Estimating Pi with a Coin

Jim Propp. arXiv:2602.14487. Submitted February 2026, revised March 2026.

A simple Monte Carlo method for estimating pi by tossing a coin. The underlying Catalan-number series identities appear implicitly in the probability theory literature, but this specific interpretation appears novel.

Method: Toss a coin until the first time the observed proportion of heads exceeds 1/2 (cumulative heads exceeds cumulative tails) and record that fraction. Example: tails, heads, tails, heads, heads -> record 3/5. Start afresh, repeat. The average of the fractions approaches pi/4.

CC BY 4.0 licensed. 3-4 pages.