Jack the Gambler has a system He is going to Las Vegas to tr

Jack the Gambler has a system. He is going to Las Vegas to try it out. He observes that a

roulette wheel lets him win half the time if he bets that the result is an even number, and he receives two times his bet if he wins. So his system works like this: He starts by betting $1, and he doubles his bet every time he loses. Eventually he must win with a net profit (for that series of bets) of $1. For example, say he bets $1 and loses, then $2 and loses, then $4 and wins $8. His net winnings over the series of 3 bets are $1. Once Jack wins a bet, he starts the process over with the next bet being for $1.

(a) Assume that the roulette wheel will run 60 times/hour. If Jack plays for two hours (give or take a few minutes to complete the current cycle), how much should he expect to win?

(b) Unfortunately, Jack has a problem. He’s only got $1000 with him. So if he loses enough times in a row, he’s out a lot of money because he won’t have enough money left to continue the cycle! If this should happen, he will start over with a new cycle (beginning with a $1 bet) if he has any money left. With this taken into consideration, how much should he expect to win or lose in 2 hours? In 20 hours?

Solution