A Mathematical Perspective on Gambling

Transcription

1 A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal distributios, the Cetral Limit Theorem, Bayesia aalysis, ad statistical hypothesis testig. These topics are applied to gamblig games ivolvig dice, cards, ad cois.. Itroductio. I 65, the well-kow gambler the Chevalier de Mere posed a set of problems cocerig gamblig odds to Blaise Pascal. These problems iitiated a series of letters betwee Pascal ad Pierre de Fermat, which established the basis of moder probability theory [, pp. 9 53]. The Chevalier de Mere was a astute gambler who observed a slight advatage i attemptig to roll a 6 i four tosses of a sigle die ad a slight disadvatage i attemptig to throw two 6s, a result referred to as a double-6, i twety-four tosses of two dice [, p. 89]. De Mere oticed that the proportio of the umber of tosses to the umber of possible outcomes is the same for both games :6 for the sigle-die game, ad :36 for the double-die game. Thus he cocluded that the odds of wiig each game should be equal. Yet from his experiece of playig these two games, he felt cofidet that there was a advatage to the sigle-die game ad a disadvatage to the double-die game. Thus he asked Pascal for a explaatio for this discrepacy. I their correspodece, Pascal ad Fermat laid the groudwork for the cocepts of sample space, probabilistic evets, expectatios, ad the biomial distributio. Sectio of this paper itroduces the ideas of sample spaces ad evets, ad applies these cocepts to de Mere s die games, poker, ad coi-tossig games. The work i this sectio cofirms de Mere s ituitio that the sigle-die game has a slight advatage p = ad the double-die game has a slight disadvatage p = 0.9. Sectio 3 defies expectatios, ad fids the expected payoff for a sigle trial of the games discussed i Sectio. Sectio presets the biomial ad ormal distributios, ad explais how these distributios ca be used to calculate the odds of makig a profit i a large umber of trials of these games. Fially, Sectio 5 develops the cocepts of Bayesia aalysis ad statistical hypothesis testig, ad shows how these methods ca be used to calculate the likelihood that a give result came from a particular distributio.. Sample Spaces ad Evets. A sample space Ω is a set of all possible outcomes of a model of a experimet [, p. 6]. This set is fiest grai ; i other words, all distict experimetal outcomes are listed separately. I additio, the elemets i the set are mutually exclusive ; each trial of the experimet ca result i oly oe of the possible outcomes. The words outcome ad evet are commoly iterchaged; however, i probability theory these two words have distict meaigs. A evet is a subset of possible outcomes. As a example, a sigle roll of a die is a experimet with 3

2 MIT Udergraduate Joural of Mathematics six possible outcomes, correspodig to the faces that lad up. Rollig a eve umber is a evet that icludes the experimetal outcomes,, ad 6. The coditioal probability is the probability of Evet A give Evet B, ad is defied by P A B P A B = P B for P B > 0; see [, pp. 3 6]. Evet B is a coditioig evet, ad redefies the sample space for Evet A. This ew sample space icludes oly the outcomes i which B occurs. Evets A ad B are called idepedet if the occurrece of oe evet does ot affect the likelihood of the occurrece of the other [, p. 7]. More formally, Evets A ad B are idepedet if P A B = P A ad P B A = P B. The cocepts of coditioal probability ad idepedece are useful for Bayesia aalysis, discussed i Sectio 5. The idea of idepedece is also importat for the biomial distributio, discussed i Sectio. We ow cosider three examples of sample spaces ad probabilities. First we discuss die games. A sigle roll of a die has six possible outcomes, correspodig to the faces that lad up. If the die is fair, the the probability of ay particular face ladig up is /6. More formally, the sample space Ω is the set {,, 3,, 5, 6}, ad the probability is P X = i = /6. Games that ivolve rollig two dice have thirty-six possible outcomes sice the roll of each die has six possible outcomes. I these games, the dice are distiguishable; therefore, the outcomes i, j ad j, i are couted separately. More formally, the sample space Ω is the set of pairs i, j where i, j {,, 3,, 5, 6}, ad the probability is PX=i,Y=j =/36. The first game that the Chavalier de Mere discussed with Pascal ivolves rollig a die four times [, pp ]. If the die lads o a 6 at least oce i four trials, the the gambler wis. If the die ever lads o a 6 i the four trials, the the gambler loses. The four tosses are idepedet. Hece, the probability of ot gettig a 6 i four trials is 5/6, or It follows that the probability of gettig at least oe 6 i four trials is equal to 0.83, or The secod game discussed by Pascal ad de Mere ivolves simultaeously rollig two dice twety-four times [, pp ]. If both dice lad o 6, a outcome called a double-6, the the gambler wis the bet. If a double-6 does ot occur, the the gambler loses the bet. As i the first game, the trials are idepedet. Hece, the probability of ot gettig a double-6 i twety-four trials is 35/36, or It follows that the probability of gettig a double-6 i twety-four trials is equal to , or 0.9. De Mere oticed the slight advatage of bettig o the first game ad the slight disadvatage of bettig o the secod game. To discer these small difffereces, de Mere must have played these games a large umber of times. A estimate for this umber is obtaied by usig statistical hypothesis testig i Sectio 5. As a secod example of sample spaces ad probabilities, we cosider card games. A stadard deck of cards cosists of fifty-two cards, divided ito four suits ad thirtee raks [, pp. 7 5]. I the simple case of drawig oe card from the deck, the sample space Ω is the fifty-two possible cards, ad the probability of gettig ay particular card is /5. Most card games ivolve hads of cards i which the order of the cards does ot matter. The umber of possible hads is the biomial coefficet x, where is the

