Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in - PDF document

Anotherwaytoaskthisverysamequestionwouldbe,\howmuchisthefairpriceforthisgame?"(Theansweris,ofcourse,$1.17.)AnotherwaytoanswerthisquestionistousethefollowingchartRoll Prot 6 $35 $14 $03 $-12 $-11 $-1Computingexpectedvalue:E(X)=1 63+1 61+1 60+1 6(�1)+1 6(�1)+1 6(�1)=1 60:167Again,youseethatyouexpectabouta$0.17prot.1.1.1NotationalnotesIntherstcomputation,wewereinterestedintheamountofmoneywewouldgetbackinasinglegame.Thus,inthiscase,XwasthisamountofmoneyandE(X)istheexpectedamountofmoneywegetback.Inthesecondcomputation,wewereinterestedinthetotalprotwewouldmake.Inthiscase,Xwastheprot.Ofcourse,itwouldhavebeennicetohaveusedadierentletter/variableforthesethings.IfwedidthisandletMbethemoneyfromonegameandPtheprot,thenwewouldhave:P=M�11.2SomeExercisesforYouDeterminetheexpectedvalueforthegames.1.Charge$1toplay.Rollonedie,withpayoutsasfollows:Roll Payout 6 $25 $24 $13 $02 $01 $1.502.Charge:$1totoss3coins.Tossthecoins.Ifyougetallheadsoralltails,youreceive$5.Ifnot,yougetnothing.3.Charge:$1.Roll2dice.Ifyouroll2oddnumbers,likea3anda5,youget$2.Ifyouroll2evennumbers,like4and6,youget$2.Otherwise,yougetnothing.4.Charge:$5.Drawtwicefromabagthathasone$10and4$1bills.Yougettokeepthebills.2 Solution:Todeterminetheprobabilityofrollinganumberyoucountthenumberofwaystorollthatnumberanddivideby36.Sum Combinations Probability 2 1-1 1 363 1-2,2-1 2 36=1 184 1-3,2-2,3-1 3 36=1 125 1-4,2-3,3-2,4-1 4 36=1 96 1-5,2-4,3-3,4-2,5-1 5 36 7 1-6,2-5,3-4,4-3,5-2,6-1 6 36=1 6 8 2-6,3-5,4-4,5-3,6-2 5 369 3-6,4-5,5-4,6-3 4 36=1 910 4-6,5-5,6-4 3 36=1 1211 5-6,6-5 2 36=1 1812 6-6 1 364CrapsInthegameofcrapsthereareawiderangeofpossiblebetsthatonecanmake.Therearesinglerollbets,linebetsandmore.Theplayerplacesthesebetsbyputtinghismoney(gamblingchips)intheappropriateplaceonthecrapstable,seeFigure1. Figure1:CrapsTableLayout4 4.1SinglerollbetsThesebetsaretheeasiesttounderstand.Inasinglerollbettheplayerisbettingonacertainoutcomeinasingleroll.4.1.1PlayingtheeldThemostobvioussinglerollbetisperhapsplayingtheeld.Thisbetisrightinthemiddleofthetable.Onarollof3,4,9,10or11,theplayerispaidevenoddsandonarollof2or12theplayerispaiddoubleodds.Thus,if$1isbetontheeldanda3,4,9,10or11isrolledtheplayerispaid$1andkeepshisoriginal$1.Ifa2or12isrolled,theplayerispaid$2andkeepshisoriginal$1.Question2.Computetheexpectedvalueofplayingtheeld.Solution:Hereistheexpectedvalueofonedollarbetontheeld.E(X)=2
7 18+3
1 18=17 180:944Inotherwords,inthelongrun$1betontheeldwillexpecttopaytheplayer$0:944.Aswewillsee,thisisbetterthansomebetsbutitisnotgoodenough.4.1.2CandEThesearethecrapsandyobets.Inthegameofcrapsarollofcrapsisarollofa2,3or12.Arollofelevenisalsocalledayo.