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Chapter 2 The workings of probability（第1页）

Chapter2Thewsofprobability

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&hesubjective，objedfrequentistapproachestoprobability，thereareotherstandpoints.Forexample，shouldonealwaysinsistonassogaprobabilitywithaitbeenoughtosaythatoneprobabilitywasgreater，reeofbeliefwasmoreihananother?Andshouldwenecessarilyofferaniofaxioms–self-evidenttruths–onwhichtoerectatheory?

Manydistiershavefeltitusefultohavetroaefreesofbeliefandoneforobjectiveprobabilities.Bothwouldhavethesamerulesoflogic，freefromtradis，buthorobabilitieswerearrivedat，aerpreted，coulddiffer.Anytheoryshouldbetwiththeclassicalview，basedoableexperimentswithequallylikelyoutes.Sowewillfothatcase，seekingahenotionofprobabilitymustobey.

&ionLaw

Dealoneawell-shuffledpack.Wetakeallcardsasequallylikely，sotheprobabilityofa，suchasobtainingaClub，oraSpade，oranAdbygtheproportionofallpossibleoutesthatleadtothoseevents.Howmightwefindtheprobabilitythat　eitheroftwosutsoccur?Ifthoseeveesinotheyaremutuallyexclusive，or　disjois‘GetaSpade’aaClub’aredisjoint，buttheeveaSpade’aadisjoint，astheAceofSpadesbelongstoboth.Whesaremutuallyexclusive，thealesthatleadtoeithereventisjustthesumofthenumbersforeatseparately，soleresult:　whewoeveuallyexclusive，

theprobabilitythatatleastoneoccursisthesumoftheirindividualprobabilities.

Thisisthe　AdditionLaw.Itplainlyholdsisouldtaketheclassicalview:usingtheballsinabaganalogy，itisthesameassayingthatthetotalnumberofballsthatareeitherRedorBlueisthesumofthenumberofRedballsandthenumberofBlueballs.Andiableexperiment，suchasrollingdice，

roulettewheels，thesumoftheindividualfrequewodisjoisisihefrequencythatatleastohemoccurs.SofrequentistsaccepttheAdditionLawtoo.

AlsoasubjectivistacceptsthisLaw.Forotherwise，therewouldbetwodisjois，callthemAadidnothold.Inthatcase，thesubjectivistcouldbetedbythreebets:oA，oB，aeitherAorB，andteaitsownasfair.　Buthecouldbeguaraolosemoneyifallthreebetswerestruck!TheAdditionLawforbidsthisincy.

ThisAdditiooaas，providednotwoofthemhaveanyoutesinon–theyarepairwisedisjoint.Theprobabilitythatatleastoneamongevenmillionsofpairwisedisjoisoccursisjustthesumoftheirindividualprobabilities.Butsupposetheesise:forexample，tossinganordinaryrepeatedlyuntilHeadsappearsforthefirsttime.

Thepossibleoutesofthisexperimeheunendinglist{1，2，3，4，….}，

eachwithitsownnon-zeroprobability.Whatisthecethatwetakesome　evehrowstogetaHead?Thateventhappenswhenaes{2，4，6，8，....}happen.Couldweputeitsprobabilitybyaddingupthedingindividualprobabilities?

Thereisicaldiffidoingthisaddingup，butthatafallsoutsidethescopeoftheclassicalviewofprobability，whichdealsonlywitha　fiofoutes.ThereisowhethertheAdditionLawforsudinglistshouldbepartsofprobability.Infavourofingit，wemaybeabletofindtheprobabilitiesofawiderclassofeventsthanwithoutit;againstin，asitisnotpartoftheclassicaltheory，weshouldbecautiousabouttakimighthavehiddenpitfalls.Thereisnanswer.

I’mapragmatist.IamttoexteheAdditionLawinthisway，andIhaveunfortablewiththeresultsofdoingso.Thispositionisastandardpartoftheatgiveninmostbooksusedtoteachthesubjeiversity.ButdeFiookthecautiousviewtoavoidmakiension，andothershavefeltthesame>

&ipliLaw

Ifyoutossanordinary，youwillexpecttoguessHeadsorTailscorrectlyhalfthetime.IfyoushuffleadeckofdpredictwhetherthetopcardisRedorBlack，youalsoexpecttobecorrecthalfthetime.Whehatossandacardcolour，howlikelyareyoutoget　bothcorrect?

Thinkofgthisdoubleexperimentahuimes.Youexpecttoguessthecorrectlyaboutfiftytimes，andwhenyoudoso，youexpecttogoohecardcolourhalfthetime.

Thatsuggestsyouexpecttogetbhtonabouttwenty-fiveos，anditlooksseooffer25%，or14，asthegrightbothtimes.Fortheseexperiments，thegesisfoundjustbymultiplyingtheirindividualces.

Tenballsofthesamesizeandpositiohtheonine，ahemisselepletelyatrandom.SoitisequallylikelytoshowaLowofhonihesenumbersarecreeareBlue，soGreenandBluearealsoequallylikely.Tryihecolour，orwhetheritisLh，wehavea50%ceeithertime.WhataboutthecethattheballwedrawisbothLowandGreen?

&abovewiththedthecardssuggests14astheaamoment’sthoughtshowsthisotbecorrect.Withtenballs，itisimpossiblethat14ofthem（twoandahalf!）willbebothLowaheswerdependsonwhiumbersarecreen，andwhichBlue.SosupposereeareBlue.

Inthatcase，fourofthetennumbers（ohree，andfour）arebothLowaheceis0.4.But，aswedidwiththefirstproblem，wealsouseatrooionsofthisexperimeogetaLowimes.FourofthefiveLreen，sowheaLowittobeGreen45ofthetime.Overall，weexpectaLowGreeimes，leadingagaintotheanswer0.4.

Withthedcards，theouteofthetosshasnonthe.WedonotindsabouttheceofaRedtoldwhetherthefallsHeads–thealprobabilityoftheset，　give，isjustitsordinaryprobability.Whenthishappewoeveobe　i，ahatbothoccuristheproductoftheirindividualprobabilities.

Withthetenballs，thecebotheventsoccuralsoarisesasaproduwhichthefirstpoisalsotheprobabilityofo（Lowhesedisthe　alprobabilityofGreehishappehetwocalsareidenti，theonlydiffereheouteofthefirsteveheceofthesed.Eachtime，wehaveusedtheMultipliLaw，>

theprobabilitythatbothoftwoeventsoccuristheprobabilityofthefirst，multipliedbytheprobabilityoftheseditiohappening.

Independence

&heterm‘iodescribethetheoceofthefirstevegeourassessmentoftheceofthesed.Supposethisholds，butwelearnthatthe　sethashappethisaffectourassessmentoftheceofthefirst?

hefaeeventhasorhasnotoccurredmakesheotherevent，itturnsoutthatwhetherornotthissetoccursmakeshecesofthefirst.Twoeventsareiheo-oceofeithermakesheprobabilityoftheother.Tofindtheultiplytheirindividualces.

&havealloher，likeraintodayinTunisahehinParis，aresurelyisometimesindepeobvious.Usinganordinaryfairdie，sidertheeveanevennumber’aipleofthree’，withrespee-halfaheonlywaybothoccuriswheaSix，havingprobabilityoh.Aiplyingone-halfahirdgivesohosetwoevents　areiheceofgettinganevengeifwearetoldwhetherornotamultipleofthreeodviceversa）.