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# An Introduction to the Multinomial Distribution

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 variableÂ X follows a multinomial distribution, then the probability that outcome 1 occurs exactly x1 times, outcome 2 occurs exactly x2 times, outcome 3 occurs exactly x3 times etc. can be found by the following formula:

Probability = n! * (p1x1 * p2x2 * â€¦ * pkxk) /Â  (x1! * x2! â€¦ *Â xk!)

where:

• n:Â total number of events
• x1:Â number of times outcome 1 occurs
• p1:Â probability that outcome 1 occurs in a given trial

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:Â 5
• x1Â (# red marbles) = 2,Â x2Â (# green marbles) = 2,Â x3Â (# blue marbles) = 1
• p1Â (prob. red) = 0.5,Â p2Â (prob. green) = 0.3,Â p3Â (prob. blue) = 0.2

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

ProbabilityÂ = 5! * (.52 * .32Â * .21) /Â  (2! * 2! * 1!)Â =Â 0.135.

## Multinomial Distribution Practice Problems

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

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

### ProblemÂ 1

Question:Â 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 isÂ 0.0504:

### Problem 2

Question:Â 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 isÂ 0.1296:

### Problem 3

Question:Â 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 isÂ 0.038272:

The following tutorials provide an introduction to other common distributions in statistics: