The Geometric Distribution. In this exercise, we discuss the geometric distribution, the probability distribution for the number of trials until the first success in Bernoulli trials. The geometric probability formula is P(X = x) = p(1 − p)

x−1

, where X denotes the number of trials until the first success and p the success probability. Using the geometric probability formula and Definition 5.4 on page 254, we can show that the mean of the random variable X is 1/p.

To illustrate, again consider the Mega Millions lottery as described in Exercise 4.267 on page 239. Suppose that you buy one Mega Millions ticket per week. Let X denote the number of weeks until you win a prize.

a. Find and interpret the probability formula for the random variable X. (Note: The appropriate success probability was obtained in Exercise 4.267(d).)

b. Compute the probability that the number of weeks until you win a prize is exactly 3; at most 3; at least 3.

c. On average, how long will it be until you win a prize?