The Trinomial Distribution

A trinomial trial is an experiment with exactly three mutually exclusive outcomes -- call them categories $1$, $2$, and $3$ -- occurring with probabilities $p_1, p_2, p_3$ satisfying

$$p_1 + p_2 + p_3 = 1.$$

Suppose $n$ such trials are performed independently, and let $X_i$ denote the number of trials whose outcome is category $i$. Then $(X_1, X_2, X_3)$ follows the trinomial distribution with parameters $(n;\, p_1, p_2, p_3)$, and its joint PMF is

$$P(X_1 = x_1,\, X_2 = x_2,\, X_3 = x_3) = \dfrac{n!}{x_1!\, x_2!\, x_3!}\, p_1^{x_1}\, p_2^{x_2}\, p_3^{x_3},$$

defined for non-negative integers $x_1, x_2, x_3$ with $x_1 + x_2 + x_3 = n$ (the probability is $0$ otherwise).

The structure mirrors the binomial PMF. The factor $p_1^{x_1} p_2^{x_2} p_3^{x_3}$ is the probability of any one specific sequence of outcomes with those counts (by independence), and the multinomial coefficient $\dfrac{n!}{x_1!\, x_2!\, x_3!}$ counts how many such sequences exist (permutations with repetition).

For instance, with $n = 3$ and $(p_1, p_2, p_3) = (0.5,\, 0.3,\, 0.2),$

$$P(X_1 = 2,\, X_2 = 1,\, X_3 = 0) = \dfrac{3!}{2!\,1!\,0!}(0.5)^2(0.3)^1(0.2)^0 = 3 \cdot 0.25 \cdot 0.3 \cdot 1 = 0.225.$$