(Atthecrapstableyouwillhearpeoplecallingfora\lucky-yo,"meaningtheywantanelevenrolled.)Aplayercanplaceaone-timebetonanyofthesenumbersandthepayosareprintedonthecrapstable.Question3.Fillinthetablebelow.Roll Oddspaid ActualOdds Probability Expectedvalueof$1bet 2 30:1 35:1 1 36 31 360:861 3 15:1 17:1 1 18 8 90:889 Yo11 15:1 17:1 1 18 8 90:889 12 30:1 35:1 1 36 31 360:861 AnyCrap 7:1 8:1 1 9 8 90:889 Any7 4:1 5:1 1 6 5 60:833 Noticethattheoddspaidareprintedonthetable.So,forexample,theoddspaidforanysevenis4to1.Thus,ifyouput$1downandasevenisrolledthiswillpayyou$4plusyouroriginalbet(thusyouwillwalkawayhaving\earned"$4).Notethatinthistableweintroducedthecolumn\ActualOdds."Thisistheoddsthatthecasinoshouldpayinordertobecompletelyfair.Inotherwords,ifthecasinopaidtheseoddsthentheexpectedvalueofadollarbetwouldbeadollar.5 Letsdothecomputationforthepoint4(thecalculationifthepointwas10isidentical).Theplayerwillwinifa4rollsbeforeaseven.So,awinningsequenceofrollscouldlooklike4playerrollsa4rightawayor3;6;5;8;4playerrollssomethingotherthana4or7andthennallya4So,thefollowingaretheprobabilitiesofwinningafterapointof4hasbeenestablished.P(4onroll1)=1 12P(4onroll2,no7)=P(nota4or7)P(4)=3 4
1 12P(4onroll3)=P(nota4or7)P(4onroll2,no7)=3 42
1 12Puttingallthistogether,theprobabilityofwinningafterapointof4hasbeenestablishedisageometricseries:P(4before7)=P(4onroll1)+P(4onroll2,no4or7beforeroll2)+


=1 12+3 4
1 12+3 42
1 12+3 43
1 12+


=1 121Xn=03 4n=1 121 1�3 4=1 3Similarly,wecancomputetheprobabilityofwinningotherpoints.(And,includedarethe\points"of2,3and12,whicharenotreallypointsbecausetheyarecraps.)P(2before7)=P(12before7)=1 7P(3before7)=P(11before7)=1 4P(4before7)=P(10before7)=1 3P(5before7)=P(9before7)=2 5P(6before7)=P(8before7)=5 11Wecannowcomputeourprobabilitiesofwinning7 Event Probability Losewitha7or11oncome-outroll 2 9Winwith2,3oncome-outroll 1 12Pushwith12oncome-outroll 1 36Establishpoint4andlose 1 12�1 3
=1 36Establishpoint4andwin 1 12�2 3
=1 18Establishpoint5andlose 1 9�2 5
=2 45Establishpoint5andwin 1 9�3 5
=1 15Establishpoint6andlose 5 36�5 11
=25 396Establishpoint6andwin 5 36�6 11
=5 66Establishpoint8andlose 5 36�5 11
=25 396Establishpoint8andwin 5 36�6 11
=5 66Establishpoint9andlose 1 9�2 5
=2 45Establishpoint9andwin 1 9�3 5
=1 15Establishpoint10andlose 1 12�1 3
=1 36Establishpoint10andwin 1 12�2 3
=1 18Computingexpectedvaluesofa$1betonthepasslinesgivesE(X)=2
1 12+1 36+41 18+1 15+5 66=217 2200:9864Whichisslightlybetterthanplayingthepassline.6CraplessCrapsThereisavariationofcraps(discoveredbytheauthoronarecenttriptoacasino)called\CrapplessCraps"or\NoMoreCraps,Craps."Whatthismeansisthatthecrapsandyoarealldoneawaywithonthecomeoutroll.Instead,the2,3,11,and12arevalidpointsjustlikethe4,5,6,8,9and10.Theoddsbetsforthesearealsopaidoattrueodds.Thislookslikeagoodthingbecausetherearelesswaystoloseonthecome-outroll.Wenowcomputesomeprobabilitiesandexpectedvalues.9 