3 A Mathematical Perspective o Gamblig 5 size of the deck of cards ad x is the size of a had. The biomial coefficiet x is defied by! = x x x! x!. The symbol x is read choose x. I a stadard poker game, each player receives a had of five cards [, p. 5 55]. The sample space Ω for poker is the set of all possible five-card hads, i which order is irrelevat. The size of this sample space is 5 5, or, 598, 960. Thus the probability of gettig a had with five particular cards is /, 598, 960, or The probability of gettig a particular type of had is computed as follows: Cout the umber of ways this type of had ca occur, the divide this result by the total umber of hads. For example, a straight flush is a ru of five cosecutive cards i a particular suit. A straight flush ca occur i ay of the four suits, ad ca begi i te possible positios, through Jack. Thus, there are forty ways to obtai a staight flush. The probability of gettig a straight flush is, therefore, 0/, 598, 960, or As aother example, a full house cosists of three of a kid ad oe pair. There are 3, or 3, possible raks for the three of a kid ad 3, or, possible combiatios of the four cards of this rak. There are, or, remaiig raks for the pair ad, or 6, possible combiatios of the four cards of this rak. Thus, the probability of gettig a full house is 3 3 3, 7 5 =, 598, 960 = Table - lists possible poker hads, the umber of ways i which they ca occur, ad their probabilities of occurrig; it was take from [, p. 53]. Table - A table of poker hads ad their probabilities HAND NUMBER of WAYS PROBABILITY Straight flush Four of a kid Full House 3, Flush 5, Straight 0, Three of a kid 5, Two pair 3, Oe pair,098, Worse tha oe pair,30, TOTALS =, 598, As the last example of sample spaces ad probabilities, we discuss coi-tossig games. The sample space Ω for a sigle toss of a coi cosists of two outcomes: heads H, ad tails T. If the coi is fair, the the probability of heads ad the probability of tails are both /. The sample space for a game cosistig of five tosses of a coi is 5, or 3,

