Last Roll Wins

What this probability-conditioning interview question tests

This is a medium-difficulty probability question that rewards careful thinking about stopping times and conditional expectation. Rather than asking for a straightforward expected value, it requires you to reason about the probability of a *specific roll value* at a particular point in a stopping process—a skill quant firms value highly.

The core challenge is recognizing that you cannot simply condition on "the process stops here" without accounting for which outcomes actually lead to stopping. Success depends on setting up the problem clearly: identifying all paths that lead to a valid stopping state, and weighing the contribution of each path to your final probability. The arithmetic is manageable once your framework is sound.

• Stopping times and their distributions
• Conditional probability under a stopping rule
• Symmetry and exchangeability arguments
• Careful enumeration of state transitions