If you enjoy a little abstract thinking, you may want to derive the formula for the negative binomial probability distribution. Use the notation of Problem 30. Consider two events, A and B.

A = {event that the first n – 1 trials contain k – 1successes}

B = {event that the nth trial is a success}

(a) Use the binomial probability distribution to show that the probability of A is P (A) = C_{n­1, k­–1} p^k­–1 q^{(n–1) – (k­–1)}.

(b) Show that the probability of B is that of a single trial in a binomial experiment, P(B) = p.

(c) Why is P(A and B) = P(A) × P (B)? Binomial trials are independent.

(d) Use parts (a), (b), and (c) to compute and simplify P (A and B).

(e) Compare P (A and B) with the negative binomial formula and comment on the meaning of your results?