How to use the Multinomial Probability Calculator
When a trial can land in three or more categories, the multinomial distribution gives the chance that n trials split into exactly the counts you have in mind. Pick 3 to 6 categories, type each category probability p and observed count k, and the tool evaluates P = n!/(k1!·k2!···)·p1^k1·p2^k2···. The total number of trials n is the sum of the counts.
If the probabilities do not add up to 100% the distribution does not exist, so the calculation is blocked and the current sum is shown instead. Factorials overflow quickly, so the page sums logarithms and takes a single exponential at the end, and the sum of counts is capped at 1,000 trials. Every figure assumes independent trials whose probabilities never shift.
This page returns the probability of one specified split. It does not cover the two-category binomial case, the geometric distribution for a first success, or goodness-of-fit testing. Treat the output as a reference. Current as of October 2026.
Frequently asked questions
A multinomial distribution is defined only when the category probabilities sum to 1. If the sum is off, the distribution does not exist, so the tool blocks the calculation and shows the current sum.
Factorials gain digits so fast that ordinary multiplication overflows. This page adds logarithms instead and converts back once at the end, and the cap of 1,000 keeps the displayed figure accurate.
Not necessarily. The expected count is the average n times p, and real counts scatter around it. A gap on its own is not proof that the probabilities you entered are wrong.