AWinningStrategyforRoulette
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1 AWinningStrategyforRoulette logreturn. Keywordsandphrases:Roulette,Bayesstrategy,Dirichletprior,convexloss,expected UniversityofWisconsinatMadison JeromeH.Klotz logcapitalafternplaysforlossfunction,weshowthatthebayesstrategyfora Dirichletpriorisasymptoticallyoptimal.Wesimulatetoillustratethebehavior ofthestrategyforabiasedfavorablewheelandanunbiasedunfavorablewheel. thegameofroulettewithanunbalancedwheel.usingnegativeaverageexpected Weexaminethestatisticalproblemofcomputingafavorablebettingstrategyfor Abstract 1IntroductionandNotation. favorablebetduetowheelimbalance.wilson(1965),presentsanicediscussionofthe Considertheproblemofbettingonaroulettegamewiththepossibilityofanunknown problemwithanecdotesaboutgroupsexploitingsuchimbalancetomakemoneyandgiving dataonhisownexperiences.morerecently,barnhart(1992)givesinterestingstoriesabout largecasinowins.ethier(1982),considerstherelatedproblemofhypothesistestingfor favorablenumbers.ofmoredirectinteresttothegambler,istheproblemofdeterminingan Xik=1iftheballfallsintocellnumberkontheithplay,andXik=0withprobability optimalbettingstrategytomaximizecapitalgain. 1?pkifitdoesnot. Xik=0,or1,PKk=1Xik=1,p=(p1;p2;:::;pK)T,PKk=1pk=1.Thuswithprobabilitypk, X1;X2;:::;XnhaveamultinomialdistributionM(1;p)whereXi=(Xi1;Xi2;:::;XiK)T,?$1:00foraloss(Xik=0).ForrouletteintheUS,thereusuallyareatotalofK=38cells Asastatisticalmodelforrouletteoutcomes,letindependentrandomcolumnvectors forawheellabeled00;0;1;2;:::;36withpayomk=$35forallsinglenumberbets. Theithplaypayofor$1:00betoncellnumberkisMkdollarsforawin(Xik=1)and 1
2 letthestrategybethecolumnvectorn=(n0;n1;:::;nk)twheren0istheproportion ofcn?1thatisnotbetandnkfork=1;2;:::;kistheproportionbetoncellnumberk forthenthgamblewithpkk=0nk=1. Startingwithinitialdollarcapital,C0,letCnbethecapitalattheendofthenthbet,and WriteX[n]=(X1;X2;:::;Xn)andassumei=(i0;i1;:::;iK)Tonlydependson (X1;X2;:::;Xi?1)(F(X[i?1])measurable). Thecapitalattheendofngamblesforthisstrategywillbe n,forknownp,tomaximizetheexpectedlogcapitalasgivenbykelly(1956).kelly's useofthelogpenalizesextremewagersthatbeteverythingonpropersubsetsandprevents Werstdiscussthesolutiontotheprobabilityproblemofndingtheoptimalstrategy Cn=Cn?1(n0+KXk=1nk(Mk+1)Xnk)=C0nYi=1(i0+KXk=1ik(Mk+1)Xik) (1) andexaminethepropertiesofthebayesstrategyforbalanceddirichletpriors. bankruptcy.wethenconsiderthestatisticalproblemofndingastrategyforunknownp 2AnOptimalStrategyforKnown(p1;p2;:::;pK). Whenthetruecellfrequenciespareknown,Kelley(1956),givestheoptimalstrategy. theorems.becauseofthelackofdetailinkelley'sderivationwepresentthesolutioninour Breiman(1960),andFinkelsteinandWhitley(1981)alsodiscusstheproblemandgivelimit notation. subjecttotheinequalityconstraintsk0fork=0;1;:::;kandequalityconstraint PKk=0k=1. initialcapital,weequivalentlymaximizetheexpectedlogreturnforasinglegamble Forknownpanoptimalstrategyn=(p)isindependentofn.Forourlossandunit Rockafellar(1970)corollary28.3.1)wecanminimizetheLagrangian UsingtheKuhnTuckertheorem(seeforexampleMangasarian(1969)section7.2.2or ()=KXk=1pkln(0+(Mk+1)k) where=(0;1;:::;k;+)tisthevectoroflagrangemultipliers. 2 (2)
