The Multinomial distribution has a long history as it has many diverse practical applications. Basically it arises as a model for the following experimental situation. An experiment or “trial” is carried out and the outcome occurs in one of k mutually exclusive categories with probabilities p i , i = 1, 2, …, k. For example, a person may be selected at random from a population of size N and their ABO blood phenotype recorded as A, B, AB, or O ( k = 4), where each phenotype is expressed in terms of genotype probabilities, e.g., phenotype A consists of genotypes AA and AO. If the trial is repeated n times such that the trials are mutually independent, and if x i is the frequency of occurrence in the ith category, then the joint probability function of the x i is \(\displaystyle P_1(x_1,x_1,\ldots ,x_k)=\frac {n!}{x_1!x_2!\cdots x_k!}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}, \) where \(\sum _{i=1}^k x_i=n\) and \(\sum _{1=1}^kp_i=1\) . This would be the correct probability function for the genetics example if further people were chosen with replacement. In practice, sampling is without replacement and the correct distribution is the multivariate hypergeometric, a difficult distribution to deal with. Fortunately, all is not lost, as when the sampling fraction f = n∕ N is small enough (say less than 0.1 or preferably less than 0.05), the Multinomial distribution is a good approximation and is used extensively in genetics. The terms of P 1(⋅) can be obtained by expanding ( p 1 + p 2 + ⋯ + p k ) n . We note that when k = 2 we have the Binomial distribution, and when n = 1 we have the multivariate Bernoulli (Categorical) distribution, which has been used, for example, in graph theory, Dai (Multivariate Bernoulli Distribution Models, PhD Thesis. University of Wisconsin-Madison, 2012).