Two players alternately flip a coin. First to get heads wins. What is probability first player wins?

Two players alternately flip a coin. First to get heads wins. What is probability first player wins?

Approach: Sum probability over rounds.

p = 1/2 + (1/2)^2*(1/2) + ... = (1/2)/(1-1/4) = 2/3. Or by symmetry: p = 1/2 + 1/2*(1-p) ⇒ p = 2/3.

Follow-up: What if coin is biased?

Key concepts: geometric series, probability, alternating.