23. A coin having probability p of coming up heads is successively flipped until the rth head appears. Argue that X, the number of flips required, will be n, n ≥ r, with probability P{X = n} = n − 1 r − 1



pr(1 − p)

n−r, n ≥ r This is known as the negative binomial distribution.

Hint: How many successes must there be in the first n − 1 trials?