stats

Hypergeometric Distribution

P(X=k) without replacement — CDF, tail probabilities, mean, variance, PMF bar chart with binomial approximation overlay.

Presets

Parameters

N — population sizeK — successes in populationn — sample size (draws)k — observed successes

PMF — P(X = k) for all k● binomial approx at k=2

Probabilities

P(X = 2)0.27428

P(X ≤ 2)0.907233

P(X < 2)0.632953

P(X ≥ 2)0.367047

P(X > 2)0.0927671

Mean E[X]1.25

Variance0.864

Mode1

Binomial Approximation (p = K/N)

Binom P(X = k)0.263672

Binom P(X ≤ k)0.896484

Sampling fraction n/N0.09615 (large — binomial is imprecise)

P(X=k) = C(K,k)·C(N−K, n−k) / C(N,n)
E[X] = nK/N · Var = nK/N·(1−K/N)·(N−n)/(N−1)

Binomial approx valid when sampling fraction n/N < 5%.