3 fork=0;1;:::;k, KXk=0kk=0 fortheequalityconstraint.solvingequation(2)k=1;2;:::;kforkintermsof0and withk0fortheinequalityconstraintsk=0;1;:::;k,and KXk=0k=1 (4) (3) gives IfwedeneG+=fk:1kK;k>0gthensubstitutingthesekvaluesinequation Ifk>0,thenk=0becauseofequation(3)andthen k=pk +?k?0 +?0 (Mk+1): (Mk+1): (5) abovevalueof0weobtain+=1. For0>0equation(3)gives0=0.Usingequation(2)fork=0,aftersubstitutingthe (4)gives Whenk=0,equation(5)with+=1gives k=1?pk(mk+1)=00 0=1?Pk2G+pk=+ 1?Pk2G+(Mk+1)?1: (7) (6) wherep0=1?pk2g+pk. themsothatp1(m1+1)p2(m2+1):::pk(mk+1).wethenhavetwocases. sok=0implies0pk(mk+1).thenforthissolution TodeterminethesetG+=fk:1kK;k>0grstsortthevaluesandrelabel ()=X (6)with+=1that01pk(Mk+1)forallk=1;2;:::;Kand()willbemaximized If1p1(M1+1)thenpk1=(Mk+1)forallk=1;2;:::;K.Itfollowsfromequation k2g+pkln(pk(mk+1))+p0ln(0) forg+empty(k=0fork=1;2;:::;k,0=1).thusforthiscasewedon'tbet. contrary,thatj62g+forj<k2g+.thenj=0and0pj(mj+1)byequation(7). contradiction. Butpj(Mj+1)pk(Mk+1)so0pk(Mk+1)and0<k=pk?0=(Mk+1)0,a If1p1(M1+1)weprovethatifk2G+thenj2G+for1jk.Assume,tothe ThusG+=f1;2;:::;rgwherer1isthelargestintegerwith1?Prk=1(Mk+1)?1>0 pr(mr+1)>0[r] 3 (8)
4 fork=1;2;:::;r,andk=0fork=r+1;:::;k.wealsohavethemaximum wherep0=pkk=r+1pk.callthismaximum(p). where0[k]=(1?pkj=1pj)=(1?pkt=1(mt+1)?1).thenwith0=0[r] ()=rxk=1pkln(pk(mk+1))+p0ln(0) k=pk?0=(mk+1) (10) (9) Forknownp,theoptimalstrategyisconstantsothatforasingleplay(n=1),wehave 3AverageReturnECnandVariance. Similarly Usingindependence,forgeneraln, E(C1=C0)=E(0+KXk=1k(Mk+1)X1k)=0+KXk=1pkk(Mk+1)) and E(Cn=C0)=EnYi=1(0+KXk=1k(Mk+1)Xik)=[E(C1=C0)]n: E[(C1=C0)2]=KXk=1pk(0+k(Mk+1))2 E[(Cn=C0)2]=fE[(C1=C0)2]gn: (11) Itfollowsthat where1=(c1=c0)=e(c1=c0)=[pkk=1pk2k(mk+1)2?(pkk=1pkk(mk+1))2]1=2 Var(Cn=C0)=fE[(C1=C0)2]gn?f[E(C1=C0)]2gn=fE(C1=C0)g2n[(1+21)n?1](12) verylargeorverysmallreturnsoccurafteraperiodofplay. isthecoecientofvariationforasingleplay. dependent,identicallydistributedrandomvariables.takinglogsinequation(1)forthe optimalstrategyn=,wecanwrite Someinsightintothevariationcanbeobtainedfromthecentrallimittheoremforin- Thestandarddeviationcanbequitelargeincomparisontotheexpectationsothatoften [0+PKk=1pkk(Mk+1)] -alargesamplelog-normalrepresentationwherezn!n(0;1)indistributionasn!1. Here 2=KXk=1pkln2(0+k(Mk+1))?[KXk=1pkln(0+k(Mk+1))]2: Cn=C0en(p)+pnZn 4
