Multinomial Distribution

TL;DR

Models the probability of a specific combination of outcomes across multiple independent, identically distributed trials.
Generalizes the binomial distribution to more than two outcomes per trial.
Used to compute probabilities of outcome counts (e.g., die toss combinations or counts of patient outcomes).

Definition

The multinomial distribution is a probability distribution that describes the probability of observing a certain combination of outcomes in a series of independent and identically distributed trials. It is a generalization of the binomial distribution, which considers only two possible outcomes (success and failure) per trial.

Explanation

Each trial can result in one of more than two categories, and trials are assumed independent and identically distributed. The multinomial distribution assigns probabilities to all possible combinations of category counts observed over a fixed number of trials. It is useful when analyzing outcomes that fall into multiple categories rather than just two.

Examples

Die tosses

Consider tossing a fair six-sided die three times. The possible outcomes of each toss are 1, 2, 3, 4, 5, and 6. The total number of possible combinations of outcomes is $6^3 = 216.$ The multinomial distribution can be used to calculate the probability of each possible combination of outcomes.

Medical study

In a medical study of a new drug on a sample of 100 patients, researchers study three health outcomes: improvement, no change, and worsening. The multinomial distribution can calculate the probability of each possible combination of outcomes in the sample of 100 patients (for example, the probability of observing 20 patients with improvement, 50 with no change, and 30 with worsening).

Use cases

Medicine
Finance
Engineering
Social sciences

Binomial distribution
Hypothesis testing
Regression analysis