This online multinomial distribution calculator finds the probability of the exact outcome of a multinomial experiment (multinomial probability), given the number of possible outcomes (must be no less than 2) and respective number of pairs: probability of a particular outcome and frequency of this outcome (number of its occurrences). All the parameters used in our multinomial calculator must be non-negative numbers. All the frequencies must be integers. The sum of all the probabilities must be equal to 1, and the sum of all the frequencies must not exceed 1000.
Multinomial formula
A multinomial experiment is a statistical experiment which involves \(n\) independent trials. Each trial has exactly \(m\) possible outcomes. The probability \(p_i\) that a particular outcome \(i\in \{ 1, 2,…, m \}\) will occur is constant for any given trial. The number of times \(k_i\) that the outcome \(i\) occurs is called its frequency. Obviously $$ k_1+k_2+\cdots+k_m = n,$$ $$p_1+p_2+\cdots+p_m = 1. $$A multinomial probability \(P\) refers to the probability of obtaining a specified set of frequencies in a multinomial experiment. It can be expressed by the following formula:
$$P = \frac{n!}{k_1!\, k_2! \cdots k_m!} p_1^{k_1} p_2^{k_2} \cdots p_m^{k_m},$$
where
$$\frac{n!}{k_1!\, k_2! \cdots k_m!} = {n \choose k_1, k_2, \ldots, k_m}$$
is the multinomial coefficient found in polynomial expansions. It can be expressed as a product of binomial coefficients:
$${n \choose k_1, k_2, \ldots, k_m} = {k_1\choose k_1}{k_1+k_2\choose k_2}\cdots{k_1+k_2+\cdots+k_m\choose k_m},$$
where factors on the right-hand side of the equality are the binomial coefficients:
$$C(n,k) = \frac { n! }{ k!(n-k)! } = {n\choose k}.$$
We use this relationship in calculation of the multinomial coefficient.
Example
Suppose we have a bowl with 10 balls – 5 red balls, 3 white balls and 2 black balls. We randomly select 5 balls from the bowl with replacement (that is important in order to keep all the probabilities constant). What is the probability of selecting 3 red balls, 1 white ball and 1 black ball?
In this case the multinomial experiment consists of 5 trials of picking a ball, so \(n\) = 5. There are balls of only three colors in the bowl, so \(m\) = 3. The 5 trials produce 3 red balls, 1 white ball and 1 black ball, thus \(k_1\) = 3, \(k_2\) = 1, and \(k_3\) = 1. In each particular trial, the probability of drawing a red, white, or black ball is 0.5, 0.3, and 0.2, respectively. So \(p_1\) = 0.5, \(p_2\) = 0.3, and \(p_3\) = 0.2. We plug this input into our multinomial distribution calculator and easily get the result \(P\) = 0.15. Thus, the probability of selecting exactly 3 red balls, 1 white ball and 1 black ball is 0.15.
Related calculators
Check out our other statistics calculators such as Binomial Probability Calculator or Hypergeometric Distribution Calculator.