Covariance of two color counts among drawn candies

Question

You draw 40 candies, each independently red, blue, or green with chances 0.5, 0.3, and 0.2. Let NR be the number of red and NB the number of blue. What is the covariance of NR and NB?

Accepted Answer

The counts share the same 40 draws, so a candy that turns out red is one that cannot be blue. That built-in competition makes the covariance negative. Build both counts from per-candy counters. Let rt = 1 if candy t is red, bt = 1 if it is blue. Different candies are independent, so only the same candy contributes to the covariance. On one candy rt bt = 0 (a candy cannot be both colors), so its contribution is \mathrm{Cov}(rt, bt) = 0 - P(	ext{red})P(	ext{blue}) = -(0.5)(0.3) = -0.15. Summing over 40 candies: \mathrm{Cov}(NR, NB) = 40 	imes (-0.15) = -6. More generally, for counts in n ind…

Covariance of two color counts among drawn candies

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