Question

1.You go to a casino with $31 and the following strategy: In the first round you bet $1, if you win, you receive double your bet, you leave. If you lose you double your bet to $2. You continue this strategy that you receive double your bet and leave if you win, and doubling you bet in the previous round if you lose, until you run out of money. Let X equal the total amount of money you win (”winning” a negative amount of money is the same as losing that amount). If the probability of winning in each round is independent and equal to 9/20:

(a) What is the pmf, pX(x), of X?

(b) What is E[X] and Var[X]?

2. You are dealt one card from a full deck of 52 cards and your opponent is dealt two cards (without replacement). If you get a face card (jack, queen or king) your opponent pays you 5, if you get an ace your opponent pays you 2. If you don’t have a face card or an ace, but you have a spade and your opponent doesn’t have a spade they pay you 1. In all other cases you pay 1. What is the expectation of your winnings?

Answer 1

1.

(a)

Since, we have $31, the maximum number of rounds you can play  

1 + 2 + 2² + ... + 2^k 31

=> 1 * (2^k - 1) / (2 - 1)   31 (By Sum of geometric series)

=>  2^k - 1 31

=>  2^k 32

=> k = 5

Thus, you can play at least 5 rounds if you lose in all rounds. If you win before 5 rounds, you will take the winning amount and leave.

Amount of winning in kth round = 2 * 2^k-1 = 2^k (As you have bet 2^k-1 in kth round)

Total you bet or loose till kth round = 1 + 2 + 2² + ... + 2^k-1 (for k = 1, 2, 3, 4, 5)

= (2^k - 1) / (2 - 1) =  2^k - 1

Profit in kth round = 2^k - (2^k - 1) = 1. So X be always 1, if you win in any of first 5 rounds.

Else you lose $31 if you lose in all 5 rounds.

Probability of winning = 9/20

Probability of loosing = 1 - 9/20 = 11/20

Probability of loosing in all 5 rounds = (11/20)⁵ = 0.05032844

Probability of winning in at least one of 5 rounds = 1 - 0.05032844 = 0.9496716

So, the PMF of X is,

P(X = 1) = 0.9496716

P(X = -31) = 0.05032844

(b)

E[X] = 0.9496716 * 1 + 0.05032844 * (-31) = -$0.61051

E[X²] = 0.9496716 * 1² + 0.05032844 * (-31)² = 49.3153

Var[X] = E[X²] - E[X]² = 49.3153 - (-0.61051)² = 48.94258