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Hack 43. Play with Dice and Get Lucky

Hack 43. Play with Dice and Get Lucky

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Figure4-2.Possibleoutcomesfortwodice



ThisdistributionresultsinthefrequenciesshowninTable4-15.

TableFrequencyofoutcomesforrollingtwodice



Totalroll

2

3

4

5

6

7

8

9

10

11

12

Totalnumberofpossibleoutcomes



Chances

1

2

3

4

5

6

5

4

3

2

1

36



Frequency

2.8percent

5.6percent

8.3percent

11.1percent

13.9percent

16.7percent

13.9percent

11.1percent

8.3percent

5.6percent

2.8percent

100percent



Thegameofcraps,ofcourse,isbasedentirelyonthese

expectedfrequencies.Someinterestingwagersmightcometo

mindasyoulookatthisfrequencydistribution.Forexample,

whilea7isthemostcommonrollandmanypeopleknowthis,

itisonlyslightlymorelikelytocomeupthana6or8.



Infact,ifyoudidn'thavetobespecific,youcouldwagerthata

6oran8willcomeupbeforea7does.Ofalltotalsthatcould

beshowingwhenthosedicearedonerolling,morethanonefourthofthetime(about28percent)thedicewilltotal6or8.

Thisissubstantiallymorelikelythana7,whichcomesuponly

one-sixthofthetime.



BarBetswithDice

MyUncleFrankusedtobetanydull-wittedpatronthathewould

rolla5ora9beforethepatronrolleda7.UncleFrankwon8

outof14times.

Sometimes,oldFrankiewouldwagerthatonanyonerollofa

pairofdice,therewouldbea6ora1showing.Though,atfirst

thought,therewouldseemtobeatleastalessthan50percent

chanceofthishappening,thetruthisthata1or6willbe

showingabout56percentofthetime.Thisisthesame

probabilityforanytwodifferentnumbers,bytheway,soyou

coulduseanattractivestranger'sbirthdaytopickthedigitsand

maybestartaconversation,whichcouldleadtomarriage,

children,orboth.

IfyouaremorehonestthanmyUncleFrank(andthereisa98

percentchancethatyouare),herearesomeeven-moneybets

withdice.TheoutcomesincolumnAareequallyaslikelyto

occurastheoutcomesincolumnB:

A

2or12

2,3,or4

5,6,or7



B

3

7

8,9,10,11,or12



Theoddsareevenforeitheroutcome.



WhyItWorks

Forthebetspresentedinthishack,herearethecalculations

demonstratingtheprobabilityofwinning:

Wager

5or9versus7

1or6showing

2or12versus3

2,3or4versus7

5,6or7versus8

orhigher



Numberofwinning

Calculation Resultingproportion

outcomes

8versus6

20

2versus2

6versus6



8/14

20/36

2/4

6/12



.571

.556

.500

.500



15versus15



15/30



.500



The"Wager"columnpresentsthetwocompetingoutcomes

(e.g.,willa5or9comeupbeforea7?).The"Numberof

winningoutcomes"columnindicatesnumberofdifferentdice

rollsthatwouldresultineithersideofthewager(e.g.,8

chancesofgettinga5or9versusonly6chancesofgettinga

7).The"Resultingproportion"columnindicatesyourchancesof

winning.

Youcanwintwodifferentwayswiththesesortsofbets.Ifitis

aneven-moneybet,youcanwagerlessthanyouropponent

andstillmakeaprofitinthelongrun.Hewon'tknowtheodds

areeven.Ifchancefavorsyou,though,considerofferingyour

targetaslightlybetterpayoff,orpicktheoutcomethatislikely

tocomeupmoreoften.



Hack44.SharpenYourCard-Sharping



InTexasHold'Emandotherpokergames,therearea

fewbasicpreliminaryskillsandabitofbasicknowledge

aboutprobabilitythatwillimmediatelypushyoufrom

absolutebeginnertothemorecomfortablelevelof

knowingjustenoughtogetintotroubleasacardsharp.

