Binomial characteristics vs hypergeometric

The binomial distribution gives the probability of x successes in n independent trials, each with the same success probability p:

P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, x = 0, 1, …, n, where q = 1 − p.

Characteristics of the binomial distribution

1. The experiment has a fixed number of trials n.
2. Each trial has only two outcomes: success or failure.
3. The probability of success p is the same in every trial.
4. The trials are independent (sampling with replacement, or from an infinite population).
5. Mean = np, variance = npq, so variance < mean.
6. It is symmetric when p = 0.5, skewed right when p < 0.5 and skewed left when p > 0.5.

Hypergeometric distribution

When n items are drawn without replacement from a finite population of N items containing K successes:

P(X = x) = C(K, x) · C(N − K, n − x) / C(N, n)

Mean = nK/N, variance = n (K/N)(1 − K/N)(N − n)/(N − 1).

Difference

When N is large compared with n (n/N < 0.05), the hypergeometric distribution is well approximated by the binomial with p = K/N.