4 6 MIT Udergraduate Joural of Mathematics sequeces of the form X X X 3 X X 5, where each X i is H with probability /, or T with probability /. The tosses are idepedet. Therefore, the probability of gettig o heads i five tosses is / 5, or Cosequetly, the probability of gettig at least oe head i five tosses of a fair coi is 0.035, or Cosider aother game that cosists of attemptig to toss exactly three heads i four trials. The sample space for this game is, or 6, sequeces of the form X X X 3 X, where each X i is H with probability /, or T with probability /. The evet of gettig exactly three heads correspods to the followig sequeces: HHHT, HHTH, HTHH, ad THHH. Thus, the probability of wiig this bet is /, or 0.5. Aother expressio for this probability is 3 /3 /, which is iterpreted as the umber of ways to choose three of the four positios for the heads multiplied by the probability of gettig three heads ad the probability of gettig oe tail. 3. Expectatios. For a iteger radom variable X, the expectatio EX is defied to be the sum of all possible experimetal values of the variable weighted by their probabilities [5, pp ]. The expectatio is writte as EX = P X =. The expectatio of a radom variable is its mea or expected value. Oe applicatio of the cocept of expectatio is to formalize the idea of the expected payoff of a game. For a iteger radom variable X with mea EX = µ, the variace σ X is defied by the formula, σ X = E[X µ ], see [3, p. 9]. The variace ca also be expressed as σ X = EX µ ; this equatio is proved i [3, p. 9]. The variace measures the spread of the distributio. The stadard deviatio σ X is the positive square root of the variace. This cocept is used i Sectios ad 5. We ow compute the expected payoffs of the games discussed i Sectio. First we fid the expected payoffs for die games. Suppose a gambler gais oe dollar for wiig a game, ad gives up oe dollar for losig. I de Mere s sigle-die game, the expected payoff is EX = $ $ 0.83 = $0.035, or approximately four cets. I his double-die game, the expected payoff is EX = $ $ = $0.07, or approximately a two-cet loss. Next we compute the expected payoff for casio poker. Suppose a gambler wages a oe-dollar bet o a five-card had. The gambler s payoff depeds o the type of had, with a rare had such as a straight flush resultig i a higher payoff tha a more commo had such as oe pair. Table 3- provides oe possible payoff scheme. Accordig to this scheme, the expected earigs for oe game are EX = $ $ $ $ $ $ $ $ $ = $ ,

5 A Mathematical Perspective o Gamblig 7 or approximately a five-cet loss. Table 3- A table of poker hads ad their payoffs HAND PROBABILITY PAYOFF Straight flush $000 Four of a kid $500 Full House 0.00 $50 Flush $5 Straight $0 Three of a kid 0.09 $5 Two pair $ Oe pair $0 Worse tha oe pair $ TOTAL Fially, we fid the expected payoffs for coi-tossig games. Suppose a gambler wis oe dollar for tossig heads ad loses oe dollar for tossig tails with a fair coi. His expected payoff is EX = $ $ 0.5 = 0. Suppose he bets oe dollar o gettig exactly three heads i four tosses. expected payoff is The his EX = $ $ 0.75 = $0.50, or a fifty-cet loss.. Repeated Trials. The previous sectio explored a gambler s expected earigs o a sigle trial of a game. A gambler may, however, wat to calculate his expected earigs if he plays the game may times. This sectio presets methods for calculatig the expected payoff ad estimatig the chaces of makig a profit o may repetitios of a game. Cosider the problem of calculatig the probability of wiig t trials out of, whe the probability of wiig each trial is set [, pp. 6]. The trials ca be expressed as idepedet, idetically distributed radom variables X,, X. Each X i has two outcomes:, which represets a wiig trial, ad occurs with probability p, ad 0, which represets a losig trial, ad occurs with probability p. The expectatio for each X i is EX i = p + 0 p = p. The radom variable S represets the umber of wiig trials i a series of trials, ad is give by S = X + X + + X.

6 8 MIT Udergraduate Joural of Mathematics Therefore, the expected umber of wiig trials is ES = EX + X + + X = EX + EX + + EX = EX i = p, sice the radom variables X i are idepedet ad idetically distributed. Furthermore, the probability of wiig exactly t trials out of ca be calculated by multiplyig the umber t of ways to choose t of the trials to be successes by the probability p t of wiig t games, ad the probability p t of losig t games. This is the biomial distributio, ad is defied by P S = t = t p t p t, for 0 t. As calculated above, the expectatio of S is Moreover, the variace of S is σ S ES = p. = p p. The probability of wiig at least t games out of is the sum of the probabilities of wiig t through games sice these evets are mutually exclusive. This probability is deoted by P S t = i=t i p i p i, for 0 t. As a example, cosider de Mere s sigle-die game, i which the probability of success o each trial is I thirty trials, the expected umber of successes is ES 30 = = 5.5, which rouds to 6. The probability of wiig exactly ietee trials out of thirty is 30 P S 30 = 9 = = Furthermore, the probability of wiig at least ietee trials out of thirty is P S 30 9 = 30 i= i i = The ormal, or Gaussia, distributio is importat because it describes may probabilistic pheomea [, pp. 07 ]. Cosider a ormal radom variable w with mea µ ad variace σ. The ormal distributio is defied by P w a = a σ π e w µ/σ / dw.

