Blue balls, red ball

Optimal stopping under penalty reversal

This is a medium-difficulty probability question that tests dynamic programming and backward induction reasoning. You face a classic optimal stopping problem where the payoff structure includes a penalty — drawing the red ball negates all your winnings — and you must decide when to lock in your gains before that happens.

To solve it, work backward from the end state. At each stage, compare the value of stopping now (keeping your current score) against the expected value of drawing one more ball. The expected value of the next draw depends on the probability the red ball appears, the immediate reward if you draw blue, and the recursive value of the resulting state. This type of sequential decision-making under risk is common in trading interview settings.

• Backward induction and value functions
• Conditional probability and hypergeometric distribution
• Comparing stopping vs. continuation value