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, outcome 3 occurs exactly x_{3} 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}:

For example, suppose there are 5 red marbles, 3 green marbles, and 2 blue marbles in an urn. If we randomly select 5 marbles from the urn, with replacement, what is the probability of obtaining exactly 2 red marbles, 2 green marbles, and 1 blue marble?

To answer this, we can use the multinomial distribution with the following parameters:

**n**: 5**x**(# red marbles) = 2,_{1 }**x**(# green marbles) = 2,_{2 }**x**(# blue marbles) = 1_{3 }**p**(prob. red) = 0.5,_{1 }**p**(prob. green) = 0.3,_{2 }**p**(prob. blue) = 0.2_{3 }

Plugging these numbers in the formula, we find the probability to be:

**Probability **= 5! * (.5^{2} * .3^{2} * .2^{1}) / (2! * 2! * 1!) = **0.135**.

**Multinomial Distribution Practice Problems**

Use the following practice problems to test your knowledge of the multinomial distribution.

**Note: **We will use the Multinomial Distribution Calculator to calculate the answers to these questions.

**Problem 1**

**Question: **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?

**Answer:** Using the Multinomial Distribution Calculator with the following inputs, we find that the probability is **0.0504:**

**Problem 2**

**Question: **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?

**Answer:** Using the Multinomial Distribution Calculator with the following inputs, we find that the probability is **0.1296:**

**Problem 3**

**Question: **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?

**Answer:** Using the Multinomial Distribution Calculator with the following inputs, we find that the probability is **0.038272:**