Chapter 5.2, Problem 39E

Expected Value. As noted in Definition 5.4 on page 232, the mean of a random variable is also called its expected value. This terminology is especially useful in gambling, decision theory, and the insurance industry, as illustrated in Exercises 5.39–5.42.

5.39 Roulette. An American roulette wheel contains 38 numbers: 18 are red, 18 are black, and 2 are green. When the roulette wheel is spun, the ball is equally likely to land on any of the 38 numbers. Suppose that you bet $1 on red. If the ball lands on a red number, you win $1; otherwise you lose your $1. Let X be the amount you win on your $1 bet. Then X is a random variable whose probability distribution is as follows.

x	1	–1
P(X = x)	0.474	0.526

1. a. Verify that the probability distribution is correct.
2. b. Find the expected value of the random variable X.
3. c. On average, how much will you lose per play?
4. d. Approximately how much would you expect to lose if you bet $1 on red 100 times? 1000 times?
5. e. Is roulette a profitable game to play? Explain.