67. A coin having probability p of coming up heads is continually flipped. Let Pj(n)

denote the probability that a run of j successive heads occurs within the first n flips.

(a) Argue that Pj(n) = Pj(n − 1) + p j

(1 − p)[1 − Pj(n − j − 1)]

(b) By conditioning on the first non-head to appear, derive another equation relating Pj(n) to the quantities Pj(n − k), k = 1,..., j.