Suppose you go to a casino with exactly $63. At this casino, the only game is ro

Question

Suppose you go to a casino with exactly $63. At this casino, the only game is roulette and the only bets allowed are red and green. In addition, the wheel is fait so that P[red] = P[green] = 1/2. You have the following strategy: First, you bet $1. If you win the bet, you quit and leave the casino with $64. If you lose, you then bet $2. If you win, you quit and go home. If you lose, you bet $4. In fact, whenever you lose, you double your bet until either you win a bet or you lose all of your money. However, as soon as you win a bet, you quit and go home. Let Y equal the amount of money that you take home. Find P_Y(y) and E[Y]. Would you like to play this game every day?

Explanation / Answer

let us assume you win nth bet which is bet of 2n-1 you will get extra 2n-1

till the you will lose until nth bet 1+21+22+..2n-2 = (2n-1-1)

you will get 63-(2n-1-1)+2n-1 = 64

You either go home with 0 or 64

maximum no of bets you can lose is is when money you will lose equals 63

money lost until nyh bet+money lost in nth bet = 63

(2n-1-1)+ 2n-1 = 63

2n = 64

n= 6

P(Y=0) i.e you should lose all bets i.e (1/2)(1/2)(1/2)(1/2)(1/2)(1/2) = (1/64)

P(Y=64) = 1-P(Y=0) = 1-1/64 = 63/64

E(Y) = 0*(1/64)+(64)*(63/64) = 63

we are not expected to gain or loss any money so there is gain in playing in this game

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