A probability distribution for the random variable X, the number of trials until a success is observed, is called the geometric probability distribution. It has the same criteria as the binomial distribution, except that the number of trials is not fixed. Its probability distribution function (pdf) is

P(x) = p(1 - p)x - 1 , x = 1, 2, 3, . . .

where p is the probability of success.

(a) What is the probability that Shaquille O'Neal misses his first two free throws and makes the third? Over his career, he made 52.4% of his free throws. That is, find P(3).

(b) Construct a probability distribution for the random variable X, the number of free-throw attempts of Shaquille O'Neal until he makes a free throw. Construct the distribution for x = 1, 2, 3, c, 10. The probabilities are small for x > 10.

(c) Compute the mean of the distribution, using the formula presented in Section 6.1.

(d) Compare the mean obtained in part (c) with the value 1/p. Conclude that the mean of a geometric probability distribution is μX = 1/p. How many free throws do we expect Shaq to take before we observe a made free throw?