Correlation between the count of ones and sixes on a die

Question

Roll a fair die 60 times. Let N1 be the number of ones you get and N6 the number of sixes. What is the correlation between N1 and N6?

Accepted Answer

Because every roll shows exactly one face, a roll spent being a one is a roll that cannot be a six. That competition for the same 60 rolls creates a negative link. To measure it, build both counts from per-roll counters. Covariance. Let at = 1 if roll t is a one, and bt = 1 if roll t is a six. Then N1 = \sum at and N6 = \sum bt. Rolls are independent, so counters from different rolls contribute nothing to the covariance. Only the same roll matters, and on a single roll at and bt can never both be 1 (a roll is not a one and a six at once), so at bt = 0. Thus each roll contributes \mathrm{Cov}(…

Correlation between the count of ones and sixes on a die

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