At a restaurant, fifteen couples are seated at a large round table with thirty s

Question

At a restaurant, fifteen couples are seated at a large round table with thirty seats. Be- fore the couples arrive, the waiter puts a card on each plate with the name of one person on it. Assume that the cards are placed randomly (with no attention to a person's gen- der or which person is a partner of which other person); that is, all rearrangements of the name cards are equally likely. What is the expected value of the number of couples who end up seated together? (Hint: let X be the random variable representing the number of couples who are seated together. You do NOT have to compute the pmf of X in order to calculate its expected value. Instead, let A, be the event that couple i are seated together, and let 1A, be the indicator function of the event A; i.e., 1 if s e A 0 else 1A, (s) Can you express X in terms of these indicator functions?)

Explanation / Answer

X is the sum of the XiXi, where Xi=1 if husband and wife of couple ii are seated next to each other, and Xi=0 otherwise. True, the random variables Xi are not independent, but the only important things are that, for every ii E(Xi)=2/29

E(Xi)=2/29, and that there are 5 couples. Hence indeed E(X)=15×2/29 = 30/29

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