# Question: In this exercise we discuss the geometric distribution the probability

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?

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?

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