The Asymmetric Coin Game

What this gambler's ruin probability question tests

This is a medium-difficulty probability problem that quant interviewers use to assess whether you can set up and solve a recurrence relation under asymmetric odds. It combines foundational probability reasoning with a practical financial intuition: what are your true odds of reaching a target before losing everything?

The key is recognizing that the winning probability from any position depends on the probabilities from neighbouring positions, which leads to a linear recurrence. Solving it requires either spotting the pattern or deriving the closed form algebraically. The biased coin (p ≠ 1/2) makes this more realistic than the fair-coin version, since real markets and bets rarely offer even odds.

• Recurrence relations and boundary conditions
• Gambler's ruin as a classic random-walk problem
• How to handle asymmetric probabilities in a martingale or algebraic framework
• Interpretation of edge and risk of ruin in practice