Against all Odds

What this conditional-expectation dice problem tests

This is a medium-difficulty probability question that probes your ability to reason about conditional expectation and filtering. It rewards clarity in setting up the problem and recognizing that you're computing an expected value over a subset of outcomes, rather than over all rolls.

The key skill is identifying which dice contribute to the final sum, understanding the distribution of those filtered values, and then calculating the expectation correctly. Candidates often benefit from thinking about linearity of expectation and what happens when you condition on a subset of the sample space.

• Conditional probability and filtering
• Expectation of a filtered random variable
• Linearity of expectation across independent trials