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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，ahemisselepletelyatrandoSoitisequallylikelytoshowaLowofhonihesenumbersarecreeareBlue，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）.
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