Taking turns to flip the first head

Question

Two players alternate flipping a fair coin, with Player 1 going first. The first person to flip heads wins immediately. What is the probability Player 1 wins?

Accepted Answer

Let p be the probability the player about to flip eventually wins. On their flip: - With probability 	frac12 they flip heads and win now. - With probability 	frac12 they flip tails, and then the opponent is in the exact same "about to flip" position, so the original player's win chance from here becomes 1 - p. This gives the equation p = \frac{1}{2} + \frac{1}{2}(1 - p). Solve: p = 	frac12 + 	frac12 - 	frac12 p, so 	frac32 p = 1, hence p = \frac{2}{3}. Player 1, who flips first, wins with probability 	frac23, a 2-to-1 edge over Player 2's 	frac13. Takeaway. Alternating "first success …

Taking turns to flip the first head

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