Problem:
Let be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of heads before one encounters a run of tails. Given that can be written in the form , where and are relatively prime positive integers, find .
Solution:
A successful string is a sequence of 's and 's in which appears before does. Each successful string must belong to one of the following three types:
those that begin with , followed by a successful string that begins with ;
those that begin with , or , followed by a successful string that begins with ;
the string .
Let denote the probability of obtaining a successful string that begins with , and let denote the probability of obtaining a successful string that begins with . It follows that
Solving these equations simultaneously, we find that
Thus the probability of obtaining five heads before obtaining tails is
and .
The problems on this page are the property of the MAA's American Mathematics Competitions