TheprofessionalTexasHold'Empokerplayerswhoappearon

televisionaredifferentfromyouandmeinjustacoupleof

importantways.(Well,theylikelydifferfromyouinjusta

coupleofimportantways;theydifferfrommeinsomany

importantwaysthatevenmycomputerbraincan'tcountthat

high.)Herearetwoareasofpokerplayingthattheyhave

mastered:

Knowingtheroughprobabilityofhittingthecardstheywant

atdifferentstagesinahand(intheflop,ontheriver,and

soon)

Quicklyidentifyingthepossiblebetterhandsthatcouldbe

heldbyotherplayers

Thishackpresentssometipsandtoolsformovingfromnovice

tosemi-pro.Thesearesomesimplehunksofknowledgeand

quickrulesofthumbformakingdecisions.Liketheotherpoker

hacksinthisbook,theyprovidestrategytipsbasedpurelyon

statisticalprobabilities,whichassumearandomdistributionof

cardsinastandard52-carddeck.



ImprovingYourHand



Halfthetime,youwillgetapairorbetterinTexasHold'Em.I'll

repeatthatbecauseitissoimportantinunderstandingthe

game.Halfthetime(alittleunder52percentactually),ifyou

stayinlongenoughtoseesevencards(yourtwocardsplusall

fivecommunitycards),youwillhaveatleastonepair.Itmight

havebeeninyourhand(apocketorwiredpair),itmightbe

madeupofonecardinyourhandandonefromthecommunity

cards,oryourpairmightbeentirelyinthecommunitycardsfor

everyonetoclaim.

Ifforthemajorityofthetimetheaverageplayerwillhavea

pairwhendealtsevencards,thenstickingarounduntiltheend

withalowpairmeansyouareonlystatisticallyspeaking,of

courselikelytolose.Inotherwords,thereisagreaterthan50

percentchancethattheotherplayerhasatleastapair,and

thatpairwillprobablybe8sorhigher(onlysixoutofthirteen

pairsare7sorlower.)

KnowinghowcommonpairsareexplainswhyAcesaresohighly

valued.Muchofthetime,heads-upbattlescomedowntoa

battleofpairversuspair.Anothergoodproportionofthetime,

theAceplaysanimportantroleasakickerortiebreaker.Aces

aregoodtohave,andit'sallbecauseoftheodds.



Probabilities

Decisionsaboutstayinginorraisingyourbetinanattemptto

lowerthenumberofopponentsyouhavetobeatcanbemade

morewiselyifyouknowsomeofthecommonprobabilitiesfor

someofthecommonlyhoped-foroutcomes.Table4-16

presentstheprobabilityofdrawingacardthathelpsyouat

variousstagesinahand.Theprobabilitiesarecalculatedbased

onhowmanycardsareleftinthedeck,howmanydifferent

cardswillhelpyou(yourouts),andhowmanymorecardswill

bedrawnfromthedeck.Forexample,ifyouhaveanAce-King

andhopetopairup,therearesixcardsthatcanmakethat



happen;inotherwords,youhavesixouts.Ifyouhaveonlyan

AcehighbuthopetofindanotherAce,youhavethreeouts.If

youhaveapocketpairandhopetofindapowerfulthirdinthe

communitycards,youhavejusttwoouts.

TableProbabilityofimprovingyourhand



Cardslefttobedealt

5(beforetheflop)

2(aftertheflop)

1(aftertheturn)



Sixouts

49percent

24percent

13percent



Threeouts

28percent

12percent

7percent



Twoouts

19percent

8percent

4percent



Thesituationsdescribedhereassumeyouhavealreadybeen

dealttwocards.Afterall,inmostpokergames,thebetbefore

theflopispredetermined,sotherearenodecisionstobemade.

Bytheway,becauseyoushouldprobablybackoutofhands

thatdidnotamounttoanythingintheflop,you'llwanttoknow

yourchancesofimprovingintheflopitself.Theyare:

Remainingouts

6

3

2



Oddsyou'llhitawinningcardintheflop

32percent

17percent

12percent



Implications

Hereareafewquickobservationsandimplicationstoetchin

yourmind,basedonthedistributiondescribedinTable4-16.

Halfthetime,youwillpairup.Thisistrueforhighcards,such

asBigSlick(Ace-King)orlowcards,suchas2-7.Youcaneven

pickfromthetwocardsyouhaveandpairthatoneup28



percentofthetime.Implication:whenlowonchipsin

tournamentplay,goall-inassoonasyougetthatAce.