5 4AnExample. Toillustratethestrategycalculationanditsexpectedreturnforaknownbiasedmodelwith asmallnumberofcells(k=4),considerbetsongreen,therstdozen,seconddozen, andthirddozen(seeforexamplescarne(1961)page365).specically,wecombinegreen outcomesf0;00gintoacellwithindexk=1,outcomesf1;2;:::;12gintoacellwithindex intoacellwithindexk=4. k=2,outcomesf13;14;:::;24gintoacellwithindexk=3,andoutcomesf25;26;:::;36g $2:00foreachofthedozens. (2=38;12=38;12=38;12=38)Tforacompletelybalancedwheel.Table1givestheoptimal strategycalculationinthiscase.table1.computation. Thepayoamountsfora$1:00betareM1=$17:00forgreen,withM2=M3=M4= Letthetruecellfrequenciesbep=(3=38;14=38;12=38;9=38)Tascomparedto k13/381754/38:= /323:=.983/95 214/38242/38:= /209:=.908/95 312/38236/38:=0.9581/95:=.853/95 49/38227/38:=0.71 pkmkpk(mk+1) 0[k]k weobtainfromequation(11) Itisinterestingtonotethat3=3=95>0despitep3(M3+1)=36=38<1. TocalculatetheexpectedreturnforstartingcapitalofC0=$1000aftern=100games y:pkj=1(1?(mj+1)?1<0,0=0[3]=81=95 y0 [(3 With 38)(395)2182+(14 E(C1=C0)=81 38)(895)232+(12 EC100=$1000[EC1=C0]100:=$7;607:50: 95+(338)(3 38)(395)232?(3 95)18+(14 38)(8 38)(395)18+(14 95)3+(12 95)3:=1: wehavethecoecientofvariation andthestandarddeviationfromequation(12)is 1=(C1=C0)=E(C1=C0):=0: =(C1=C0):=0: )(8 95)3+(12 38)(395)3)]1=2 Notetheextremevariability. (C100)=(EC100)[(1+21)100?1]1=2:=22;139:21: 5
6 5TheCaseofUnknown(p1;p2;:::;pK). Wenowconsiderthestatisticalproblemofdeterminingastrategy[n]=(1;2;:::;n)(K+1)n fornconsecutivegambles,tomaximize,forunknownxedp,theexpectedlogreturn ThomasKurtzmentionedbyEthier(1982),weconsidertheBayesstrategyfortheDirichlet priorjointdensityfp(p)= Asameansofderivingaclassofinterestingstrategies,andnotingthesuggestionby EX[n]jpln(Cn([n];X[n])): byraiaandschlaifer(1961)(seeforexamplewilks(1962)ordegroot(1970)).the convenientpriorforthemultinomialmodelandistheconjugatepriorgenerallyrecommended posteriordistributionalsobelongstothisdirichletclassofdistributions. where0<pk<1,pkj=1pj=1,k>0,+=pkj=1j.thisisanaturalandmathematically?(1)?(2)?(k)p1?1?(+) 1p2?1 2pK?1 ABayessolutionwillminimizetheBayesriskormaximizetheexpectedlogreturnaveragedwithrespecttotheDirichletprior: TondtheBayesstrategy,rewriteequation(13)as EPEX[n?1]jPEXnjX[n?1];Pfln(Cn?1)+ln(n0+KXk=1nk(Mk+1)Xnk)g EPEX[n]jPln(Cn([n];X[n])): (13) K where =EX[n?1]EPjX[n?1]fln(Cn?1)+KXk=1pkln(n0+nk(Mk+1))g istheposteriorexpectationofpkbasedonn?1observationswithsk[n?1]=pn?1 =EX[n?1]fln(Cn?1)+KXk=1^pnkln(n0+nk(Mk+1))g forthestartingstrategy(i=1). byequation(9)withpkreplacedbythebayesestimates^pikfori=1;2;:::;nwheresk[0]=0 ItfollowsthattheithcolumnofaBayesstrategy^[n]=(^1;^2;:::;^n)(K+1)nisgiven ^pnk=e(pkjx[n?1])=k+sk[n?1] ++n?1 i=1xik. 6
7 6LargeSampleOptimality. givenby Intheorem1weshowitattainstheexpectedlogreturn(p)fortheoptimalstrategywhen FortheBayesstrategy,considerthelimitingaverageexpectedlogreturnoverngambles pisknown. Write n!1ex[n]jp1nln(cn(^[n];x[n]): andconsiderthenthterminthesumontheright =1nln(C0)+1nnXi=1EX[n]jpln(^i0+KXk=1^ik(Mk+1)Xik) n=ex[n]jpln(^n0+kxk=1^nk(mk+1)xnk) (14) =EX[n?1]jpEXnjX[n?1];pln(^n0+KXk=1^nk(Mk+1)Xnk) =EX[n?1]jpKXk=1pkln(^n0+^nk(Mk+1)): (15) Lemma1Forxedp,Bayesestimator^pnandstrategy^[n]=(^1;^2;:::;^n)(K+1)nwe Torstshow havekxk=1pkln(^pnk(mk+1))kxk=1pkln(^n0+^nk(mk+1))kxk=1pkln(mk+1): Proof.Therightinequalityholdssince(^n0+^nk(Mk+1))(Mk+1). n!(p)asn!1weprovethefollowing Fortheleftinequality, =rxi=1pjiln(^pnk(mk+1))+kx since^n0^pnk(mk+1)fork>rwhererisdenedinequation(8)withprobabilities^pnk. Thiscompletestheproof. KXk=1pkln(^pk(Mk+1)) k=r+1pkln(^n0) Nextweprove 7
8 asn!1. Lemma2For0<pk<1,k=1;2;:::;KandtheBayesestimates^pnkwehave thedenitionof^pnkwehave Choose",0<"<pkanddenethesetAnk=f!:Sk[n?1](n?1)(pk?")g.Using Proof.Byadditivity,itsucestoproveEln(^pnk)!ln(pk)asn!1. EX[n]jpKXk=1pkln(^pnk(Mk+1))!KXk=1pkln(pk(Mk+1)) and 0<k=(++(n?1))(k+Sk[n?1])=(++(n?1))=^pnk<1 ln ++(n?1)!p(ank)eln(^pnk)iank0: usingfeller(1950page140,(3.6)).itfollowsasn!1that Forsn=b(n?1)(pk?")cthegreatestintegernotexceeding(n?1)(pk?"),wehave n?1 sn!psn k(1?pk)n?1?snp(ank) sn!psn k(1?pk)n?1?sn (n?sn)pk npk?sn! (16) using(16). OnthecomplementsetAcnkwehaveSk[n?1]>(n?1)(pk?")sothat ln 1^pnk=Sk[n?1]+k ++(n?1)!p(ank)!0andeln(^pnk)iank!0 k vergencetheoremtoobtain fornsucientlylarge.thusln(^pnk)isboundedonacnkandweapplythedominatedcon- pk?" 1++=(n?1)(pk?")=2 n?1++>(n?1)(pk?")+k Eln(^pnk)IAcnk!ln(pk) since^pnkconvergesalmostsurelytopkbythestronglawoflargenumbersandiacnk!1 tonishtheproofoflemma2. almostsurely.finally Nowusingthelemmasweconclude Eln(^pnk)=Eln(^pnk)IAnk+Eln(^pnk)IAcnk!0+ln(pk) 8
9 Theorem1Forxedpk,0<pk<1andBayesstrategy^[n]wehave n!1,where (^pn),usingequations(9)and(10),convergesalmostsurelytoanoptimalstrategyforp Proof.Since^pnkconvergesalmostsurelytopk,itfollowsthataBayesstrategy^n= nandaredenedbyequations(15)and(10),and n!1ex[n]jp1nln(cn(^[n];x[n])=(p): lim n!(p)as known almostsurelysinceitisalsoacontinuousfunction.ifwedenotetherandomfunction asn!1sincetheequationsarecontinuousinp.thusthefunction hn=kxk=1pkln(^n0+^nk(mk+1))!(p) ^n!(p) gnhng.sincegn!g=pkk=1pkln(pk(mk+1)),andbylemma2,egn!g,wecan applythetheoremofpratt(1960)toconclude gn=pkk=1pkln(^pnk(mk+1))andtheconstantg=pkk=1pkln(mk+1),thenbylemma1, appliedtoequation(14). 7PerformanceoftheBayesStrategy. (p)usingn?1ln(c0)!0andtoeplitzlemma(seeforexampleash(1972page270)) Forthesecondpartofthetheorem,theaverageexpectedlogreturnalsoconvergesto n=ehn!(p). forallcellnumbers(k=).ifisselectedtobelarge,thebayesstrategyobservesfor quiteafewgameswithoutbetting.ifthewheelisfavorablybiased,eventuallythebayes Forsinglenumberbets,symmetryconsiderationssuggestusingpriorparametersthesame estimates^pnkwilldiscoverthisaftermanygamesandbettingwillbegin.ifasmallis astheinitialcapital.sampleaveragesandsamplestandarddeviations used,chanceuctuationsinthecountsleadtoearlybettingonunfavorablecellsresultingin capitalnearzero. ideaofperformance.table2givesresultsfornplaysusingasthepriorparameterandc0 BecauseofthecomplexityofCnwhen[n]dependsonX[n?1],wesimulatetogetsome forr=10;000samplesaregivenwherecn(j)isthecapitalattheendofnplaysforthejth samplereplication.abiasedwheelwithp1=1=30,p2=:::=p38=29=1110wasused.this Wilson(1965). degreeofbiasinthewheelisconsistentwiththeestimatesforwheelimbalancediscussedby Cn=RXj=1Cn(j)=R;S(Cn)=[RXj=1(Cn(j)?Cn)2=(R?1)]1=2 9
