The multinomial distribution describes the probability of obtaining a specific number of counts for *k* different outcomes, when each outcome has a fixed probability of occurring.

If a random variable *X* follows a multinomial distribution, then the probability that outcome 1 occurs exactly x_{1} times, outcome 2 occurs exactly x_{2} times, etc. can be found by the following formula:

**Probability = ****n! * (p _{1}^{x1} * p_{2}^{x2} * … * p_{k}^{xk}) / (x_{1}! * x_{2}! … * x_{k}!)**

where:

**n:**total number of events**x**number of times outcome 1 occurs_{1}:**p**probability that outcome 1 occurs in a given trial_{1}:

The following examples show how to use the scipy.stats.multinomial() function in Python to answer different probability questions regarding the multinomial distribution.

**Example 1**

In a three-way election for mayor, candidate A receives 10% of the votes, candidate B receives 40% of the votes, and candidate C receives 50% of the votes.

If we select a random sample of 10 voters, what is the probability that 2 voted for candidate A, 4 voted for candidate B, and 4 voted for candidate C?

We can use the following code in Python to answer this question:

from scipy.stats import multinomial #calculate multinomial probability multinomial.pmf(x=[2, 4, 4], n=10, p=[.1, .4, .5]) 0.05040000000000001

The probability that exactly 2 people voted for A, 4 voted for B, and 4 voted for C is **0.0504**.

**Example 2**

Suppose an urn contains 6 yellow marbles, 2 red marbles, and 2 pink marbles.

If we randomly select 4 balls from the urn, with replacement, what is the probability that all 4 balls are yellow?

We can use the following code in Python to answer this question:

from scipy.stats import multinomial #calculate multinomial probability multinomial.pmf(x=[4, 0, 0], n=4, p=[.6, .2, .2]) 0.1295999999999999

The probability that all 4 balls are yellow is about **0.1296**.

**Example 3**

Suppose two students play chess against each other. The probability that student A wins a given game is 0.5, the probability that student B wins a given game is 0.3, and the probability that they tie in a given game is 0.2.

If they play 10 games, what is the probability that player A wins 4 times, player B wins 5 times, and they tie 1 time?

We can use the following code in Python to answer this question:

from scipy.stats import multinomial #calculate multinomial probability multinomial.pmf(x=[4, 5, 1], n=10, p=[.5, .3, .2]) 0.03827249999999997

The probability that player A wins 4 times, player B wins 5 times, and they tie 1 time is about **0.038**.

**Additional Resources**

The following tutorials provide additional information about the multinomial distribution:

An Introduction to the Multinomial Distribution

Multinomial Distribution Calculator

What is a Multinomial Test? (Definition & Example)