Ifyoudon'thitthethirdcard,youneedtoturnapairintoaset

(threeofakind)ontheflop,andthereisonlyan8percent

chanceyouwillhititdowntheroad.Implication:don'tspend

toomuchmoneywaitingaroundforyourlowpairtoturnintoa

gangbusterhand.

YourAce-KingorAce-Queenthatlookedprettygoodbeforethe

flopdiminishesinpotentialasmorecardsarerevealedwithout

pairinguporgettingstraightdraws.87outof100times,that

greatstartinghandremainsameaslyhigh-card-onlyhandif

youhaven'thitbeforetheriver.Implication:stayinwiththe

unfulfilleddreamthatisAce-Kingonlyifyoucandosocheaply.



ReadingtheCommunityCardsinaFlash

Herearesomecommon-sensestatementsaboutyour

opponents'handsthatmustbetruebutaren'talwayssaidout

loud:

Ifthecommunitycardsdonot

have...

Apair

Apair

Threecardsofthesamesuit

Threecardswithinafive-cardrange



Youropponent(s)cannot

have...

Fourofakind

Afullhouse

Aflush

Astraight



Youcanmakequickerdecisionsaboutwhatyouropponents

mighthavebylearningtheserules.Then,youcanautomatically

ruleoutkillerhandswhenthesituationissuchthattheyare

impossible.Youmaynotbeworriedaboutspeed,butyoucan

spendyourtimeconcentratingonmoreimportantdecisionsif



youdon'thavetowastementalenergyfiguringthesethingsout

fromscratcheachtime.







Hack45.AmazeYour23ClosestFriends



Whatarethechancesofatleasttwopeopleinagroup

sharingabirthday?Dependingonthenumberofpeople

present,surprisinglyhigh.Impressyourfriendsat

parties(andperhapswinsomemoneyinabarbet)using

thesesimplerulesofprobability.

Someeventsthatseemlogicallyunlikelycanactuallyturnout

tobequiteprobableinsomecases.Onesuchexampleis

determiningtheprobabilitythatatleasttwopeopleinagroup

shareabirthday.Manypeopleareshockedtolearnthataslong

asthereareatleast23peopleinthegroup,thereisabetter

than50percentchancethatatleast2ofthemwillhavethe

samebirthday!Byusingafewsimplerulesofprobability,you

canfigureoutthelikelihoodofthiseventoccurringforgroups

ofanysize,andthenamazeyourfriendswhenyourpredictions

cometrue.



Youcouldalsousethisresulttomakesomecashinabarbet(aslong

asthereareatleast23peoplethere).



So,howdoyoufigureouttheprobabilityofatleasttwopeople

sharingabirthday?Tosolvethisproblem,youneedtomakea

coupleofassumptionsabouthowbirthdaysaredistributedin

thepopulationandknowafewrulesabouthowprobabilities

work.



GettingStarted

Todeterminethechancesofatleasttwopeoplesharinga

birthday,wehavetomakeacoupleofreasonableassumptions

abouthowbirthdaysaredistributed.First,let'sassumethat

birthdaysareuniformlydistributedinthepopulation.This

meansthatapproximatelythesamenumberofpeopleareborn

oneverysingledayoftheyear.

Thisisnotnecessarilyperfectlytrue,butit'scloseenoughfor

ustostilltrustourresults.However,thereisonebirthdayfor

whichthisisdefinitelynottrue:February29,whichoccursonly

everyfouryearsonLeapYear.Thegoodnewsisthatfew

enoughpeoplearebornonFebruary29thatitiseasyforusto

justignoreitandstillgetaccurateestimates.

Oncewe'vemadethesetwoassumptions,wecansolvethe

birthdayproblemwithrelativeease.



ApplyingtheLawofTotalProbability

Inourproblem,thereareonlytwomutuallyexclusivepossible

outcomes:

Atleasttwopeopleshareabirthday.

Noonesharesabirthday.

Sinceoneofthesetwothingsmustoccur,thesumofthetwo

probabilitieswillalwaysbeequaltoone.Statisticianscallthis

theLawofTotalProbability,anditcomesinhandyforthis

problem.



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