The negative binomial probability distribution can be used to compute the probability of the random variable X, the number of trials necessary to observe r successes of a binomial experiment. The probability distribution function is given by

P(x) = (x - 1Cr - 1)pr(1 - p)x - r

x = r, r + 1, r + 2, . . .

Consider a roulette wheel. Remember, a roulette wheel has 2 green slots, 18 red slots, and 18 black slots.

(a) What is the probability that it will take x = 1 trial before observing r = 1 green?

(b) What is the probability that it will take x = 20 trials before observing r = 2 greens?

(c) What is the probability that it will take x = 30 trials before observing r = 3 greens?

(d) The expected number of trials before observing r successes is r/p. What is the expected number of trials before observing 3 greens?