the hypergeometric distribution

The hypergeometric distribution is the model for sampling without replacement from a finite pool. Picture an urn with N balls, of which K are red and N - K are white. You scoop out a handful of n balls all at once and count how many are red. Because each ball you remove is gone — it cannot be drawn again — the chance of red changes with every pick. That is the defining feature, and it is exactly what the binomial assumes away.

The pmf is built straight from counting. The number of ways to choose k red ones from the K available is C(K, k); the number of ways to fill the remaining n - k slots with white balls is C(N - K, n - k); and the total number of equally likely handfuls of size n is C(N, n). So P(X = k) = C(K, k) C(N - K, n - k) / C(N, n) — favourable outcomes over total outcomes, the classical definition of probability applied to combinations. The mean is clean and intuitive: E[X] = n times K/N, where K/N is the fraction red in the pool, so on average a sample of n contains the same fraction red. The variance is n times K/N times (1 - K/N) times the factor (N - n)/(N - 1). That last factor is the finite population correction; it is less than 1, shrinking the variance below the binomial's because sampling without replacement is more 'informative' and less variable.

This is the right model whenever the pool is finite and you do not put items back: quality-control acceptance sampling (how many defectives in a sample of 20 from a lot of 200), card hands (how many aces in a poker hand), capture-recapture estimates of animal populations. Use the binomial here only as an approximation, and only when the pool dwarfs the sample.