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Chapter 6 Games people play（第2页）

Theoneywasbilledasthemoststressfulshowosurvivedonlyafewepisodesin2009.ButitdoesgiveasplendidopportunitytoillustrateusesoftheAdditionandMultipliLawsinfindingaprobability.

Thesums￡1，000，￡2，000，...，￡20，000wererandomlyallocatedtotwentyboxesofdifferentcolours，andtheplayer，Paula，soughttoreaepre-assig，say￡64，000.Todoso，shecouldselectuptoteatime.Ifshe（unknowingly）chosethe￡14，000box，theamounts￡1，000，￡2，000，...upto￡14，000earinthatorderatastatelypace:shecouldcallStopatanystage.Ifshemadethattime，shebalastshowing，butifshewaitedtoolong，shebahing.If，aftertenboxes，shehadarget，shewotacticsshouldsheuse?

Colourapart，alltheboxesareidentiakesapletelyraionfromthoseleftinead.Iround，witheleve，herstrategywillbeobvious:forexample，ifsheher￡9，000tetalysixboxesareworth￡9，000ormore，shewillhopetocallStopwhen￡9，000isshown，andherwinningceis611.Butwhatshouldshedoinearlierrounds?

&hetwelveboxesleftwithtworoundstogo（inunitsof￡1，000）theamounts1，4，5，6，9，10，12，13，15，17，19，20，andsherequiresanother￡15，000.ItmakesocallStopwhenshesees￡7，000;ifthatfigureeverappears，sheknowsthatherboxsatleast￡9，000，soshecouldStopatthatsum，plaiics.SherestrictheroptioingfromthetwelvevaluesihesameargumentalsoappliesattheearlierroucallofStopwillalwaysbeatavaluediheremainingboxes.

IfPauladoesiopat￡9，000here，sheargue:‘Eightofthetwelveboxeshaveatleastthatamount，somyceofsuccessis812.AndifIdosucceed，I’ll￡6，000inthefinalround，ahelastelevenboxeswillwork.TheMultipliLawtellsmethatceofbothoftheseis（812）*（811）=64132.Alsofourboxeshavelessthan￡9，000，sotheknois412;Ithenneed￡15，000fromthelastbox，withce411.BytheMultipliLawagaihispathwillworkis（412）*（411）=16132.Thesewaysofwi，sotheAdditioheoverallceofsuccessas80132.’

Shemakeasimilaranalysisforherotherchoices，sugfor￡6，000，or￡12，000.Iiodothesums–theAppendixdescribesherbestchoice.

Intheplanhisshow，theideaofusihematitadvicewasmooted.PaulacouldsuggestshewilltrytocallStopat￡8，000，theexpertmightsay‘Notabadchoice.Ifyoudothat，you’vegota75%ingthemoifyouplantoStopat￡11，oesupto80%.’

Youwellimagicouldhappeheexpertsaidwascorrect，Paulagedherdfailedtowinthemoney，whileherinalinstinctwouldhaveworked.Sometabloidneerwouldsurelyscream‘MathsBoffinRobsArmyHero’sWidowof￡64，000’.

AllofuswhoihsofTVgameshowsarerelievedthatnosuchmathematicaladvicewasevergivenonthisshow!

Cardgames

TheLaweyouwillreceiveyourfairshareofgoodorbadthelongrun，sodifferenskilllevelswilltelleventually.ames.InBlackjaustfollowfixedrulesaboutwhentodrawcards，theplayerdowhathelikes.Uhorpstartedwinningsignifitsums，osbelievedthatnosystemcouldbeattheirbuilt-inadvaheirlogichadafatalflaw:althoughtheycouldexpe1–2%ofstakeswithafullstackhtdecksofcards，afterafewdealstheoddsmightshiftinfavouroftheplayer.Theittedtousethealprobabilitiesbasedonwhatcardsremaiherthaheprobabilitiescalculatedforafullstack.

Thorpdevelopedaingtrackofwhichcardswereleftiathereisahighproportionofhighvaluecardsremaining，itbeorelikelythattherulespelthehousetodrawacardthatleadstoalosingtotalofabovetwehesameces，theplayerottodraw.Thorpwouldbettheminimumamountsoloackhadaleproportionofhighvaluecards，thesshouldthestapositionshiftinhisfavour.Simple，buteffective.

&apositiondoesgiveanadvaheplayer，howmuchshouldhebet?JohnKellyhadansweredpreciselythatquestionafewyearsbeforeThorp’sanalysis:heshouldbetthatproportionofhiscapitalthatisequaltothesizeofhisadvahisizestherateatwhichhiscapitalwillgrow.