7 A Mathematical Perspective o Gamblig 9 A stadard ormal radom variable z has µ = 0 ad variace σ =. The stadard ormal distributio is defied by P z a = a π e z / dz = Φa. Most probability ad statistics textbooks provide tables for Φx; for example, see [3, pp ] ad [5, p. 77]. The Cetral Limit Theorem states that the probability distributio for a sum S of idepedet, idetically distributed radom variables X i approaches the ormal distributio as the umber of radom variables goes to ifiity [, pp. 5 9]. This result is remarkable as it does ot deped o the distributio of the radom variables. The Cetral Limit Theorem states that t ES lim P S t = Φ Hece, for fiite, but large values of, the followig approximatio holds: t ES P S t Φ. For example, i de Mere s sigle-die game, the radom variable S 30 represets the umber of wiig games out of thirty. The radom variable σ S z = S 30 ES 30 σ S30 is approximately stadard ormal. Therefore, the probability of wiig at least ietee trials out of thirty is σ S. P S 30 9 = P S ES = P z σ S Φ = = 0.3. This probability provides a close estimate of the actual value 0.388, which was foud usig the biomial distributio. Most probability ad statistics textbooks provide graphs of the biomial ad ormal distributios for various values of ; for example, see [, p. 7] ad [3, p. 7]. The graphs idicate that as, the ormal distributio forms a good approximatio for the biomial distributio. 5. Bayesia Aalysis ad Statistical Hypothesis Testig. So far we have looked oly at the probabilities of observig particular outcomes ad the expected payoffs for various games with kow distributios. This last sectio focuses o estimatig the likelihood that a observed outcome came from a particular distributio. For example, a gambler tosses forty-oe heads i oe-hudred tosses of a coi, ad wishes to kow

8 30 MIT Udergraduate Joural of Mathematics if the coi is fair. Bayesia aalysis ad statistical hypothesis testig are two methods for determiig the likelihood that a outcome came from a particular distributio. Bayesia aalysis tests the likehlihood that a observed outcome came from a particular probability distributio by treatig ukow parameters of the distributio as radom variables [, pp ]. These ukow parameters have their ow probability distributios, which ofte are derived from the assumed probability distributio of the outcome. The derivatios use Bayes Theorem, which states that P A i B = P B A i P A i N i= P B A i P A i. For example, let θ be a ukow parameter that has two possible values, correspodig to the evets A = {θ = θ } ad A = {θ = θ }. The a priori probabilities for A ad A are P A ad P A. They do ot take the iformatio gaied from the occurrece of evet B ito accout. Evet B is a observed outcome, which has a assumed probability ditributio that depeds o the ukow parameter θ. Cosider the evet A = {θ = θ }. By Bayes Theorem, the probability of A give B is P A B = P B A P A P B A P A + P B A P A. Bayesia aalysis ca be applied to the example of the gambler who tossed forty-oe heads i oe-hudred coi tosses, ad wishes to kow if his coi is fair. Assume that the gambler has two cois, a fair coi p = / ad a biased coi p = /. Evet A correspods to his selectig the fair coi, ad Evet A correspods to his selectig the biased coi. He radomly selects the coi; therefore, the a priori probabilites, P A ad P A, are both equal to /. Evet B is the observed outcome that S 00 =. By Bayes Theorem, the probability that the gambler used the fair coi is P p = S 00 = = P S 00 = p = = 00 = P S 00 = p = P p = P p = + P S00 = p = P p = Bayesia aalysis combies prior kowledge of the ukow parameter i this case, the a priori probabilities P p = / ad P p = / with experimetal evidece to form a better estimate of this parameter. I statistical hypothesis testig, a observed outcome is compared to a assumed model to determie the likelihood that the outcome came from the model [5, pp ]. If the outcome deviates sigificatly from the expected outcome of the assumed model, the it is ulikely that the result came from this model. Statistical hypothesis testig, therefore, ivolves determig what level of deviatio from the expected outcome is sigificat. Cosider a radom variable Y that has a probability distributio that relies o the parameter θ. This method tests a ull hypothesis H 0 : θ = θ 0 agaist a alterative

