485.Closed Groups at the Party
★★★expected valueCh. 6 — Random Variables and Expectation444+ Problems in Probability
At a party, each of 50 guests puts their name in a hat and draws one name uniformly at random (a random permutation of the 50 names). Guests who draw each other in a cycle form one closed group; drawing your own name makes a group of one. What is the expected number of closed groups?
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For each , what is the probability that a fixed guest belongs to a cycle of length — and how much does one cycle of length contribute to the group count?
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