A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING

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1 A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING MATTHEW ACTIPES Abstract. This paper begis by defiig a probability space ad establishig probability fuctios i this space over discrete radom variables. We the defie the expectatio ad variace of a radom variable with the ultimate goal of provig the Weak Law of Large Numbers. We the apply these ideas to the cocept of gamblig to show why casios exist ad whe it is beeficial to a idividual to gamble as defied by idividual utility. Cotets 1. The Probability Space 1 2. Radom Variables ad Probability Fuctios 2 3. Expectatio ad Variace 4 4. Casio Ecoomics 8 5. Gambler s Utility 9 Ackowledgmets 10 Refereces The Probability Space To motivate the probability space we ll begi by statig that a radom experimet is oe whose results may vary betwee trials eve uder idetical coditios. To try to predict how ofte each trial will produce a certai result we eed to look at the set Ω of all possible outcomes of the radom experimet. We the deote A as a subset of the power set of Ω, which is the collectio of subsets of Ω. Fially, we eed a probability measure P that measures the likelihood of each evet i A occurrig as a result of a radom experimet. With these ideas i mid, we ca defie the otio of a probability space. Defiitio 1.1. A Probability Space is a triple <Ω,A,P> represetig the probability of every possible evet i a system occurrig, where Ω is the set of all possible outcomes (the sample space), A is a collectio of subsets of Ω (evets) which satisfies: A, for a evet a i A, (Ω\a i ) A, if a i A for i=1,2,... the i a i A, Date: August 24,

2 2 MATTHEW ACTIPES ad P is the probability measure where P : A [0, 1] is a fuctio ( satisfyig P(Ω) = ) 1 ad for a coutable set of disjoit evets a 1, a 2...a, P a i = P(a i ). i=1 i=1 Because it is clear each evet a i is a subset of A, we ca exted the otio of set theory to the evets. By itroducig the biary operatios of uio ad itersectio, for ay two evets a j ad a k, we say that a j a k is the evet either a j or a k or both, a j a k is the evet both a j ad a k. Note that if the evets a j ad a k are mutually exclusive, the a j a k =, meaig that both evets caot occur simultaeously. Lookig further at these coditios, if the probability that a j a k occurs depeds oly o the probability that a j occurs disjoit from the probability that a k occurs, the two evets are said to be idepedet. Defiitio 1.2. Two evets are idepedet if ad oly if P(a j a k ) = P(a j )P(a k ). We would ow like to exted the probability measure to our sample space to motivate the probability mass fuctio ad distributio fuctio. First, we will eed to defie the radom variable. 2. Radom Variables ad Probability Fuctios We are ow goig to take the sample space ad assig a umber to each outcome w Ω where each umber cotais some iformatio about the outcome. This fuctio X is called a radom variable. Defiitio 2.1. A radom variable X is a fuctio X: Ω R. Example 2.2. Suppose we toss a coi three times. The Ω = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}, where H represets the coi ladig o heads ad T represets the coi ladig o tails. Let s defie our radom variable X to be the fuctio that seds each evet cotaiig a eve umber of heads to 1 ad each evet cotaiig a odd umber of heads to 0. Thus a table of our fuctio looks like: Table 1-1 Outcome HHH HHT HTH HTT THH TTH THT TTT X Defiitio 2.3. A discrete radom variable is a radom variable that takes o a fiite or coutably ifiite umber of values. Remark 2.4. For a discrete radom variable X, we ca arrage the possible values that X takes o as x 1, x 2... where x i x j for i>j. Each of these values of X will have a probability associated with it, P(X = x i ), where 1 i for distict values of X ad [1, ). From this we ca defie the probability mass fuctio ad distributio fuctio of X. Defiitio 2.5. For a discrete radom variable X, f(x i ) is the probability mass fuctio of X if f(x i ) = P(X = x i ) for all i.

