The multinomial distribution is a probability distribution that generalizes the binomial distribution to experiments with more than two possible outcomes. It describes the probabilities of obtaining a specific number of counts for ( k ) different outcomes when each outcome has a fixed probability of occurring. This distribution is particularly useful in scenarios where you conduct ( n ) independent trials, and each trial can result in one of ( k ) outcomes ¹².

In a multinomial experiment, the outcomes are categorized, and the distribution provides the likelihood of each outcome occurring a certain number of times. The parameters of the multinomial distribution include the number of trials ( n ) and a vector ( p ) that contains the probabilities of each outcome, which must sum to 1 ³⁴.

The multinomial distribution is widely used in various fields, including machine learning, particularly in classifiers like the multinomial naive Bayes classifier, which relies on this distribution to model the probabilities of different classes based on feature counts ⁵.