10 (seeforexamplemarsaglia(1972)).adierentstartingseedintegera06=0wasusedto thejthrandomnumberon(0;1)givenbyxj=aj=232.theperiodis232=4=1;073;741;824 computeeachentry. Themultiplicativerandomnumbergeneratoraj=(aj?169069)mod232wasusedwith (S(Cn))n=100 =1 p=(1=30;29=1110;:::;29=1110);mk=$35;c0=$1000;r=10;000. Cn 20.12Table2.BayesStrategySimulation ( )( ) ( )( )(272.36)(1.15)(0.04) (906.92)(36.39)(0.00)(0.00)(0.000) (0.51)(0.00)(0.00) (0.00) (613.25)( )( )(296.89)(54.02) (213.69)(424.02)(915.12)( )( )(832.58) (29.13)(85.02)(323.98)( )( )( ) (5.88) (1.48)(10.55)(67.40)(227.77)( )( ) (0.00)(0.20)(3.34)(30.05)(230.72)( ) usinglogarithmicscaleclassintervalsandpositivecountsasgivenintable3. Figure1givesthehistogramof10;000valuesofCnfor=100,n=5000,C0=$1000 (0.00)(0.00)(0.00)(0.18)(5.44)(174.80) Interval[26;27)[27;28)[28;29)[29;210)[210;211)[211;212)[212;213)[213;214)[214;215) Interval[215;216)[216;217)[217;218)[218;219)[219;220)[220;221)[221;222)[225;1) Count Alog-normaldistributionissuggestedasinthecaseofknownp Table3.Countsfor10,000valuesofCn
11 knownwecalculate andtocomparetheseresultswiththoseforp=(1=30;29=1110;:::;29=1110)tassumed Figure1.Countsof10;000valuesofCnwith=100,n=5000,C0=$1000. E(C1=C0)=(174=175+(1=30)(1=175)36):=1: =(174=175;1=175;0;:::;0)T ^2 2^5 2^8 2^11 2^14 2^17 2^20 2^23
12 Table4givescorrespondingexpectationsandstandarddeviations. 1=[(1=30)(1=175)2362?((1=30)(1=175)36)2]1=2 Table4.OptimalStrategyValues ne(c1=c0) ECn (Cn) :=0: : Inadditiontoresultsforabiasedwheel,itisofinteresttoseehowtheBayesstrategy performsintheequiprobablecasep=(1=38;1=38;:::;1=38)whenthereisnofavorablebet (Mk=$35).Table5givesCnand(S(Cn))forR=10;000simulationsforsome(;n) values. (S(Cn))n=1000 = Cnpk=1=38,R=10;000. Table5.BayesSimulation (460.16)(570.81)(113.28) (148.20)(294.36)(349.26) (15.15)(61.25)(204.69) $5:714)canbebet.InrealitythereisusuallyaminimumbetandCnkmustbean 8PracticalConsiderations. TheBayesstrategyassumesthatanarbitraryfractionofthecapital(e.g.$10001=175= integermultipleofthisminimumbet.thisrestrictionshoulddiminishtheexponentialrate (0.00)(0.12)(7.39) possibletobringacomputerintosomecasinosalthoughtoday'spalmtopportablesare betoneachplayaswellasamaximumbetlimitationfurthercomplicatesimplementingthe strategy. alargeinitialcapitalc0.inaddition,arequirementtobetatleastfourtimestheminimum ofcapitalincreaseforafavorablybiasedwheel.theeectcanbereducedbystartingwith Anotherdicultyisthecomputationrequiredtodeterminebets.Itmaybenotbe 12
