A bold play

What this gambler's ruin problem tests

This is a hard probability question that combines elements of optimal betting strategy, recursive probability, and dynamic programming reasoning. It asks you to compute the likelihood of reaching a target wealth under a specific betting rule and fixed odds—a classic setup in quantitative finance and applied probability.

The challenge lies in tracking how the betting rule (bet as much as possible, but never more than needed to reach exactly 5) constrains your decisions at each state, and then setting up and solving the system of equations that relates the winning probability from each token count to the winning probabilities from subsequent states. You must account for both win and loss outcomes, weight them by their probabilities, and handle the boundary conditions (ruin at 0 tokens, success at 5 tokens) correctly.

• State-space formulation and boundary conditions
• Recursive probability equations and backwards induction
• Solving systems of linear equations in probability
• Optimal vs. constrained betting strategies