3 A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING 3 Defiitio 2.6. The distributio fuctio F(x) for a discrete radom variable X is defied as F (x) = P(X x) = y x f(y), where x R. Note: For ay y x j, where x j is a value take o by X, P(y) = 0. Therefore, for discrete radom variables, f(y) is equivalet to writig y x f(x j ). x j x Note: Aside from discrete radom variables, there are also cotiuous radom variables which do ot satisfy Defiitio 2.3. Defiitio 2.7. A radom variable X is cotiuous if its distributio fuctio ca be writte as F (x) = P (X x) = x where f X is the probability desity fuctio of X. f X (u)du, Remark 2.8. The probability desity fuctio is the cotiuous radom variable s aalog to the discrete radom variable s probability mass fuctio. While this defiitio is importat i the field of probability, this paper will oly focus o discrete radom variables. The previous ideas ca be exteded to the case with two or more discrete radom variables. Oce we defie the joit distributio for the discrete radom variables X ad Y, geeralizatios ca be made to the case with more tha two radom variables. Defiitio 2.9. For two discrete radom variables X ad Y, f(x i, y i ) is the joit probability mass fuctio of X ad Y if for all i. f(x i, y i ) = P(X = x i, Y = y i ) Defiitio For two discrete radom variables X ad Y, the joit distributio fuctio of X ad Y is defied by F (x, y) = P(X x, Y y) = f(x i, y i ), x i x y i y where x i are the values of X less tha or equal to x ad y i are the values of Y less tha or equal to y. Just as with two distict evets, the cocept of idepedece also applies to radom variables. If each pair of evets take from the radom variables X ad Y is idepedet, where oe evet is take from X ad oe from Y, we say that X ad Y are idepedet.

4 4 MATTHEW ACTIPES Defiitio If for all x i ad y i the evets {X = x i } ad {Y = y i } are idepedet, we say that X ad Y are idepedet radom variables. The which meas P(X = x i, Y = y i ) = P(X = x i )P(Y = y i ), f(x i, y i ) = f X (x i )f Y (y i ), where f X ad f Y are the probability mass fuctios of X ad Y. 3. Expectatio ad Variace Now that we are able to calculate the probability that a certai value of X will result from a trial, we wat to kow beforehad what the expected outcome of the trial will be. For a discrete radom variable X we ca write the possible values of X as x 1, x 2,...x ad associate with each the probability P(X = x i ) to fid the expectatio of X. Defiitio 3.1. For a discrete radom variable X the Expectatio of X, deoted E(X), is defied by E(X) = x 1 P(X = x 1 ) x P(X = x ) = x i Ω x i P(X = x i ) = x i Ω x i f(x i ), for [1, ). Note: The expectatio of X is also called the expected value of X or the mea of X ad is ofte deoted by µ. Example 3.2. Suppose that a fair coi is about to be flipped. If it lads o heads, you wi 30 dollars. If it lads o tails, you lose 20 dollars. We ca defie X to be the radom variable that seds the result of the flip to the amout of moey that you wi. The for heads, x 1 is +30 ad f(x 1 ) is For tails, x 2 is -20 ad f(x 2 ) is Thus the expectatio is E(X) = (+30)(0.50) + ( 20)(0.50) = +5. Therefore, o each trial of this game, you are expected to ear 5 dollars i profit. Remark 3.3. The cocept of expectatio is of extreme importace to mathematical gamblig theory. We will see later that expectatio is closely related to et profit whe playig a large umber of games i a casio. For two radom variables X ad Y it ca be show that the expectatio of the sum or product of the variables is equal to the sum or product of the expectatios uder certai coditios. The followig two theorems will oly be proved i the discrete case. Geeralizatios ca easily be made i the case that X ad Y are cotiuous variables, but these geeralizatios are ot ecessary to this paper. Theorem 3.4. If X ad Y are radom variables, the E(X + Y ) = E(X) + E(Y ).

