PL-OM 1984 Problem 4

A coin is tossed $n$ times, and the outcome is written in the form ( $a_1,a_2,...,a_n$ ), where $a_i = 1$ or $2$ depending on whether the result of the $i$ -th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$ , and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$ . Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$ .