---
title: "probablistic models"
type: concept
related: [Binomial Distribution, Multinomial Coefficient, Probability]
source: https://www.jemoka.com/posts/kbhprobablistic_model/
confidence: high
status: active
---

multinomial distribution A probability distribution to model specific outcomes like a binomial distribution but for multiple variables.
like binomial distribution, we have to assume independence and same probability per trial.
&ldquo;what&rsquo;s the probability that you get some set of assignments xj=nj&rdquo;:
\begin{equation} P(X_1=c_1, X_2=c_2, \dots, X_{m}=c_{m}) = {n \choose c_1, c_2, \dots, c_{m} } p_{1}^{c_1} \cdot \dots \cdot p_{m}^{c_{m}} \end{equation}
where the big choose is a multinomial coefficient, and \(n\) is the number of different outcomes, and \(p_{j}\) is the probably of the $j$th outcome.
IMPORTANT: \(\sum_{j=0}^{m} c_{j} = n\): that is, you MUST provide an assignment for each type of outcome.
