],{"class":9642},[47,11018,10768],{"class":9658},[47,11020,10736],{"class":9650},[47,11022,9740],{"class":9642},[47,11024,9743],{"class":9650},[47,11026,11028,11031,11033],{"class":9638,"line":11027},61,[47,11029,11030],{"class":9650},"  console.",[47,11032,10128],{"class":9646},[47,11034,10326],{"class":9650},[47,11036,11038,11041,11044,11047,11049,11052],{"class":9638,"line":11037},62,[47,11039,11040],{"class":10133},"    `Probability of winning a prize in the top ${",[47,11042,11043],{"class":9650},"i",[47,11045,11046],{"class":9642}," *",[47,11048,10768],{"class":9658},[47,11050,11051],{"class":10133},"}%: ${",[47,11053,10326],{"class":10133},[47,11055,11057,11060,11063,11065,11067,11069,11072,11074],{"class":9638,"line":11056},63,[47,11058,11059],{"class":9650},"      successProbabilities",[47,11061,11062],{"class":10133},"[",[47,11064,11043],{"class":9650},[47,11066,10642],{"class":9642},[47,11068,10645],{"class":9658},[47,11070,11071],{"class":10133},"] ",[47,11073,9779],{"class":9642},[47,11075,11076],{"class":9658}," 100\n",[47,11078,11080,11083,11086,11088,11091,11093],{"class":9638,"line":11079},64,[47,11081,11082],{"class":10133},"    ).",[47,11084,11085],{"class":9646},"toFixed",[47,11087,387],{"class":10133},[47,11089,11090],{"class":9658},"2",[47,11092,405],{"class":10133},[47,11094,11095],{"class":10133},"}%`\n",[47,11097,11099],{"class":9638,"line":11098},65,[47,11100,11101],{"class":9650},"  );\n",[47,11103,11105],{"class":9638,"line":11104},66,[47,11106,10034],{"class":9650},[2839,11108,11110],{"id":11109},"results-from-the-program","Results from the Program",[9628,11112,11114],{"className":10221,"code":11113,"language":10223,"meta":3110,"style":3110},"Probability of winning a prize in the top 10%: 66.96%\nProbability of winning a prize in the top 20%: 70.67%\nProbability of winning a prize in the top 30%: 74.42%\nProbability of winning a prize in the top 40%: 78.06%\nProbability of winning a prize in the top 50%: 81.70%\nProbability of winning a prize in the top 60%: 85.45%\nProbability of winning a prize in the top 70%: 89.14%\nProbability of winning a prize in the top 80%: 92.84%\nProbability of winning a prize in the top 90%: 96.60%\nProbability of winning a prize in the top 100%: 100.00%\n",[9634,11115,11116,11121,11126,11131,11136,11141,11146,11151,11156,11161],{"__ignoreMap":3110},[47,11117,11118],{"class":9638,"line":9639},[47,11119,11120],{},"Probability of winning a prize in the top 10%: 66.96%\n",[47,11122,11123],{"class":9638,"line":9688},[47,11124,11125],{},"Probability of winning a prize in the top 20%: 70.67%\n",[47,11127,11128],{"class":9638,"line":9705},[47,11129,11130],{},"Probability of winning a prize in the top 30%: 74.42%\n",[47,11132,11133],{"class":9638,"line":9712},[47,11134,11135],{},"Probability of winning a prize in the top 40%: 78.06%\n",[47,11137,11138],{"class":9638,"line":9746},[47,11139,11140],{},"Probability of winning a prize in the top 50%: 81.70%\n",[47,11142,11143],{"class":9638,"line":9785},[47,11144,11145],{},"Probability of winning a prize in the top 60%: 85.45%\n",[47,11147,11148],{"class":9638,"line":9822},[47,11149,11150],{},"Probability of winning a prize in the top 70%: 89.14%\n",[47,11152,11153],{"class":9638,"line":9827},[47,11154,11155],{},"Probability of winning a prize in the top 80%: 92.84%\n",[47,11157,11158],{"class":9638,"line":9834},[47,11159,11160],{},"Probability of winning a prize in the top 90%: 96.60%\n",[47,11162,11163],{"class":9638,"line":9856},[47,11164,11165],{},"Probability of winning a prize in the top 100%: 100.00%\n",[15,11167,11168],{},"We see that, using this strategy, we maximize the chances of getting a high-value prize.",[10,11170,11172],{"id":11171},"conclusion","Conclusion",[15,11174,11175,11176,230],{},"The simulation verifies that selecting 37% of the boxes first and then choosing the next box with a prize greater than the maximum from the first 37% is the optimal strategy. This maximizes the probability of winning the highest prize, known as the ",[36,11177,11178],{},"37% rule",[15,11180,11181],{},"This strategy can also be applied in real-life scenarios, such as making optimal choices in uncertain situations.",[11183,11184,11185],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .snl16, html code.shiki .snl16{--shiki-default:#F97583}html pre.shiki code .svObZ, html code.shiki .svObZ{--shiki-default:#B392F0}html pre.shiki code .s95oV, html code.shiki .s95oV{--shiki-default:#E1E4E8}html pre.shiki code .s9osk, html code.shiki .s9osk{--shiki-default:#FFAB70}html pre.shiki code .sDLfK, html code.shiki .sDLfK{--shiki-default:#79B8FF}html pre.shiki code .sAwPA, html code.shiki .sAwPA{--shiki-default:#6A737D}html pre.shiki code .sU2Wk, html code.shiki .sU2Wk{--shiki-default:#9ECBFF}",{"title":3110,"searchDepth":9688,"depth":9688,"links":11187},[11188,11189,11190,11191,11198,11203],{"id":12,"depth":9688,"text":13},{"id":41,"depth":9688,"text":42},{"id":1990,"depth":9688,"text":1991},{"id":2745,"depth":9688,"text":2746,"children":11192},[11193,11194,11195,11196,11197],{"id":2841,"depth":9705,"text":2842},{"id":4002,"depth":9705,"text":4003},{"id":5712,"depth":9705,"text":5713},{"id":6219,"depth":9705,"text":6220},{"id":9291,"depth":9705,"text":9292},{"id":9619,"depth":9688,"text":9620,"children":11199},[11200,11201,11202],{"id":10146,"depth":9705,"text":10147},{"id":10310,"depth":9705,"text":10311},{"id":11109,"depth":9705,"text":11110},{"id":11171,"depth":9688,"text":11172},false,"md",null,{"author":3110},"2024-09-21","\u002Fblog\u002F2024\u002F2024-09-21-the-37-rules-in-making-choice",{"title":5,"description":3110},"blog\u002F2024\u002F2024-09-21-the-37-rules-in-making-choice","This article proposes the classic 37% rule in game theory and its mathematical derivation process.",[11214,11215],"investment","game-theory","JJ3gvoV7gVToCYH24uwU6sfLjI6BD-nPvZ01LBxzqas",1777090296782]