Suppose you flip a biased coin (probability of heads is , ). What is the expected number of flips to get consecutive heads, where is a positive integer?
Let be the number of flips until consecutive heads are observed. Let , . Clearly for and . For , we have
Let be the probability generating function of . Multiplying by and summing over , the left-hand side is
and the right-hand side
Reversing the order of summation ( implies ) yields
Equating the two, we have
Differentiating, we have
In particular, when , we have