Forexample，supposehehas￡1，000andthegameisslightlyinhisfavour–hiswinningceis51%，hislosingceis49%.Hisadvantageis2%，sohebets2%ofhistcapital，i.e.￡20.ime，hewillhaveeither￡980or￡1，020，soifhisadva2%，hisbetwillbeeither￡19.60or￡20.4towhiepertaioogreedy–betting10%ofhiscapitalwhenKellyi2%–thenhewouldeventuallyberuiehisadvantage.Hiscapitalisfihestakewouldbetoohightostaablelosingstreak.

ostakestepstoidentifyandbaters.etothepowerprobabilityhaseverbeenpaid.

&hatBayes’RuleistheproperwaytoseehowpiecesofevidenceshouldgeourbeliefsaboutGuiltorInnoacourtcardgameslikeWhiste，usingthisRuleimproveyourakidesduringplay.Force，Iretainthelegalvocabulary，ahewuiltytomeanthataparticularoppoholdscards，saybothKingandQuees，whileIsheholdsatmostohosecards.

Byg，wedtheproportionofallpossibledealswheresheholdsbiveaninitialassessmentoftheprobabilityof‘Guilt’.Itturovertthisprobabilityiodds，iandardmahisadeattheoutset，wesaythatwehavefoundthe　Priorodds（ofGuilt）.

Ascardsareplayed，relevant　Evidenceemerges–perhapssheplaystheKisonatrick.ToseehowsuceaffectstheoddsofGuilt，aquahe　LikelihoodRatioisfound:first，assesstheprobabilityoftheEvidengGuilt（sheholdsbothKihenfinditsprobabilityassumingInnoce（shehasatmostoheLikelihoodRatioisjusttheratioofthefirsttothesed.

WeowdeducethePosteriorodds，i.e.theoddsofGuilt，takingatofthisEvidengBayes’Rule

Posteriorodds=Priorodds≈LikelihoodRatio.

ItsformatisplaihebiggertheLikelihoodRatio（i.e.themorelikelyistheEvideheoppoisGuilty），themoretheoddsofGuiltihisRuletellsyou　preuchtheEvidehecesofGuilt.

&iion，siderarealisticsituation:ouroppoherholdsjusttheKi），orshehastheKingonly（IhePrioroddsarethatthosealternativesarejustaboutequallylikely.IfsheisGuiltyyoudobesttoplaytheAce，ifsheisIyoushouldplaysomeothercard.Evidenoears–sheplaystheKing.

WithouttheEvidenustguess，andyouwillmakethewinningplayhalfthetime.WithInnoce（shehasKiheprobabilityoftheEvideheKing）is100%;butwithGuilt（shehadbothKingandQueeequallylayedtheQueeheKingthatyousaw，sotheprobabilityoftheEvidenly50%.Theirratioisonehalf，sotheRuletellsyouthatthePosterioroddsareonehalf，i.e.sheistwiceaslikelytobeIasGuilty–sheistwiceaslikelytohavetheKiplayiherightdetwothirdsofthetime.

If，bythisproperuseofprobability，youwillmakethewinningplaytwothirdsofthetime，ratherthanjusthalfthetime，youshouldexpeuchbetter.Youotguaraomakethewinningplay，butyouimproveyourcesofdoingso.

BridgeplayersrefertothissarioasthePririctedChoice–iftheoppohadKingalooplayit，withbothKingandQueenshehadachoice.ThefactthatshedidplaytheKingshiftstheoddsttodoso.

Today，themostpopularformof　PokerisTexasHold’Em.Eachplayerisdealttwodseekstomakethebestpokerhandpossiblefromherowndfiveunalcardsthataredealtfaceuplater.Whichofthefollowinghahesenseofbeiobeateitheroftheothertwowhenthoseunalcardsaredealt?

HandA:TwoofClubs，T>

HandB:AceofSpades，KingofDiamonds

HanddTes.

Trickquestion，ofcourse:aftercarefulg，itturnsoutthatHandAwillbeatBabout52%ofthetime，Bbeatse，whiletheceCbeatsAisaround53%.SoyouwouldratherholdAthanB，andratherholdBthanC，butalsoyoupreferCtoA!Youringexceeds50%ifyouletyouroppopiyofthethreehands，providedyoumaytheheroftheothersforyourself.

Thereisfarmoretopokerthanfacilitywithprobabilities.Youmustmakejudgementsaboutwhatcardsyouroppoohold，andwhenyoumightbluff.Butsometimesprobabilityisveryuseful.Supposethepothas50doneunalcardremai.Youseethat，ifthisfinalcardisaSpade，youwillmakeaFlush，whichmustwin;ifitisnotaSpade，awillwin.Shouldyoubetmoreaininthegame?

Ignorehowmuchyouhavealreadyputi.Allthatmattersisthefuture.Youseesixcards–twoinyourhand，fourunalthetable.Ofthe46unknowneareSpadesthatgiveyouvictory，therestleadtodefeat.With50chipsalreadyi，isitw10moretoseethefinalcarddealt?20more?

Bywoutthemeanprofit（orloss）ifyoumustpay　xchipstoseethefinaldthecut-offvalueof　xthatwill，iveaprofit.TheAppeheanswer.