Multinomial Probability Calculator
The multinomial distribution extends the binomial to experiments with more than two possible outcomes per trial. Enter the probability and target count for each of three outcomes; the counts should sum to the number of trials. The calculator returns the multinomial coefficient and the probability of that exact set of counts.
Multinomial formula
n = x1 + x2 + x3
Coefficient = n! / (x1! * x2! * x3!)
P = Coefficient * p1^x1 * p2^x2 * p3^x3
(probabilities are normalised so p1 + p2 + p3 = 1)
The coefficient counts the orderings of n trials that yield the given counts. Multiplying by each outcome probability raised to its count gives the probability of that exact breakdown.
Worked example
With p1 = 0.5, p2 = 0.3, p3 = 0.2 and counts 3, 2, 1 (so n = 6), the coefficient is 6! divided by (3! times 2! times 1!) = 60. The probability is 60 times 0.5^3 times 0.3^2 times 0.2^1 = 60 times 0.125 times 0.09 times 0.2 = 0.135, about 13.50%.
Multinomial probability: frequently asked questions
What is the multinomial distribution?
It generalises the binomial to more than two outcomes. Across n independent trials, each with fixed outcome probabilities, it gives the probability of seeing exactly a specified count of each outcome. This calculator handles three outcomes.
What is the formula?
P = n! divided by (x1! times x2! times x3!) times p1^x1 times p2^x2 times p3^x3, where the counts x1, x2, x3 sum to n and the probabilities p1, p2, p3 sum to 1. The first factor is the multinomial coefficient.
What must the inputs satisfy?
The three counts must be non-negative whole numbers that add up to the number of trials n. The three probabilities must be non-negative and add up to one. The calculator normalises the probabilities and checks the counts before computing.
How does it relate to the binomial distribution?
With only two outcomes, the multinomial reduces exactly to the binomial. The multinomial coefficient becomes the binomial coefficient C(n, x1), and the formula matches the binomial probability mass function.
Sources
- NIST Digital Library of Mathematical Functions: Multinomials and multinomial coefficients.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.