5 A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING 5 Proof. Let f(x,y) be the joit probability mass fuctio of X ad Y. The we have E(X + Y ) = (x + y)f(x, y) x y = xf(x, y) + yf(x, y) x y x y = E(X) + E(Y ). Theorem 3.5. If X ad Y are idepedet radom variables, the E(XY ) = E(X)E(Y ). Proof. Let f(x,y) be the joit probability mass fuctio of X ad Y. X ad Y are idepedet, so f(x,y)=f 1 (x)f 2 (y). Therefore, E(XY ) = xyf(x, y) = xyf 1 (x)f 2 (y) x y x y = [xf 1 (x) yf 2 (y)] x y = x [xf 1 (x)e(y)] = E(X)E(Y ). Also oteworthy is the fact that due to the ature of sums, a costat c ca be pulled through the expectatio operator. Theorem 3.6. Let X be a radom variable. If c is ay costat, the Proof. E(cX) = ce(x). E(cX) = x i Ω cx i f(x i ) = c x i Ω x i f(x i ) = ce(x). Now that we have begu workig with the expectatio of a radom variable X, we d like to have a idea of how widely the values of X are spread aroud E(X). Recallig that E(X) is deoted by µ, we ca defie the cocepts of variace ad stadard deviatio. Defiitio 3.7. For a radom variable X with mea µ the variace of X, deoted Var(X), is defied by Var(X) = E[(X µ) 2 ].

6 6 MATTHEW ACTIPES Defiitio 3.8. The stadard deviatio of X is defied as the positive square root of the variace of X ad is give by σ = Var(X) = E[(X µ) 2 ]. Remark 3.9. For a discrete radom variable X with probability mass fuctio f(x), we ca write the variace of X as Var(X) = E[(X µ) 2 ] = σ 2 = x i Ω(x i µ) 2 f(x i ). With the previous defiitios established, we are ow able to prove a few theorems ecessary i provig the Law of Large Numbers, which is essetial to our justificatio of usig the expected value of a radom variable i determiig logterm ecoomic success of a casio. Theorem For ay costat c, Var(cX) = c 2 Var(X). Proof. Let E(X)=µ ad E(cX)=cµ. The Var(cX) = E[(cX cµ) 2 ] = E[c 2 (X µ) 2 ] = c 2 Var(X). Theorem If X ad Y are idepedet radom variables, the Var(X + Y ) = Var(X) + Var(Y ) = Var(X Y ). Proof. Let µ X be the expectatio of X ad µ Y be the expectatio of Y. The we have Var(X + Y ) = E[[(X + Y ) (µ X + µ Y )] 2 ] = E[[(X µ X ) + (Y µ Y )] 2 ] = E[(X µ X ) 2 + 2(X µ X )(Y µ Y ) + (Y µ Y ) 2 ] = E[(X µ X ) 2 ] + 2E[(X µ X )(Y µ Y )] + E[(Y µ Y ) 2 ]. From here we otice that because X ad Y are idepedet, (X µ X ) ad (Y µ Y ) are also idepedet. Also, because E(X) is defied as µ X ad E(Y) is defied as µ Y, the E(X µ X ) = E(Y µ Y ) = 0. Therefore we have Var(X + Y ) = E[(X µ X ) 2 ] + 2E[(X µ X )(Y µ Y )] + E[(Y µ Y ) 2 = E[(X µ X ) 2 ] + 2[E(X µ X )E(Y µ Y )] + E[(Y µ Y ) 2 = E[(X µ X ) 2 ] + E[(Y µ Y ) 2 = Var(X) + Var(Y ). To complete this proof, we otice that Var(X Y ) = Var(X) + Var(( 1)Y ). The by Theorem 3.10 this equals Var(X) + ( 1) 2 Var(Y ) = Var(X) + Var(Y ) = Var(X + Y ). Remark Notice that the above proof ca be geeralized to ay umber of idepedet radom variables. Therefore, we ca see that the variace of a sum of idepedet radom variables is equal to the sum of the variaces.

