Probability Distributions > Multinomial Distribution

The multinomial distribution is used to find probabilities in experiments where there are more than two outcomes.

## Binomial vs. Multinomial Experiments

The first type of experiment introduced in elementary statistics is usually the binomial experiment, which has the following properties:

- Fixed number of
*n*trials. - Each trial is an independent event.
- Only two outcomes are possible (Success and Failure).
- Probability of success (p) for each trial is constant.
- A random variable Y= the number of successes.

A multinomial experiment is almost identical with one main difference: a binomial experiment can have two outcomes, while a multinomial experiment can have multiple outcomes.

**Example**: You roll a die ten times to see what number you roll. There are 6 possibilities (1,2,3,4,5,6), so this is a multinomial experiment. If you rolled the die ten times to see how many times you roll a three, that would be a binomial experiment (3 = success, 1,2,4,5,6 = failure).

A binomial experiment will have a binomial distribution. A multinomial experiment will have a multinomial distribution.

## Multinomial Distribution Example

Three card players play a series of matches. The probability that player A will win any game is 20%, the probability that player B will win is 30%, and the probability player C will win is 50%. If they play 6 games, what is the probability that player A will win 1 game, player B will win 2 games, and player C will win 3?

Use the following formula to calculate the odds:

where:

- n = number of events
- n
_{1}= number of outcomes, event 1 - n
_{2}= number of outcomes, event 2 - n
_{3}= number of outcomes, event x - p
_{1}= probability event 1 happens - p
_{2}= probability event 2 happens - p
_{x}= probability event x happens

Using the data from the question, we get:

- n = 12 (6 games total).
- n
_{1}= 1 (Player A wins). - n
_{2}= 2 (Player B wins). - n
_{3}= 3 (Player C wins). - p
_{1}= 0.20 (probability that Player A wins). - p
_{2}= 0.30 (probability that Player B wins). - p
_{3}= 0.50 (probability that Player C wins).

Putting this into the formula, we get:

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