Find the Correlation Between Two Die-Face Counts

A fair six-sided die is tossed n independent times, with n a fixed positive integer. Let X count the rolls that land on 1, and let Y count the rolls that land on 5. Compute Corr(X, Y) and explain why these two counts are dependent even though the individual rolls are independent.

Constraints & Assumptions

• Each roll produces exactly one result from {1, 2, 3, 4, 5, 6}.
• The die is fair, and different rolls are mutually independent.
• n > 0, so both variances are positive and the correlation is well-defined.

Clarifying Questions to Ask

• Is n a fixed quantity rather than a random one?
• Can a single roll add to both X and Y at the same time?

Hint — Examine a single roll closely: Compare the dependence between the two face indicators within one roll against the independence across separate rolls.

What a Strong Answer Covers

• Accurate expectations, variances, covariance, correlation, and an explanation of the negative dependence.
• A derivation showing whether and how n cancels out.

Follow-up Questions

• What does the correlation become if the two counted categories have unequal probabilities?
• How does the covariance change when n itself is random?

Overview: A fair six-sided die is tossed n independent times, with n a fixed positive integer. State the assumptions and derivation clearly, examine edge cases, and describe how the result changes when those assumptions are relaxed.