Biased Urn Theory

Two different probability distributions are both known in the literature as “the” noncentral hypergeometric distribution. These two distributions will be called Fisher’s and Wallenius’ noncentral hypergeometric distribution, respectively. Both distributions can be associated with the classical experiment of taking colored balls at random from an urn without replacement. If the experiment is unbiased then the result will follow the well-known hypergeometric distribution. If the balls have different size or weight or whatever so that balls of one color have a higher probability of being taken than balls of another color then the result will be a noncentral hypergeometric distribution. The distribution depends on how the balls are taken from the urn. Wallenius’ noncentral hypergeometric distribution is obtained if n balls are taken one by one. Fisher’s noncentral hypergeometric distribution is obtained if balls are taken independently of each other. Wallenius’ distribution is used in models of natural selection and biased sampling. Fisher’s distribution is used mainly for statistical tests in contingency tables. Both distributions are supported in the BiasedUrn package. The difference between the two noncentral hypergeometric distributions is difficult to understand. I am therefore providing a detailed explanation in the following sections.