Noncentral hypergeometric distribution

Two different probability distributions are both known in the literature as "the" noncentral hypergeometric distribution. The nomenclature problems are discussed. Wallenius' noncentral hypergeometric distribution can be described as an urn model with bias. Fisher's noncentral hypergeometric distribution is the conditional distribution of independent binomial variates given their sum. Several different methods for calculating probabilities from Wallenius' noncentral hypergeometric distribution are derived. Range of applicability, numerical problems and efficiency are discussed for each method. Approximations to the mean and variance are also discussed. This distribution has important applications in models of biased sampling and in models of evolutionary systems.