完本神战

Chapter 3 Historical sketch（第1页）

Chapter3Historicalsketch

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%oftheballsinaherestareBlaeballisdrawnatrandom。Thatballisrepladtheexperimeimes。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

&ofoutesofaofBernoullitrialsasasequenceofSudFailures，e。g。FFFSFFFSSFSFF。。。NowreplaceeachSbythenumberone，andeachFbyzero，giving0001000110100。。。ThisindicatesagwaytothinkaboutthetotalnumberofSuthesetrials:itisjustthe　sumofthesenumbers（agreed?）。DeMivenagoodapproximationthatdescribedhowthissumwhisso-alcurve。