五二小说网

手机浏览器扫描二维码访问

Chapter 3 Historical sketch（第1页）

Chapter3Historicalsketch

banner"

>

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