Multinomial Probability Calculator

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 the counts must sum to n and the probabilities must sum to 1.

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.