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 5.9 on page 217, we can show that the mean of the random variable X is 1/p.
To illustrate, again consider the Arizona state lottery Lotto, as described in Exercise 5.154. Suppose that you buy one Lotto 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.
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?