Chapter 4, Problem 4.25TE

Here is another way to obtain a set of recursive equations for determining $P_n$, the probability that there is a string of k consecutive heads in a sequence of n flips of a fair coin that comes up heads with probability p:

a. Argue that for $k<n$ there will be a string of k consecutive heads if either 1, there is a string of k consecutive heads within the first $n-1$ flips, or

2. there is no string of k consecutive heads within the first $n-k-1$ flips, flip $n-k$ is a tail, and flips $n-k+1$ ,….,n are all heads.

b. Using the preceding, relate $P_n$ to $P_{n-1}$. Starting with $P_k=p^k$ the recursion can be used to obtain $P_{k+1}$, then $P_{k+2}$, and so on, up to $P_n$.