Gambler's Ruin

A gambler starts with $10. Each round they bet $1 on a fair coin flip, winning or losing $1. They stop when they reach $0 (ruin) or $20 (goal). What is the probability they reach $20? Express as a fraction.

• optimal stopping
• random walks

Source: The gambler's ruin problem, from the Pascal-Fermat correspondence (1656) and Christiaan Huygens (1657). Statement written for AxiomIQ.