EXAMPLE 2!: Expected number of matches Suppose that N people throw their hats into the center of a room. The hats are mixed up, and each person randomly selects one. Find the expected number of people that select their own hat. Solution. Letting X denote the number of matches, we can compute E[X] most eas ily by writing X=X1+X2+'-'+XN where 1 if the ith person selects his own hat X,- = . 0 otherWISe Since, for each i, the ith person is equally likely to select any of the N hats, Thus, E[X]=E[X1] + + E[X ]= (i)N=1 Hence, on the average, exactly one person selects his own hat. I Exercise 5. In Example 2h (Section 7 .2 of Ross' textbook), say that i and j, 1' 7E j, form a matched pair if i chooses the hat belonging to j and j chooses the hat belonging to 1'. Find the expected number of matched pairs