Question

6) At a restaurant, fifteen couples are seated at a large round table with thirty seats. Before 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 gender 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 8 E Ai 0 else 1 , (s) = Can you express X in terms of these indicator functions?)

Answer 1