7 A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING 7 Theorem (Chebyshev s Iequality) Suppose X is a radom variable with fiite mea ad variace. The for ay ε>0, P( X µ ε) σ2 ε 2. Proof. Let f(x) be the probability mass fuctio of X. The σ 2 = E[(X µ) 2 ] = x i Ω(x i µ) 2 f(x i ). But if we limit the sum over the x such that x µ ε, the value of the sum ca oly decrease sice we are oly workig with oegative terms. Therefore we have σ 2 (x µ) 2 f(x) ε 2 f(x) = ε 2 f(x) = ε 2 P( x µ ε). x µ ε x µ ε x µ ε Dividig both sides by ε 2 gives the desired result of P( X µ ε) σ2 ε 2. Usig Chebyshev s Iequality, we get a iterestig result whe we plug i ε = kσ, amely that P( X µ kσ) 1 k 2. This result shows that for k 1, the probability that a certai value of X lies outside of the rage (µ kσ, µ + kσ) is bouded above by 1 k 2 for ay fiite µ ad σ. Example Let k=3. The usig Chebyshev s Iequality, P( X µ 3σ) This meas that the probability of ay value of X beig withi three stadard deviatios of the mea is at least about 89%. Cosequetly, there is at most about a 11% chace that the value lies 3σ away from µ. Remark Chebyshev s Iequality provides oly a geerous upper boud o the probability that a radom variable lies withi a certai umber of stadard deviatios. For may distributios, the probability is much smaller. For example, i the ormal distributio there is about a.13% chace that the radom variable lies three stadard deviatios from the mea. We are ow i positio to state oe of the most importat theorems i casio ecoomics. Usig the cocept of variace ad Chebyshev s Iequality, we ca prove the Weak Law of Large Numbers, showig that the expected value of a casio game is directly related to the game s log-term profit. Theorem Weak Law of Large Numbers Let X 1, X 2,, X be mutually idepedet radom variables, each with fiite mea ad variace. The for ay ε>0, if S = X 1 + X X, the lim P ( S µ ε ) = 0.

8 8 MATTHEW ACTIPES Proof. By Theorem 3.4, we have ( ) ( ) S X1 + X X E = E ad by a extesio of Theorem 3.11, = 1 [E(X 1)+E(X 2 )+ +E(X )] = 1 (µ) = µ, Var(S ) = Var(X 1 + X X ) = Var(X 1 ) + Var(X 2 ) + + Var(X ) = σ 2, so by Theorem 3.10, we have that Var ( S ) = 1 2 Var(S ) = σ2. Now we ca ivoke Chebyshev s Iequality. By lettig X = S, we have ( P S ) µ ε σ2 ε 2. Now we eed oly take the limit as approaches ifiity of both sides to get ( lim P S ) µ ε = 0. This theorem states that as the umber of trials approaches ifiity, the probability that the average of the results deviates from the expected value approaches zero. A good example is that of a large umber of coi flips. While the disparity betwee heads ad tails may be large for a small sample space, as more ad more flips occur, the total amout of heads will ed up very close to the total amout of tails. This idea is essetial to the busiess model of a casio. From this theorem, we ca deduce why casios choose to ope ad why they stay i busiess. 4. Casio Ecoomics Busiess firms work i the log ru. That is, they are willig to sped moey or forgo profits ow to esure a higher profit i the big picture of their compay. Casios are o differet. They are willig to face days where they make a egative profit as log as they ed up i the positive at the ed of some larger amout of time. Therefore, casio operators eed to esure that they actually will see a positive et icome if they stay i busiess log eough. Because of this, they rely heavily o the Law of Large Numbers. Let s look at oe of the simplest games a casio ca offer: roulette. A roulette wheel cosists of 38 evely spaced pockets umbered 0, 00, 1, 2,, 36, where 0 ad 00 are colored gree, ad the other 36 umbers are split ito two groups of eightee, with oe group colored black ad the other colored red. The wheel is spu ad a ball is dropped ito it while bets are placed o where the ball will lad. Whe played accordig to desig, this game relies completely o the laws of probability. Therefore, it makes sese to quote the expected value of the bets o the table. Example 4.1. The payout for bettig o red is 1 to 1. Sice there are 38 pockets o the wheel, 18 of them beig red, we otice that the odds of wiig this bet are

