First Head Wins

Question

You and I take turns flipping a fair coin, and you flip first. The first person to flip a head wins. What is the probability that you win?

Accepted Answer

Let p be the probability the first mover wins. Either the first flip is a head (you win immediately), or both opening flips are tails and the game restarts identically with you flipping first again: p = frac12 + frac12 cdot frac12 cdot p implies frac34 p = frac12 implies p = frac23. (The middle case, your tail, my head, contributes zero to your win probability.) Approach 2, geometric series You win on flip 1, 3, 5, dots, i.e. after k full rounds of double tails: p = sum_k=0^infty left(frac14right)^k cdot frac12 = frac1/21 - 1/4 = frac23. General rule If each attempt succeeds with probability q, the first mover wins with probability fracq1-(1-q)^2 = frac12-q. The first-mover edge shrinks as success gets rarer: for a coin it is frac23 vs frac13, but for "first to roll a six" it is only frac611 vs frac511. Rare events make turn order nearly irrelevant, a useful intuition for who-gets-there-first races in general.

First Head Wins

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