\(P(X=i)=\frac{\binom{m}i\binom{N-m}{n-i}}{\binom{N}{n}}, i=0,1,\ldots, \min(n,m).\)

The above probability is for a single "replicate".

In statistics, "Repeated measures" involves measuring the same cases (subjects, people, silicon chips, whatever) multiple times.

"Replication" involves running the same study on different subjects but identical conditions.

For a requirement of a process, I have to "replicate" the above same experiment \(2\) times and hence find the probability.

But the question itself is not clear to me that what probability I have to calculate if the experiment is replicated twice?

By replicating the above experiment twice, does it mean I need to calculate the probability of having exactly \(2i\) white balls from a sample of \(2n\) balls, which is sampled from \(2N\) balls, of which \(2m\) are white and \(2(N-m)\) are black?

Also it seems to me, the probability of getting exactly \(i\) white balls is same for each replicate. But what is the definition of the following probability

\(\big(\frac{\binom{m}i\binom{N-m}{n-i}}{\binom{N}{n}}\big)^2\)?