Solution
Gambler's Ruin
Show the problem again
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.
Worked solution
The answer is 1/2. For a fair game (p = 1/2), the probability of reaching goal N starting from position k is exactly k/N. Here k = 10 and N = 20, so the probability is 10/20 = 1/2. This elegant result reflects the martingale property of a fair random walk.
Source: The gambler's ruin problem, from the Pascal-Fermat correspondence (1656) and Christiaan Huygens (1657). Statement written for AxiomIQ.