Two researchers independently select simple random samples f

Two researchers independently select simple random samples from a population of size N, with sample sizes m and n (for each researcher, the sampling is done without replacement, with all samples of the prescribed size equally likely). Find the expected size of the overlap of the two samples

Solution

The probability of being in the first and second samples are m/N and n/N, respectively.

Thus, the probability of a single data point being in both samples is

P(being in both) = (m/N)(n/N)

P(being in both) = (mn / N^2)

Thus, those we expect in both are

E(number in both) = N P(being in both)

E(number in both) = N (mn / N^2)

E(number in both) = mn / N [ANSWER]