7.1PasslineplayerBasically,thismeansthatifyouarebettingonthepasslinetheneveryhouryouwillplacearound17bets.Ifyouaremaking$100betsthenthismeansyouwillhaveplaced$100onthetable17timesandonyouwillreceive$98.59backfromthesebets.Inotherwords,inanhouryouwillhavelostonlyabout$23.97.7.2FieldplayerOveranhour,theeldplayerwillmakeabout112betsand,ifplaying$100abetthentheeldplayershouldreceive$94.44fromeverybet.Thismeansthatthiseldplayerwillloseabout$622.72.7.3Howtominimizeyourlosses?Thereareclearlysomebadbets,suchastheeld(and,theeldisnoteventheworstbet).So,rstthingtodoistonotplaythebadbets.Whenplayingthepass-line,thehouseedgeisonlyabout1.41%.Ifyouplaywithoddsandplaceasmanyoddsbehindyourbetthenyoucanexpecttodecreasethehouseadvantage.Themoreoddsyouplaceonyourpass-linebetthemoreyoudecreasetheadvantagethatthehousehas.7.4Pass-linewithoddsHereweassumeaplayerplacesapass-linebetof$1andthen,whenapointisestablished,placesanoddsbetsbehindthepass-line.Mostcasino'slimittheamountofthisoddsbetand10isahighamount.Whatthismeansisthatifyourpass-linebetis$1thenyouroddsbetcanbeatmost$10.Herearethepayosonsuchabetifthenumberishit.Point Bet Payout 4 1+10 1+20 5 1+10 1+15 6 1+10 1+12 8 1+10 1+12 9 1+10 1+15 10 1+10 1+20 So,ifthepointis4andthatpointishitthentheplayerwouldhave$1onthepass-lineand$10forodds.Theplayerwouldbepaid$1forhispass-linebetand$20forhisoddsbet.Howdoyoucomputethehouseadvantagenow?7.5Methodstowin?IfoundtheseontheInternet:http://casinogambling.about.com/cs/craps/a/5count.htmhttp://homepage.ntlworld.com/dice-play/CrapsSystems.htm11 7.5.5ThePlaceBettingSystemAbigtimegamblerssystemwhichhasevendeceivedsomeCrapsoperators.Youplacethemaximumlimit(althoughitcanbeless)onallofthesixplacebets:4,5,6,8,9and10.Ifthelimitis$200thenthismakesatotalof$1,200.Assoonasoneofthesebetsiswonyoucollectonitandcallotheremainingveplacenumberbets.Yes,mostcasinosallowyoutocalloplacenumberbetsatanytime.Thetheoryisthatthebettorhassixnumbersagainstthehouse'sone,the7.Thereare24waystomaketheplacenumbersandonly6waystomakea7sotheoddsareworkedoutas4to1infavorofthegamblerwiththesystem.7.5.6TheRightandWrongWaySystemInthissystemyouplaceyourbet(say$60)onthe\Don'tPass"linebeforethecomeoutthrow.Theshooteristhensupposedtothrowapointonthecomeoutroll,either4,5,6,8,9or10.Youthenput$60onthepointsplacenumber.Theideaisiftheshooterdoesn'tthrowthepointyouhavestillwononthe\Don'tPass"betandyouarestilleven.You'velost$60ontheplacenumberbutwon$60onDon'tPass.Iftheshootermakeshispointyoulose$60ontheDon'tPassbutyourupbybetween$48and$10dependingonwhichplacenumberitis.8MoreProbabilityandExpectedValueChallengesQuestion4.Supposeyourolladieandreceivethenumberofdollarsthatyouroll.Whatisthefairvaluetoplaythisgame?Question5.Ifyou ipacointwice,whattheprobabilitythatitwillcomeupheadseachtime?Whatifyou ipthecoin3times?ntimes?Question6.Supposeweplayagameinwhichyou ipacoinuntilaheadscomesupforthersttime.Whatistheprobabilitythattherstheadswillcomeupontherst ip?Second ip?Third ip?Whataboutthenth ip?Question7.Playagamewhereyou ipacoinuntilyougetaheads.Iftherstheadscomesuponthekth ip,thenyouwin2kdollars.SoifyougetHonthersttry,youwin$2.IfyougetTandthenH,youwin$4.IfyougetTTTTH,youwin$32.Whatistheleastamountofmoneyaplayercanwininthisgame?Whatisthemost?Howmuchwouldyoubewillingtopaytoplaythisgame?Whatisafairamounttopaytoplaythisgame?13