Let n greaterthanorequalto 2 be an integer and consider a group P_1, P_2, ..., P

Question

Let n greaterthanorequalto 2 be an integer and consider a group P_1, P_2, ..., P_n of n people. Each of these people has a uniformly random birthday, which is independent of the birthdays of the other people. We ignore leap years; thus, the year has 365 days. Define the random variable x to be the number of unordered pairs f {P_i, P_j} of people that have the same birthday. What is the expected value E(X) of X? (a) 1/365 middot (n 2) (b) (1/365)^2 middot (n 2) (c) 1/365 middot n^2 (d) (1/365)^2 middot n^2

Explanation / Answer

The random variable X is defined as the number of unordered pairs {Pi, Pj}. So X=nC2

Now the probability that two people will have the same birthday is [(365/365)*(1/365)] = 1/365

Therefore the expected value of X i.e.

E(X) = 1/365*nC2

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