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.
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.