9 A Mathematical Perspective o Gamblig 3 hypothesis H, which may take the form θ < θ 0, θ > θ 0, or θ θ 0 ; see [3, pp ]. A sigificace level α is chose for the test, ad a critical value c that depeds o α is calculated. The critical regio cosists of the critical value, ad all values that are more extreme i the directio of the alterative hypothesis. If the observed outcome of Y falls withi the critical regio, the H 0 is rejected ad H is accepted. However, if the observed outcome of Y does ot fall withi the critical regio, the H 0 is accepted. The coi-tossig example ca be worked out usig statistical hypothesis testig. The hypothesis H 0 : p = / is tested agaist a alterative hypothesis H : p /. I the model where H 0 is true, the expectatio is ES 00 = 00 / = 50, ad the variace is σ S 00 = 00 / / = 5. By the Cetral Limit Theorem, the variable S 00 has a almost ormal distributio. Therefore, the variable z = S has a approximately stadard ormal distributio. Thus, P.96 S = 0.95, or P 0. S = Sice the observed outcome, S 00 =, falls withi this iterval, H 0 is accepted at the α = 0.05 sigificace level. The sigificace level is the probability that the assumed model H 0 is erroeously rejected. This error is commoly referred to as a type I error. Likewise, a type II error is the evet that the assumed model H 0 is erroeously accepted. The probability of this type of error is deoted as β. Ideally, we wat α ad β to be as small as possible; however, decreasig both ivolves icreasig the sample size. For example, whe H 0 : p = / is tested agaist H : p = / i a coi-tossig model, H 0 will be rejected if the umber of heads S 00 is below some value c, kow as the critical value. I this case, α represets the probability that S 00 is less tha c whe p = /, ad β represets the probability that S 00 is greater tha c whe p = /. For istace, if α = 0.00 ad β = 0.00, this critical value c ad the sample size ca be foud by solvig the equatios for α ad β. Namely, α = 0.00 = P S c p = S = P c = P z z Because z 0.00 = 3.090, it follows that Likewise, c β = 0.00 = P S c 3.090, ad c p = = P Because z 0.00 = 3.090, it follows that c 3 S , ad c c 3 Combiig Expressios 5- ad 5- yields that = P z z

10 3 MIT Udergraduate Joural of Mathematics Thus, is equal to 3.8, which is rouded to 33, ad c is equal to 8.6, which is rouded to 9. To reduce the probabilities of Type I ad Type II errors, α ad β, to 0.00, the umber of trials must be icreased to 33. As metioed i the Itroductio, the Chevalier de Mere oticed that his bet o the double-die game was less tha chace [, pp. 83 8]. Some of the techiques i this last sectio ca provide a estimate for the umber of trials de Mere must have played to legitimately claim that the probability of success was ideed less tha /. Assume that de Mere played the game a large umber of times, ad foud that he rolled a double-6 i 9% of the games. To determie whether he could legitimately accept that p < /, de Mere would eed to test the hypothesis H 0 : p = / agaist the hypothesis H : p < /. Let Y be the umber of times de Mere rolled a double-6 i trials. From the Cetral Limit Theorem, it follows that Y has a almost ormal distributio. Thus, the variable z = Y p p p = Y p p p has a approximate stadard ormal distributio. For a 0.05 sigificace level, it follows that If de Mere estimated that Y Cosequetly, Y P is 9 00, the 9 00 z 0.05 = 0.05 = α. z 0.05 =.65. = To legitimately accept the hypothesis H : p < / for the double-die game, de Mere must have played 6765 games. Refereces [] David, F. N., Games, Gods ad Gamblig, Hafer Publishig Compay, 96. [] Drake, A. W., Fudametals of Applied Probability Theory, McGraw-Hill, Ic., 988. [3] Hogg, R. V., ad Tais, E. A., Probability ad Statistical Iferece, Pretice-Hall, 997. [] Packel, E. W., The Mathematics of Games ad Gamblig, The Mathematical Associatio of America, 98. [5] Rota, G. C., with Baclawski, K., Billey, S., ad Sterlig, G., Itroductio to Probability Theory, Third Prelimiary Editio, Birkhauser Press, 995.