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) + pj (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 .