Find population mean probability

Question

Find the probability: We have N persons and Y boxes. What is the mean person’s probability of choosing the same box with another one? (I want the probability of the whole population, not the case on which at least one of the N people choose the same box with another one).

Answer 1

The probability that any 2 people match is 1/Y

Thus, the expected number of matches for a particular person is (N - 1)/Y and the expected total number of matches is N(N-1)/(2Y) (a particular match occurs for 2 people)

The probability that a particular person has at least 1 match is 1 - (1 - 1/Y)^(N - 1)

The probability that there is at least one match across all n people is

1 - P(N,Y)/N^Y where P (N,Y) is N permute Y.