Penney's game: HH versus TH

Question

A fair coin is flipped repeatedly. You win if HH appears first; I win if TH appears first. What is the probability that TH beats HH?

Accepted Answer

Watch the first flips. The decisive observation: HH can win only if the game opens with HH. Why? Suppose the first two flips are not HH. Then a tail appears at some point before any HH is completed. Consider the first H that comes after that tail (one must come before any HH can form). That tail plus that head form TH, finished before HH can even start, since any later HH must begin with an H that follows our tail, and the first such H already completes TH. So: - First two flips HH: probability 	frac14, HH wins. - Anything else (	frac34): a T occurs before HH is complete, and TH must finish…

Penney's game: HH versus TH

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