Exercise 7.C.10. If you want to gamble at a game of pure chance, you are better

Question

Exercise 7.C.10. If you want to gamble at a game of pure chance, you are better off playing craps than roulette. The probability of winning a bet on red or black in roulette is approximately .4737, resulting in an expected return on a one dollar bet of -.0526. If you place a one dollar bet on "Don't Pass" at the craps table, you win a dollar with probability 4930; otherwise you lose your dollar a. You plan to bet one dollar at a time at craps. Let X, represent the returns on your ith bet. What is the distribution of X,? What is its expected value and variance? b. What random variables represent your total winnings after n bets and your win- nings per bet after n bets? What are the expected value and variance of these random variables? number of bets? number of bets? positive? What is the probability that your winnings per bet are positive? probability that your total winnings are positive? What ranges of total winnings c. What does the law of large numbers say about your total winnings after a large d. What does the law of large numbers say about your winnings per bet after a large e. Suppose you place 100 bets. What is the probability that your total winnings are f. Suppose you make 100 trips to the casino, placing 100 bets each time. What is the represent the best 50%, 10%, 5%, and 1% of your possible outcomes? g. In Las Vegas, craps is a far more popular casino game than roulette. Why might this be so?

Explanation / Answer

Question 1 (a)

Here at craps

Pr(Winning a dollar) = 0.4930

Pr(Losing a dollar) = 1 - 0.4930 = 0.5070

Here xi represents the returns on my bet here distribution of x is discrete distribution

p(x) = 0.493 ; x = 1

= 0.507 ; x = -1

E[X] = 0.493 * 1 + 0.507 * (-1) = -0.014

VaR [X] = 0.493 * (1 + 0.014)2 + 0.507 * (-1 + 0.014)2 = 1.000

(b) Here random variables represents total winning after n bets and your winnings per bet after n bets

Expected value = -0.014n

Variance = n * 1 = n

(c) The law of large numbers is a principle of probability according to which the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances. As the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes. So, here it say that after large number of bets our total winnings will tend to go (-0.014n) iwth the expected variance.

(d) Here in this case of large numbers, our winnings per bet after a large number of bets will go towards -0.014 and variance is 1/sqrt(n).

(e) Here n = 100

total winnings = 100 * (-0.014) = $ -1.4

Variance = 100

standard deviation = sqrt(100) = 10

Pr(Winnings per bet are positive) = Pr(x > 0 ; -$ 1.4 ; 10)

Z = (0 + 1.4)/10 = 0.14

Pr(Winnings per bet are positive) = Pr(x > 0 ; -$ 1.4 ; 10) = 1 - Pr(Z < 0.14) = 1 - 0.5557 = 0.4443

Pr(Total winning per bet are positive) = Pr(x > 0 ; $ - 0.014 ; 0.1)

Z = (0 + 0.014)/0.1 = 0.14

Pr(Winning per bet is positive) = 1 - Pr(Z <0.14) = 0.4443

(f) Here n = 100 trips to the casino, placing 100 bets each time.

Pr(Total winnings are positive) = 0.4443

Range of total winnings represent the best 50% is above $ -1.4

for 10% , Z value = 1.282

so here best 10% winnings are = -1.4 + 1.282 * 10 = $ 11.42

best 5% winning are = -1.4 + 1.645 * 10 = $ 15.05

best 1% winning are = -1.4 + 2.326 * 10 = $ 21.86

(g) Here in Las Vegas, craps is more population , it might be because it has high probability of winnings.

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