Question

In this activity, you are investigating the financial viability of the martingale betting strategy that was introduced in class.

Recall how a martingale works. You place a bet on any "even money" event (that is, a wager where winning means gaining the money bet). If you win, you stop — you have won money. If you lose, you keep doubling the bet until you win, and then stop. Your winnings will cover the previous losses. Unfortunately, a long losing streak means that you run out of money and can no longer bet – hence losing a lot.
Balph Snerdwell has $31 in his wallet. He is going to be betting on red or black, at the roulette table. Recall that, in an American casino, there are 38 numbers — 18 red and 18 black (plus 0 and 00). So Balph has an 18/38 probability of winning, each time he bets.

a) Balph plays a martingale — doubling his bet after each loss. What’s the maximum number of times he can lose? Why?

b) Recall that Balph might win a dollar from playing a martingale … or he might lose all $31 in his wallet. What’s the probability that he wins a dollar? That he loses everything?

c) And so, what is the expected value of Balph’s net winnings, from one martingale?

d) Sometimes Balph will bet only one dollar. But sometimes he will bet three dollars (if he loses, then wins). Or seven dollars (lose, lose, win). Or some other amount. What are the possible values for total amount wagered? Their probabilities? Therefore, what is the expected value of the numbers of dollars wagered?

e) In Part D you found how much money Balph won, on average. In Part E, you found how much money Balph bet, on average. How much did Balph win, per dollar bet? (Does this number look familiar?)



The answers are:
b) 0.9596, 0.0404
c) expected value of dollars won = -$0.29236
d) expected value of dollars wagered = $5.55475

Can someone please break it down and explain how they got the answers

Answer 1

a) Balph can lose a maximum of 6 times in a row before running out of money and not being able to place any more bets. This is because if he loses 6 times in a row, he would have to bet $64 (1+2+4+8+16+32) which is more than the $31 he has in his wallet.

b) The probability that Balph wins a dollar is 0.9596 and the probability that he loses everything is 0.0404. These probabilities are calculated by assuming a 18/38 probability of winning each bet and using the martingale betting strategy.

c) The expected value of Balph's net winnings from one martingale is -$0.29236. This means that on average, Balph can expect to lose $0.29236 each time he plays the martingale betting strategy.

d) The possible values for the total amount wagered are $1, $3, $7, $15, $31, and $64. These values correspond to the amount wagered after losing 0, 1, 2, 3, 4 and 5 times in a row. The probability of each value depends on the probability of losing that many times in a row, calculated using the 18/38 probability of winning each bet. The expected value of the number of dollars wagered is $5.55475.

e) The amount Balph wins per dollar bet is -0.0527. This number is calculated by dividing the expected value of Balph's net winnings by the expected value of the number of dollars wagered. This number is less than zero, indicating that on average, Balph will lose money for every dollar he bets.