6. Suppose that N people attend a party. Each person is wearing a coat when they

Question

6. Suppose that N people attend a party. Each person is wearing a coat when they come to the party, and takes their coat off as they enter. Unfortunately, however, the power goes out and the guests must leave he party early. Since the power is out, however, no guest can see which coat is theirs as they grab their coats to leave, and so each guest just randomly grabs one coat as they leave the party. What is the expected number of guests that leave the party with the same coat they were wearing when they arrived? (Hint: Let X, be the Bernoulli random variable that is 1 if the i-th guest leaves with the correct coat, and 0 if they leave with someone else's coat. The number of guests that leave with the correct coat is then X1 + X2++XN.)

Explanation / Answer

Every Xi is a bernoulli random varibale , with p_success = 1/ N (only see what options is available to a single person , N coats in which for success he has to choose 1 )

Now Y = X1+X2 + ..... XN is the sum of N bernoullie random variable which is nothing

but Binomial random with paramters N and p .

Now , in case of a Binomial random variable E(Y) = Np

in our case p = 1/N

E(Y) = N * 1/N = 1

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