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Chapter3Historicalsketch
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Beginnings
AgamepopularinFlorend1600restedoalsthreeordihescoresofThree，whenalldicese，aheyallscoredsix，aroserarely，withmostsearthemiddlee.YoushouldcheckthattherearesixdifferentwaysNine（e.g.6+2+1，5+2+2，etdalsosixwaysTen.Itwasothis‘ought’tomaketotalsofenequallyfrequent，butplayers，overaperiodoftime，thetotalofTenoccurredappreciablymoreoftenthaheyaskedGalileoforaion.
Galileopoihattheirmethodofgwasflawed.ColourthediceasRed，Green，andBlue，andlisttheoutesinthatorder.Tosefrom3+3+3requiresallthreedievalue，andthatinonewayonly，（3，3，3）.Butthe5+2+2binationcouldariseas（5，2，2），（2，5，2），
or（2，2，5），sothisbinatiooarisethreetimesasoftenastheformer;and6+2+1arisesvia（6，2，1），（6，1，2），（2，6，1），（2，1，6），（1，6，2），and（1，2，6），sothisbinationhassixwaystooccur.AvalidapproachtohowoftethedifferenttotalstakesthisfatoaddoesiomorewaysofobtaihaheFlamblerslearallessoninprobability–youmustlearntot properly.
Inthesummerof1654，PasParis）a（inToulouse）hadaersonthe problemofpoints.SupposeSmithaoplayaseriesoftests，thevigthefirsttames;ueiheustehleadsJonesby2-1.Howshouldtheprizebesplit?
Suchquestionshadbeeleast150yearswithoutasatisfaswer，butPasdFermatilyfou，fetsdahetestwasabandoned，woulddividetheprize fairlybetweeookdifferentapproaches，butreachedthesame，andeachshoweredpraiseoherforhisbrilliahespestated，thesplitshouldbeiio3:1，withSmithgetting34oftheprize，Jones14.
Theesseionosethatbothplayerswereequallylikelytowinaheyanyofthepossibleoutesofthesehypotheticalgameswouldgiveoverallvictorytoeitherplayer，andproposeddividingtheprizeiioofthesetwonumbers.Ilaheprizeshouldbesplitastheratioofthetwo probabilitiesofeitherplayerwinningtheseries，assumingtheywereeveuregames.Thesystematicstudyofprobabilityhadbegun.
Thisissuewassettledviatheobjectiveapproachtoprobability，butPascalalsothoughtmorewidely.HesuggestedawagerabouttheexistenceofGod.‘Godis，orisnot.Reasonswer.AgameisoherendofaandHeadsorTailsisgoingtoturnup.illyoubet?’
&hatifGodexists，thediffereweenbeliefabetweenattaininginfinitehappinessiernaldamnationinHell.IfGoddoes，belieforuoonlyminordifferehlyexperiehusanagnosticshouldleanstronglytobeliefinGod.
Inthisgame，thevaluesofthecesof‘Heads’or‘Tails’arepersoderivablefromsymmetryarguments.ThusPascaliohesubjectiveapproachtoprobabilitytoo.
TheSwissFamilyBernoulli
Durihauries，membersoftheBernoullifamilyfromBaslemadesignifimathematigprobability.Rivalryur:ohemwouldposeges，anotherwouldrespoorofthegewouldclaimtofindflawsinthesupposedsolution，andsoon.
Gamesofspiredmuchoftheearlyihewsofprobability.Inthesegames，beitrollingdigcards，e‘experiment’iscarriedoutrepeatedlyuhesames.Thenaturalquestion，raisedearlier，is:howdoesthe observedfrequenerelatetoits objectiveprobability?
JaoulligaveananswerinhisposthumouslypublishedTheArtofjeg（1713），ratedbyhisexample.Suppose60%oftheballsinaherestareBlaeballisdrawnatrandoThatballisrepladtheexperimeimes.Bernoullishowedthat，solongasatleast25，55saremade，foreverytimetheproportionofWhiteballsfalls outsidetherangefrom58%to62%，itwillfall irahousandtimes.Informally，theobservedfrequencyofWhiteballsis，inthelongrun，lylikelytobeclosetoitsobjectiveprobability.
Asimilaranalysisappliestoahatberepeatedielyuiditioheresultofohasheothers.Eachtime，esdeheirobjectiveprobabilityissomefixedvalue p.（Thisnotionhelabel Bernoullitrials.）Takeanyinterval，assmallasyoulike，aroundthevalue p–plusorminus2%，plusorminus0.1%，itmattersnot.Also，sayhowmuchmoreoftenyouwanttherunningfrequencyofSuccessestobeierval，rathertha–ahuen，amillioever.Bernoulli’smethodsshowthatanysudalwaysbemet，providedtheexperimeeheobservedfrequencywillbeascloseto
&iveprobabilityasyoulike，givea.Thisassertionisknownasthe Lawehefamily’sfamewashonouredin1975bythenamechoice‘TheBernoulliSociety’foraioyoseistofosteradvahestudyofprobabilityaicalstatistics.
