Question

A roulette wheel is divided into 38 labeled pockets, 18 of which are red, 18 are black, and 2 are green. Each round, you can place a $1 bet on one specific red or black pocket with odds of 35 : 1. That is, if you are correct you receive $35 in addition to the dollar you bet, and if you are incorrect, you get nothing. Your friend has developed what he describes as a “foolproof system” for winning. It involves him repeatedly betting $1 on red pocket number 7. You try to discourage him, but he is insistent. (a) What are your friend’s expected winnings after one spin of the wheel? What are his expected winnings after 36 spins of the wheel? (b) Despite showing him the outcome of the above calculation, he is unwavering in his opinion, so you decide to at least try to make some money for yourself. You bet your friend $20 that at the end of 36 spins he will have lost money, i.e. have less money than he began with. What are your expected winnings in this bet? What are your friend’s expected winnings after 36 spins taking this side-bet into account as well? (c) Explain what went wrong.

Please show in steps with explanations!

Thanks!

Answer 1

a)

For one spin,

Probability of winning = 1/38

Winning amount = $35

Probability of losing = 1 - 1/38 = 37/38

Losing Amount = -$1

Expected winning in 1 spin = E[X] = (1/38) * 35 - (37/38) * 1 = -$0.0526 -$0.05

For 36 spins, winning amount = X1 + X2 + ... + X36

where X1, X2, ..., X36 are the winning amount for 1, 2, .., 36 spin.

Expected winning amount after 36 spins = E[X1 + X2 + ... + X36]

= E[X1] + E[X2] + ... + E[X36]

= -0.0526 - 0.0526 - ... - 0.0526 = 36 * -0.0526

= -$1.89

b)

For one spin,

E[X²] = (1/38) * 35² + (37/38) * (-1)² = 33.21053

Var(X) = E[X²] -E[X]²   = 33.21053 - 0.0526² = 33.20776

Variance of winning amount after 36 spins = Var[X1 + X2 + ... + X36]

= Var[X1] + Var[X2] + ... + Var[X36]

= 33.20776 + 33.20776 + .. + 33.20776 = 36 * 33.20776   

= 1195.479

Let Y be the amount of money after 36 spins. Assuming normal approximation, Y ~ N(-1.89, 1195.479)

Probability of less money after 36 spins = P(Y < 0)   

= P[Z < (0 - (-1.89))/1195.479]

= P[Z < 0.0016]

= 0.5006

Expected winnings of your bet = 0.5006 * $20 - (1 - 0.5006) * $20 = $0.024

Friends expected winnings = -1.89 - 0.02 = -$1.91