Puzzles

Two gamblers with unequal fortunes flip coins until one goes broke. Who wins more often?

Player A starts with £9, Player B with £1. Each round, they flip a fair coin for £1, stopping only when one player is wiped out. It's a fair coin, so surely the game itself is fair. Is Player A's chance of eventually bankrupting Player B closer to 50%, or much higher?

Reveal the answer

Player A wins about 90% of the time — exactly proportional to their starting share of the total money (£9 of £10). This 'gambler's ruin' result grew out of a 1656 correspondence between Blaise Pascal and Pierre de Fermat, later formalized by Christiaan Huygens. It shows that with a fixed stopping point, a fair bet still overwhelmingly favours whoever starts richer.

— Christiaan Huygens, De Ratiociniis in Ludo Aleae — 1657, building on the Pascal–Fermat correspondence of 1656