AbrahamdeMoivre
DeMoivresettledinEnglandasaHuguenee，andmadealivingfromdfromhisknowledgeofprobability.Isaa，thenover50yearsoldandwithmanye，deflequiriesaboutmathematicswiththewotoMrdeMoivre，hekhihanIdo.’DeMoivre’sDoeofcesappearedinEnglishin1718，aion，in1738，edamajoradvanoulli’sreciatewhathedid，sidersomethingspecific:ifafairdieisrolled1，000times，howfarfromtheaveragefrequeneumberofSixestobe?
DeMoivredevelopedasimpleformulathatwaswidelyusefulforquestionsofthisnature.OneofhissuperbinsightswastorealizethatthedeviatioualnumberofSixesfromtheaverageexpectedwasbestdescribedbyparingittothe squarerootofthenumberofrolls.
Itishardtooverplaythesighisdiscovery.Wheanopinionpollhasputsupportforapoliticalpartyat40%，itisoftenapaniedbyaremihisisoe，butthatthetruevalueis‘verylikely’tobeinselike38%to42%.Thewidthofsugetellsyouaboutthepreoftheinitialfigureof40%，andifyouwanthigherpre，youneedalargersample:thissquarerootfaeansthatto doublethepre，thesampleobe fourtimesaslarge!Wehavealawofdimihaveodotemustspendfourtimesasmuch.
DeMoivre’sapproabeillustratedbylookingathowmanyHeadswillotwentythrowsofafair.Takingallsequeh20suchasHHHTH...HTHTasequallylikely，westruct Figure 1，wheretheheightsoftheverticalbarsshowhowmanyoftheonemillionorsodifferentsequencesproduceexactly0，1，2，...，19，20Heads.Therespectiveobjectiveprobabilitiesarethenproportios.DeMoivreshowedthatthebest-fittinghthetopsofthesebarsisveryclosetoaparti，nowoftehe normal distribution.
Acurveofthisnaturearisesfenumberofthrows，andalsowhentheceofHeadsdiffersfromonehalf.Allthesecurvesbearasimplerelatioher，sodeMoivrecouldprodugleableforjustonebasicduseiteverywhere.AgoodestimateoftheproportioheoverallfrequencyofSuccesseswouldbewithiainlimitsoweasilybefound–allthatwasheceofSudtheimestheexperimentwastobeducted.Ytorollafairdie200timesandyouwanttoknowhowlikelyitisthatthenumberofSixeswillbebetween
&ivefrequenciesofHeadsin20throws
30and40?OrhowlikelyisitthatafairwillfallHeadsmorethan60timesin100tosses?Noproblem–deMoivrehadthesolution.
Supposeweknowtheagesofdeathfroupofmen，allofwhomreachedatleasttheirfiftiethbirthday.DeMoivre’sworkswerthequestion:‘Ifamanaged50ismorelikelythannottodiebef70，howlikelyisitthatthefiguresobservedfroupwouldarise?’Usefulthoughthiswas，itdidhekeyquestiohelifeiry:‘Howsurewebethata50-year-oldmahannottodiebeforehereachestheageof70?’
Inverseprobability
TheideasofThomasBayes，aPresbyterianministerwhodabblediics，arefarbetterappreowthaniime.His Essaytowardssolvingaproblemirineofces，publishedin1764，threeyearsafterhedied，givesthebeginningsofageneralapproachtosubjectiveprobability，andawaytheactuaries’problemabprobabilitiesfromdata.Italsoinessentialtwithprobabilities，termedBayes’Rule.
Toillustratethelatter，supposewethrowafairdietwithatthesthefirstthrowisthree，itiseasytofihatthetht，asthishappenspreciselywhenfiveissthesedthroause，wegivetheanswer16.Butturntheproblemround，andask:givealscht，whatisthecethatthefirstthrowyieldedthree?Theanswerisfarlessobvious，butbefoundbyapplyingBayes’Rule.Uandardmodelofdicethrows，thesouttobe15.
Thisnotionof inverseprobabilityistraltothewayevidenceshouldbesideredinaltrials.Supposefisfoundataeareidentifiedasbelongingtoaknownindividual，Smith.Theprobabilityoffindingthisevidehisi，islikelytobeverylow.Butitisnot‘HowlikelyisthisevideSmithisitheCourtpassesjudgmenton:itis‘HowlikelyisSmithtobei，giventhisevidence?’Bayes’RuleistheonlysoundwaytoobtainanansillseeiershowthisRulehelpsinmakingsensibledes.
&sshownbyBayeswereoverlookedformahedidideralproblem:iftheceofSuaseriesofBernoullitrials，likedicethrows，isunknowiverialsandSuccessesareknown，howlikelyisitthatthisunknowncefallsbetwees?Laplace，afarsuperiormathemati，wasabletocarryouttheputationsthathaddefeatedBayes.
Fromtentativebeginningsin1774toasynthesisin1812，Laplacesteadilyimprovedhisanalysis，aformulaetoaion.Forexample，usingdataonthenumbersofmaleahsinParis，hecludedthatitwasbeyonddoubtthatthealebirthexceededthatforafemale–heputtheprobabilitythiswasfalseasabout10–42!
BayesiheLoeryofBunhillFields，atisticalSociety.Thevaulthasbeeored，anddisplaysatributetoBayespaidforbystatistisworldwide.
&ralLimitTheorem
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