The Multinomial Distribution: When There Are More Than Two Outcomes

The binomial distribution handles trials with exactly two possible outcomes — success or failure, heads or tails. Plenty of real situations have more than two categories: a survey respondent can prefer Candidate A, B, or C; a rolled die lands on one of six faces; a manufactured part is graded excellent, acceptable, or defective. The multinomial distribution is the direct generalization of the binomial to exactly this case, and it shows up any time an experiment repeats a fixed number of independent trials, each falling into one of several distinct, fixed-probability categories.

The Formula

For n independent trials, each landing in exactly one of r categories with fixed probabilities p₁, p₂, ..., pᵣ (which must sum to 1), the probability of observing exactly k₁ outcomes in category 1, k₂ in category 2, and so on, is:

The coefficient n!/(k₁!k₂!···kᵣ!) — the multinomial coefficient — counts the number of distinct ways to arrange n trials into groups of the specified sizes; it's the direct r-category generalization of the binomial coefficient C(n,k), which is itself just the 2-category case of this same formula (with k and n−k as the two group sizes).

Worked Example: A Three-Candidate Survey

Problem: In an election, true voter support is 50% for Candidate A, 30% for Candidate B, and 20% for Candidate C. In a random sample of 10 voters, what's the probability of getting exactly 5 supporters of A, 3 of B, and 2 of C?

Here n=10, and (k₁,k₂,k₃) = (5,3,2) with (p₁,p₂,p₃) = (0.5,0.3,0.2).

Multinomial coefficient = 10! / (5!×3!×2!) = 3,628,800 / (120×6×2) = 2,520

P(5,3,2) = 2,520 × (0.5)⁵ × (0.3)³ × (0.2)² = 2,520 × 0.03125 × 0.027 × 0.04 ≈ 0.0851

There's roughly an 8.5% chance a sample of 10 voters splits exactly 5-3-2, even though that split matches the true population proportions closely (50%-30%-20% would suggest 5-3-2 out of 10 exactly) — a useful reminder that even the single most likely specific outcome in a multi-category distribution often still carries a fairly modest probability, since there are many other individual outcomes competing for probability mass alongside it.

A Second Worked Example: A Fair Die

Problem: A fair six-sided die is rolled 12 times. What's the probability of getting exactly 2 of each face value (two 1s, two 2s, ..., two 6s)?

Here n=12, r=6 categories, each kᵢ=2, and each pᵢ=1/6.

Multinomial coefficient = 12! / (2!)⁶ = 479,001,600 / 64 = 7,484,400

P(2 of each face) = 7,484,400 × (1/6)¹² ≈ 0.00344

Despite "2 of each face in 12 rolls" being the single most evenly balanced, seemingly most representative outcome, it still occurs only about 0.34% of the time — a vivid illustration of how many distinct outcomes a multi-category distribution spreads its total probability across; even the most "expected-looking" specific result remains individually rare once there are enough categories and trials in play.

Mean and Variance of Each Category

Each individual category, considered on its own, actually follows an ordinary binomial distribution (that category "succeeding" versus everything else being treated as "failure"), so the familiar binomial formulas apply directly to each Xᵢ:

For the survey example, the expected number of Candidate A supporters is μ = 10×0.5 = 5, exactly matching the intuitive proportional expectation — the multinomial distribution's joint formula answers a more specific question (the probability of one exact full combination of all categories at once), while these simpler per-category formulas answer the more familiar single-category question.

Connection to the Chi-Square Goodness-of-Fit Test

The multinomial distribution is the theoretical foundation underneath the chi-square goodness-of-fit test: that test compares an observed multinomial count split (how many respondents actually fell into each category) against the counts a hypothesized set of category probabilities would predict, and the chi-square statistic quantifies how far the observed split has drifted from that expectation. Anywhere a chi-square goodness-of-fit test is being used, the multinomial distribution is the underlying probability model making that test mathematically valid in the first place — the chi-square test can be thought of as a practical shortcut for assessing multinomial fit without needing to compute exact multinomial probabilities for every possible outcome combination directly.

Real-World Uses

Market research and polling: as in the worked example, surveys with more than two response options (candidate preference, product choice among several brands, satisfaction ratings across several categories) are naturally modeled with the multinomial distribution.

Genetics: classic Mendelian genetics problems, where offspring fall into several distinct phenotype categories with theoretically predicted ratios, are directly multinomial — testing observed offspring counts against theoretical ratios is a textbook chi-square-goodness-of-fit application built on this exact distribution.

Natural language processing: the distribution of word categories or token types in text-generation models is frequently modeled using the multinomial distribution, since each generated token falls into exactly one of many possible categories (words or subword units) at each step.

Quality control with multiple grades: inspecting a batch of manufactured parts and classifying each one into more than two quality categories (excellent, acceptable, defective) rather than a simple pass/fail extends naturally from the binomial into the multinomial framework, letting a single test evaluate the full distribution across all grades at once rather than running several separate two-category tests.

Common Mistakes

Forgetting that the category probabilities must sum to exactly 1. Every trial must land in exactly one of the r categories — if the pᵢ values don't sum to 1, either a category is missing or the probabilities themselves are wrong.

Confusing the joint probability of one exact combination with a single category's marginal probability. The full multinomial formula answers "what's the probability of this exact full split across all categories," a much more specific and typically much smaller number than "what's the probability Candidate A gets exactly 5 votes" (which uses the simpler per-category binomial formula instead, ignoring how the remaining votes split between B and C).

Treating a small individual probability as evidence something unusual happened. As the die-roll example shows, even the single most representative-looking outcome can have a low individual probability once there are several categories — a small P(X) on its own doesn't mean the outcome is surprising relative to the full range of alternatives it's being compared against.