First to flip heads, taking turns

Question

Two players alternate flipping a fair coin, with the first player going first. Whoever flips heads first wins. What is the probability the first player wins?

Accepted Answer

Let P be the win probability of the player about to flip. They win now with probability 	frac12; otherwise (probability 	frac12) the opponent becomes the flipper and wins with probability P, so the current player wins with probability 1 - P there: P = 	frac12 + 	frac12(1 - P) = 1 - 	frac12 P \;\Longrightarrow\; 	frac32 P = 1 \;\Longrightarrow\; P = \frac23. Takeaway. With p = 	frac12, the first mover wins \frac{1}{2-p} = \frac23: going first is a real advantage when the event is likely.

First to flip heads, taking turns

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