Three heads before two tails in a row

Question

You flip a fair coin until either three heads in a row appear (you win) or two tails in a row appear (you lose). What is the probability you win, meaning three heads in a row before two tails in a row?

Accepted Answer

The relevant "current run" states are H1, H2 (one or two heads so far) and T1 (one tail); a third head wins, a second tail loses. Let a1, a2, b be win probabilities from H1, H2, T1. a2 = 	frac12(1) + 	frac12 b, \quad a1 = 	frac12 a2 + 	frac12 b, \quad b = 	frac12 a1 + 	frac12(0). From the third, b = 	frac12 a1. Then a2 = 	frac12 + 	frac12 \cdot 	frac12 a1 = 	frac12 + 	frac14 a1, and a1 = 	frac12 a2 + 	frac12 b = 	frac12\big(	frac12 + 	frac14 a1\big) + 	frac12 \cdot 	frac12 a1 = 	frac14 + 	frac18 a1 + 	frac14 a1 = 	frac14 + 	frac38 a1. So 	frac58 a1 = 	frac14, giving …

Three heads before two tails in a row

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