Chapter 17, Problem 79E

In American roulette there are 18 red numbers, 18 black numbers, and 2 white numbers (0 and 00) as illustrated in Fig. 17-20. The probability of a red number coming up on a single play of roulette is $p=18/38\approx 0.47$. Suppose that we go on a binge and bet $1 on red 10,000 times in a row (A $1 bet on red wins $1 if a red number comes up, if a black or white number comes up, the bettor loses $1.)

a. Let Y represent the number of times we lose (i.e., the number of times that red does not come up). Use the dishonest-coin principle to describe the distribution of the random variable Y.

b. Approximately what are the chances that we will lose 5300 times or more?

c. Approximately what are the chances that we will lose somewhere between 5150 and 5450 times?

d. Explain why the chances that we will break even or win in this situation are essentially zero.