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Multinomial Probability Calculator

Enter outcome counts and their probabilities to compute the multinomial probability.

How it is calculated

Enter the observed count for each category and the chance of each category, both as comma-separated lists of the same length (at least two entries). The tool computes

P = n! ÷ (x₁! · x₂! · … · xₖ!) × p₁x₁ · p₂x₂ · … · pₖxₖ

where xᵢ are the counts, n = Σxᵢ is the number of trials and pᵢ are the category probabilities. Two things happen before the formula: counts are rounded to whole numbers, and the probabilities are divided by their total, so 50, 30, 20 behaves exactly like 0.5, 0.3, 0.2. The sum is taken in log space to avoid overflow with large n. The result is shown as a decimal with six places and as a percentage. Negative counts, negative probabilities, a different number of counts and probabilities, or non-numeric entries give no result.

Worked example

Draw 10 balls with replacement from a bag where red, blue and green have probabilities 0.5, 0.3 and 0.2. What is the chance of exactly 5 red, 3 blue and 2 green?

  1. Arrangements: 10! ÷ (5! × 3! × 2!) = 2520.
  2. Probability of one such arrangement: 0.5⁵ × 0.3³ × 0.2² = 0.03125 × 0.027 × 0.04 = 0.00003375.
  3. Multiply: 2520 × 0.00003375 = 0.08505.

The tool shows 0.085050, which is 8.51% over 10 trials.

Try this example →

Common mistakes

  • Entering probabilities as percentages (50, 30, 20) and expecting an error. The tool rescales them to sum to one, so it works, but a typo such as 50, 30, 30 is silently rescaled as well instead of being flagged.
  • Thinking the counts need to match a target total. The number of trials is simply the sum of the counts you enter, so forgetting one category quietly changes n.
  • Using this for draws without replacement. The multinomial assumes independent trials with fixed probabilities; drawing cards from a deck needs the hypergeometric distribution.
  • Reading the result as the chance of approximately that split. It is the probability of exactly these counts; a split like 6, 2, 2 has its own separate value.
  • Separating values with spaces. Only commas split the lists (or semicolons, if you use them throughout), so 5 3 2 gives no result.

About this calculator

The multinomial probability calculator extends the binomial distribution to trials with more than two possible outcomes. For n independent trials split into k categories with probabilities p₁, p₂, … pₖ, it computes the probability of observing exactly x₁, x₂, … xₖ occurrences of each category using P = n! / (x₁!x₂!…xₖ!) × p₁^x₁ × p₂^x₂ × … × pₖ^xₖ, where n is simply the sum of the counts you enter (rounded to whole numbers). Enter counts and probabilities as comma-separated lists of equal length; the probabilities are scaled by their total, so they don't need to add up to exactly 1 — 2, 1, 1 works the same as 0.5, 0.25, 0.25.

This distribution is the natural model whenever outcomes fall into several discrete buckets rather than just success/failure — genetics uses it for Mendelian ratios across multiple phenotypes, quality control uses it when defects are classified into several categories, and pollsters use it to model how a sample splits across more than two candidates or choices. Statisticians, geneticists, and quality-assurance analysts use this calculator instead of doing the factorial arithmetic by hand.

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Comments (1)

  • Editorial team

    It is easy to expect the 'expected' split to come out as a high probability, but with several categories it rarely does. In the 10-trial example with 0.5, 0.3 and 0.2, the split 5, 3 and 2 is the single most likely outcome, yet it is only about 8.5%; the rest is spread over neighbouring splits like 6, 3, 1. To judge a result, compare it with the exact-match chance of the most likely split. Which situation are you modelling?

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