Example 3.37 The 2n seats around a circular table are numbered clockwise. The guests at dinner form n king/queen pairs. The queens sit at random in the odd-numbered seats, with the kings at random between them. Let N be the number of queens sitting next to their king. Find the mean and variance of N.

Solution Let Ai be the event that the i th king/queen pair are seated adjacently. Then N =

Xn i=1 1Ai , (3.38)

so that E(N) =

Xn i=1 E(1Ai ) =

Xn i=1 P(Ai ) = nP(A1), by symmetry. It is easily seen (by conditional probability, or simply by counting) that P(A1) =

2/n, and hence E(N) = n(2/n) = 2 regardless of the value of n.

3.6 Problems 47 In order to find the variance, we should calculate E(N2). By (3.38), E(N2) = E

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"

X i

1Ai

#2

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 = E

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X i

12 Ai + 2 X

i

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