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 trials, 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 = 0.6, k = 3, find P8.