The Multinomial Distribution

The multinomial distribution generalizes the trinomial distribution to any number of outcomes $k \geq 2.$

Suppose we perform $n$ independent trials, where each trial results in exactly one of $k$ possible outcomes with respective probabilities $p_1, p_2, \dots, p_k$ satisfying

$$p_1 + p_2 + \cdots + p_k = 1.$$

Let $X_i$ denote the number of trials that produce outcome $i.$ Then for any nonnegative integers $x_1, x_2, \dots, x_k$ with $x_1 + x_2 + \cdots + x_k = n,$

$$P(X_1=x_1,\, X_2=x_2,\, \dots,\, X_k=x_k) = \dfrac{n!}{x_1!\, x_2! \cdots x_k!}\, p_1^{x_1}\, p_2^{x_2} \cdots p_k^{x_k}.$$

The coefficient $\dfrac{n!}{x_1!\, x_2! \cdots x_k!}$ is called the multinomial coefficient.

For example, roll a fair $4$-sided die $5$ times. The probability of getting two $1$s, one $2,$ one $3,$ and one $4$ is

$$\begin{align*} P(X_1{=}2,X_2{=}1,X_3{=}1,X_4{=}1) &= \dfrac{5!}{2!\,1!\,1!\,1!}\left(\dfrac{1}{4}\right)^2\!\left(\dfrac{1}{4}\right)\!\left(\dfrac{1}{4}\right)\!\left(\dfrac{1}{4}\right) \\[3pt] &= 60 \cdot \dfrac{1}{1024} \\[3pt] &= \dfrac{15}{256}. \end{align*}$$