9 18 38 A PROBABILISTIC VIEW ON THE ECONOMICS OF GAMBLING Therefore the odds of losig the bet are 38. We the see that E(Red) = ( ) ( ) a i p i = (1) + ( 1) = This says that for every oe dollar bet we place o red, we are expected to lose about 5.26 cets, or the casio makes about 5.26 cets. This calculatio ca be carried out for all of the bets to show that every bet has a egative expected value. Furthermore, this holds true for every chace-based game offered by the casio. Recall that the Weak Law of Large Numbers states for ay ε>0, ( lim P S ) µ ε = 0. Therefore as the casio remais i busiess loger ad the umber of bets customers place icreases, the probability that the casio does ot ear their expected value approaches zero. Seeig as the expected value of the casio is positive, the casio ca expect with probabilistic certaity that they will make a positive et profit. I other words, the house always wis. 5. Gambler s Utility So if the house is always expected to come out o top i the log ru, why do people gamble? To aswer this we eed to defie a gambler s utility fuctio. Let s deote the utility fuctio of a gambler as U(x i ). We ca work uder the assumptio that all ratioal people would prefer, uder idetical circumstaces, to have more moey tha ot. Therefore, U(x i ) is icreasig. For every evet x i there is a associated utility U(x i ) that will add or subtract from the gambler s total utility, depedig o the outcome. Therefore each gamble has with it a associated expected utility. Defiitio 5.1. For a discrete radom variable X, let P(X = x i ) = f(x i ). The the Expected Utility of X, deoted ϕ(x), is defied as ϕ(x) = x i Ω U(x i )f(x i ). Remark 5.2. We otice that a gambler will accept a gamble if ϕ(x)>0 ad will declie a gamble if ϕ(x)<0. I the case that ϕ(x) = 0, the gambler is idifferet as to whether or ot he accepts the gamble. Let s ow go back to our roulette example. The gambler who plays roulette is willig to do so eve though he has a egative expected retur. This shows that for him, ϕ(x) 0, which implies that he receives some amout of positive utility from the act of gamblig itself. This ca be likeed to thikig of gamblig as a form of etertaimet, such as goig to a baseball game. The expected retur of goig to a baseball game is egative sice there is a cost for tickets, parkig etc., but there is also positive utility gaied from the experiece. Sice the gambler was willig to accept the bet with a egative expected retur, he would surely opt to take part i a similar gamble with a expected retur of 0. Because of this, we say that he is risk lovig.

10 10 MATTHEW ACTIPES Defiitio 5.3. A Risk Lovig perso will always accept a fair gamble. I other words, give the choice of earig the same amout of moey through gamble or through certaity, the risk lovig perso will opt for the gamble. Remark 5.4. From this defiitio we ca see that a risk lovig perso has icreasig margial utility. This meas that the perso derives more utility from wiig a certai amout of moey tha from losig that same amout. Therefore, his margial utility is icreasig ad cocave up. Returig to the previous defiitio, we kow that for ay positive amout of moey c, U(x i ) U(x i c) U(x i + c) U(x i ). But if the expected retur of a gamble is ot 0, such as i roulette, we ca defie the utility that a perso receives from gamblig. Defiitio 5.5. The Utility of Risk, deoted by U R, is defied as U R = max[(b c)] for b,c satisfyig U(x i ) U(x i b) U(x i + c) U(x i ). Usig this equatio we ca determie for each idividual how much utility they derive from the experiece of gamblig. Example 5.6. We ve quoted the expected value of a roulette bet to be about cets per dollar. We kow that someoe who makes this bet has a expected utility greater tha or equal to zero, so usig expected value we ca say U(x i ) U(x i 1) U(x i ) U(x i ). Thus, this perso has a utility of risk greater tha or equal to Therefore while gamblig agaist the house seems illogical mathematically, the cocept of utility ca be applied to show that gamblig is ideed logical. Furthermore, usig a idividual s utility of risk ad their utility fuctio we ca calculate the maximum expected loss a bet ca carry that will still ot lower the idividual s expected utility. The combiatio of probability ad ecoomics ca therefore justify the cocept of gamblig both for the casio ad for the gambler. Ackowledgmets. I d like to thak my metor Ya Zhag for his help throughout this process. His guidace ad support has bee essetial to my work ad I appreciate all he s doe for me. I d also like to thak Peter May for ivitig me to participate i this program ad to everyoe who helped orgaize ad ru the 2012 REU. Refereces [1] Murray R. Spiegel Probability ad Statistics McGraw-Hill Ic [2] Vai K. Borooah Risk ad Ucertaity Uiversity of Ulster [3] James H. Stapleto Models for Probability ad Statistical Iferece: Theory ad Applicatios Joh Wiley ad Sos 2008