Counts belong to labeled categories
A multinomial model counts outcomes in m mutually exclusive and exhaustive categories across n independent trials, with fixed category probabilities p₁,…,pₘ summing to 1. Count vector (x₁,…,xₘ) must sum to n.
Swapping counts between categories can change probability when their probabilities differ. A specific labeled count vector differs from an unordered pattern that sums many such vectors. Older versions of this page confused those questions; the tables below consistently use labeled counts.
Three labeled categories: n = 6, p = (0.5, 0.3, 0.2)
The following are selected vectors, not the complete distribution. Rounded probabilities do not sum to 1 across these selected rows.
| (x₁, x₂, x₃) | Probability |
|---|---|
| (6, 0, 0) | 0.0156 |
| (5, 1, 0) | 0.0563 |
| (5, 0, 1) | 0.0375 |
| (4, 2, 0) | 0.0844 |
| (4, 1, 1) | 0.1125 |
| (3, 3, 0) | 0.0675 |
| (3, 2, 1) | 0.1350 |
| (2, 2, 2) | 0.0810 |
Worked example
For exactly (3,2,1), the arrangement count is 6!/(3!2!1!) = 60. The probability is 60 × 0.5³ × 0.3² × 0.2 = 0.1350. This is the chance of those labeled category totals in any trial order, not just one specific sequence.
For six independent fair die rolls, the probability of observing each face exactly once is 6! × (1/6)⁶ ≈ 0.0154. A particular ordered sequence such as 1,2,3,4,5,6 has probability (1/6)⁶, without the 6! multiplier.
For a category i, E[Xᵢ] = npᵢ and Var(Xᵢ) = npᵢ(1 − pᵢ). For different categories, Cov(Xᵢ,Xⱼ) = −npᵢpⱼ: the counts are dependent because their total is fixed, even though the underlying trials are independent.
Assumptions and next steps
Check category coding, fixed probabilities and independence of trials. Sampling without replacement from a finite categorized population needs a multivariate hypergeometric model. A goodness-of-fit test can compare observed category counts with a specified probability model, subject to its own approximation conditions.
Read chi-square tests and counting principles. SciPy documents the multinomial distribution. All examples here are mathematical or hypothetical, not estimates of demographic rates.