17. A coin is tossed repeatedly; on each toss a head is shown with probability p, or a tail with probability 1− p. The outcomes of the tosses are independent. Let E denote the event that the first run of r successive heads occurs earlier that the first run of s successive tails. Let A denote the outcome of the first toss. Show that P(E | A = head) = pr−1 + (1 − pr−1)P(E | A = tail).

Find a similar expression for P(E | A = tail), and hence find P(E). (Oxford 1981M)