13 betting.apracticalapproximationmightinitiallyobservethewheelforalongperiod withoutbetting,ifthisispermitted,andthenuseaxedstrategyforpestimatedfrom isafelonyinnevada. theinitialcounts-a\wheelclocking"approach. unobtrusive.accordingtobarnhart(1992),theuseofanelectronicdevicetoaidgambling isnoguaranteethatthecasinowillnotchangethewheeliftherearelargewinnings. ones.evenifafavorablewheelisfoundsuchasdescribedbywilson(1965page33),there TheBayesstrategiesthatdowellintable2havealongperiodinitiallywithverylittle (e.g.food,lodging,parking,etc.).includingsuchanoverheadcostwouldchangethebayes solutionandmakefavorablereturnsevenmoredicult. Amajordicultyisndingawheelwithsucientfavorablebiasandavoidinggaed 9Conclusions. Iftruefrequenciesareknownaccuratelyandafavorablebiasexists,theoptimalstrategy Wedidnotincludeaxedoverheadcosttoobservethewheelwhennobetsareplaced earningpossibilitiesusingpriorparametervaluesintherange200500arequite frequenciesareunknown,anexponentialincreasealsooccursonfavorablybiasedwheelsbut interestingfor2000ormoreplays.however,becauseofextremevariation,largelossesas wellaslargewinningsarepossible. aconsiderablenumberoftrialsarerequired(seetable2).despiteallthedicultiesthe expectedreturnincreasesexponentiallywiththenumberofgamesplayed(seetable4).when Acknowledgements. References ThanksgotoStewartEthierforhelpfulcorrectionsandsimplicationstoanearlierdraft. ThanksalsogotoMarkFinkelstein,RichardJohnson,TomLeonard,BinYu,andDavid Blackwellforcommentsandsuggestionsforimprovement. [1]Ash,RobertB.(1972)RealAnalysisandProbability.SanDiego:AcademicPress,Inc. [2]Barnhart,R.T.(1992)BeatingtheWheel.WinningStrategiesatRoulette.NewYork: [4]Ethier,S.N.(1982)Testingforfavorablenumbersonaroulettewheel.Journalofthe [3]Breiman,L.(1961)Optimalgamblingsystemsforfavorablegames.Proceedingsofthe FourthBerkeleySymposiumonMathematicalStatisticsandProbabilityI65-78.UniversityofCaliforniaPress LyleStuart. AmericanStatisticalAssociation77,
14 [5]Feller,William(1950)AnIntroductiontoProbabilityTheoryandItsApplications,Vol [6]Finkelstein,MarkandWhitley,Robert(1981)Optimalstrategiesforrepeated [7]Kelley,J.L.(1956)Anewinterpretationofinformationrate.BellSystemTechnicalJournal36, games.advancesinappliedprobability13, I.NewYork:JohnWiley&Sons. [8]Mangasarian,O.L.(1969)NonlinearProgramming.NewYork:McGrawHill. [9]Marsaglia,George(1972)Thestructureoflinearcongruentalsequences,inApplicationsofNumberTheorytoNumericalAnalysis.EditedbyS.K.Zaremba.NewYork: AcademicPress, [10]Pratt,JohnW.(1960)Oninterchanginglimitsandintegrals.AnnalsofMathematical [11]Raia,H.andSchlaifer,R.(1961)AppliedStatisticalDecisionTheory.DivisionofResearch,GraduateSchoolofBusinessAdministration,HarvardUniversity,Boston. [12]Rockafellar,R.T.(1970)ConvexAnalysis.Princeton:PrincetonUniversityPress. [13]Scarne,John(1961)Scarne'sCompleteGuidetoGambling.NewYork:Simon&Schuster. York:NorthHolland. Statistics31, [14]Srivastava,M.SandKhatri,C.G.(1979)AnIntroductiontoMultivariateStatistics.New [15]Wilks,S.S.(1962)MathematicalStatistics.NewYork:JohnWiley&Sons. [16]Wilson,AllanN.(1965)TheCasinoGambler'sGuide.NewYork:Harper&Row. 14
( ) = ( ) = {,,, } β ( ), < 1 ( ) + ( ) = ( ) + ( )
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