Solution
First Heads Wins
Show the problem again
Alice and Bob alternately flip a fair coin, with Alice going first. The first person to flip heads wins. What is the probability Alice wins? Express as a fraction.
Worked solution
The answer is 2/3. Let p be Alice's win probability. She wins immediately with probability 1/2; if both flip tails (probability 1/4), the game restarts with Alice first. So p = 1/2 + (1/4)p, giving p = 2/3. Equivalently, Alice wins on flips 1, 3, 5, ...: p = 1/2 + 1/8 + 1/32 + ... = 2/3.
Source: Standard geometric-distribution exercise from introductory probability texts. Statement written for AxiomIQ.