54. Independent trials, each resulting in a success with probability p, are performed until k consecutive successful trials have occurred. Let X be the total number of successes in these trial, and let Pn = P(X = n).

(a) Find Pk .

(b) Derive a recursive equation for the Pn, n  k, by imagining that the trials continue forever and conditioning on the time of the first failure.

(c) Verify your answer in part

(a) by solving the recursion for Pk .

(d) When p = .6, k = 3, find P8.