Here is another way to obtain a set of recursive equations for determining Pn, the probabi...

Here is another way to obtain a set of recursive equations for determining Pn, 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

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1. there is a string of k consecutive heads within the first n − 1 Hips, or

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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.

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(b) Using the preceding, relate Pn to Pn-1. Starting with Pk − pk , the recursion can be used to obtain Pk+1, then Pk+2, and so on, up to Pn.