The multinomial distribution is a probability distribution that models the outcomes of repeated experiments where each trial results in one of three or more categories. It is a generalization of the binomial distribution, which only applies to two-category outcomes.
Each trial must be independent, and the probability of each outcome must remain constant across trials. The multinomial distribution tells you the probability of observing a specific count for each category after a fixed number of trials.
Use the multinomial distribution when:
- You repeat an experiment a fixed number of times.
- Each trial has more than two possible outcomes.
- The probabilities of each category are known and fixed.
- You want to calculate the probability of a specific combination of outcomes.
It is commonly used in survey research, categorical experiments, and genetics, where outcomes fall into multiple non-overlapping groups.
Example
Suppose a survey asks people to choose their favorite fruit: apple, banana, or orange. Each person picks one fruit. You know that in the population, 40% prefer apples, 35% prefer bananas, and 25% prefer oranges. If you survey 10 people, the multinomial distribution can calculate the probability of getting exactly 4 apple votes, 3 banana votes, and 3 orange votes.
The distribution uses the number of trials, the category probabilities, and the observed counts to compute the result.
In this case, the probability of observing that specific combination is approximately 7.2%. Typically, you’ll use statistical software to calculate the answer for a multinomial distribution.
« Back to Glossary Index