The multinomial distribution describes the counts you get when the same random experiment is repeated many times and each trial can land in one of several categories. It generalizes the binomial distribution, which has only two categories, to three or more possible outcomes. This matters in statistics because many real data sets are counts across groups, such as survey choices, dice results, genetics categories, or customer preferences.
Understanding Statistics: The Multinomial Distribution
A multinomial model needs more than a list of categories. Each trial must produce one category only, and the categories must cover every possible result. The trials should be independent, meaning one result does not change the next result.
Their chances should stay the same from trial to trial. These conditions are often reasonable for repeated rolls of a die or randomly generated data. They can fail in real surveys, though.
People in the same family may give similar answers. Drawing objects from a small box without replacing them changes the chances after every draw. A model is useful only when its assumptions are close enough to the real situation.
The probability calculation has two parts. First, find the chance of one particular order of results. For example, a sequence containing certain numbers of red, blue, and green outcomes has a chance found by multiplying the relevant category chances repeatedly.
But the same set of counts can occur in many different orders. Red, blue, red is different from blue, red, red, even though both give two red outcomes and one blue outcome.
The factorial part of the multinomial rule counts all these possible orders. This is why count patterns with many mixed arrangements can be more likely than patterns where almost every trial lands in one category.
The category counts are linked because the total number of trials is fixed. If a class survey has one hundred responses and more students choose one option than expected, fewer responses remain for the other options. This creates a negative relationship between different category counts.
It does not mean the categories cause one another. It comes from the fixed total. Expected counts give the long-run average over many repeated samples, not a prediction that every sample will match exactly.
Variation is normal, especially for categories with small probabilities. A category expected to appear only a few times can easily have a noticeably higher or lower count in one sample.
Students often meet this model when comparing observed counts with a claimed pattern. A teacher may test whether answers on a multiple-choice item are spread as expected, or a researcher may compare survey responses across several choices. In these settings, observed counts are compared with expected counts.
Large differences may suggest that the claimed probabilities are not suitable, though random variation can produce some differences by chance. The chi-square goodness-of-fit test is built on this idea.
When learning it, keep track of the total, make categories non-overlapping, and check that expected counts are not too small. Combining rare categories is sometimes necessary before making a reliable conclusion.
Key Facts
- For k categories, the counts are X1, X2, ..., Xk and must satisfy X1 + X2 + ... + Xk = n.
- The category probabilities are p1, p2, ..., pk and must satisfy p1 + p2 + ... + pk = 1.
- Multinomial probability formula: P(X1 = x1, ..., Xk = xk) = n!/(x1!x2!...xk!) p1^x1 p2^x2 ... pk^xk.
- The expected count in category i is E(Xi) = npi.
- The variance of category i is Var(Xi) = npi(1 - pi).
- For two different categories i and j, the covariance is Cov(Xi, Xj) = -npipj.
Vocabulary
- Multinomial distribution
- A probability distribution for the counts in several categories after a fixed number of independent trials.
- Trial
- One repetition of the random experiment, such as one roll of a die or one selected survey response.
- Category probability
- The probability that a single trial falls into a particular category.
- Count vector
- The list of category counts, such as (x1, x2, x3), recorded after all trials are complete.
- Multinomial coefficient
- The factor n!/(x1!x2!...xk!) that counts how many orderings produce the same category counts.
Common Mistakes to Avoid
- Using probabilities that do not add to 1. The multinomial model requires all category probabilities to sum exactly to 1.
- Using counts that do not add to n. The formula only applies when the listed category counts account for every trial.
- Forgetting the multinomial coefficient. Multiplying only p1^x1 p2^x2 ... pk^xk gives the probability of one specific order, not all possible orders with the same counts.
- Treating category counts as independent. If the total number of trials is fixed, a high count in one category leaves fewer trials available for the others.
Practice Questions
- 1 A bag contains balls that are red with probability 0.5, blue with probability 0.3, and green with probability 0.2. If 6 balls are drawn with replacement, what is the probability of getting 3 red, 2 blue, and 1 green?
- 2 A fair six-sided die is rolled 12 times. What is the probability of getting exactly 2 ones, 2 twos, 2 threes, 2 fours, 2 fives, and 2 sixes?
- 3 A survey has four response categories with fixed probabilities. Explain why increasing the observed count in one category tends to decrease the possible counts in the other categories when the sample size is fixed.