Variance of the number of matching pairs among five guesses

Question

Five people each independently pick a number from $\{1, 2, 3, 4\}$ , all choices equally likely. Look at all $\binom{5}{2} = 10$ pairs of people. Let $X$ count how many pairs picked the same number.

What is the variance of $X$ ?

Accepted Answer

Make a counter for each of the 10 pairs: I{ij} = 1 if persons i and j chose the same number. A specific pair matches with chance 	frac14 (whatever the first picks, the second matches it one time in four), so each counter has variance 	frac14\cdot	frac34 = 	frac{3}{16}. Covariance between counters. Two pairs can relate in two ways. - Disjoint pairs (four different people, like \{1,2\} and \{3,4\}): the two matches involve separate people, so they are independent and the covariance is 0. - Overlapping pairs sharing one person (like \{1,2\} and \{1,3\}): both match means persons 1, 2, 3 all …

Variance of the number of matching pairs among five guesses

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