Find the Correlation Between Two Die-Face Counts

# Find the Correlation Between Two Die-Face Counts A fair six-sided die is rolled independently `n` times, where `n` is a fixed positive integer. Let `X` be the number of rolls showing `1`, and let `Y` be the number of rolls showing `5`. Calculate `Corr(X, Y)` and explain why the two counts are not independent even though the rolls themselves are independent. ### Constraints & Assumptions - Each roll has exactly one outcome in `{1, 2, 3, 4, 5, 6}`. - The die is fair and rolls are mutually independent. - `n > 0`, so both variances are positive and correlation is defined. ### Clarifying Questions to Ask - Is `n` fixed rather than random? - Can one roll contribute simultaneously to both `X` and `Y`? ```hint Check one roll carefully Contrast dependence between the two face indicators on the same roll with independence across different rolls. ``` ### What a Strong Answer Covers - Correct expectations, variances, covariance, correlation, and interpretation of the negative dependence. - A derivation that shows whether and how `n` cancels. ### Follow-up Questions - What is the correlation if the two counted categories have unequal probabilities? - How does the covariance change if `n` itself is random?