2026-03-22

Problem — 2026-03-22

Medium

Mediumprobability

Alice and Bob take turns flipping a fair coin, starting with Alice. The first person to flip heads wins. What is the probability that Alice wins?

Express as a fraction

Show solution

Alice wins on her first flip with probability

\frac{1}{2}

. For the game to continue, both must flip tails (probability

\frac{1}{4}

), and the situation resets. So

P = \frac{1}{2} + \frac{1}{4}P

, giving

\frac{3P}{4} = \frac{1}{2}

, so

P = \frac{2}{3}