完本神战

Chapter 2 The workings of probability（第1页）

Chapter2Thewsofprobability

&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）。

Nowsiderthesameproblemwhe-sidedfairdie，oraten-sidedfairdie，withthesideslabelledooroively。Dothearithmetic:youshouldfindthatthetwoevents　areiihesecases，but　iionaboutindependenotalwaysenough。