Urn models have a long history, starting with Laplace suggesting in 1786 that France's population be estimated by an urn-sampling scheme. They are conceptually relatively easy to understand, which also makes them easy to recognize and apply to a variety of real-world situations.

Urn with Black and White Balls

BernoulliDistribution — sampling of a single ball

BinomialDistribution — sampling with replacement until balls are drawn

HypergeometricDistribution — sampling without replacement until balls are drawn

GeometricDistribution — sampling with replacement until a white ball is drawn

NegativeBinomialDistribution — sampling with replacement until white balls are drawn

WalleniusHypergeometricDistribution — biased sampling without replacement

FisherHypergeometricDistribution — biased sampling at a time without replacement

BetaBinomialDistribution — number of white balls sampled from Polya's urn

PoissonDistribution — sampling from urn with an infinite number of balls

Urn with Multiple Colored Balls

DiscreteUniformDistribution — sampling of a single ball

MultinomialDistribution — sampling with replacement until n balls are drawn

MultivariateHypergeometricDistribution — sampling without replacement until balls are drawn

NegativeMultinomialDistribution — sampling with replacement until white balls are drawn