The describes the probability of obtaining a specific number of counts for k different outcomes, when each outcome has a fixed probability of occurring.

If a  X follows a multinomial distribution, then the probability that outcome 1 occurs exactly x₁ times, outcome 2 occurs exactly x₂ times, etc. can be found by the following formula:

Probability = n! * (p₁^x₁ * p₂^x₂ * … * p_k^x_k) /  (x₁! * x₂! … * x_k!)

where:

• n: total number of events
• x₁: number of times outcome 1 occurs
• p₁: probability that outcome 1 occurs in a given trial

To calculate a multinomial probability in R we can use the dmultinom() function, which uses the following syntax:

dmultinom(x=c(1, 6, 8), prob=c(.4, .5, .1))

where:

• x: A vector that represents the frequency of each outcome
• prob: A vector that represents the probability of each outcome (the sum must be 1)

The following examples show how to use this function in practice.

Example 1

In a three-way election for mayor, candidate A receives 10% of the votes, candidate B receives 40% of the votes, and candidate C receives 50% of the votes.

If we select a random sample of 10 voters, what is the probability that 2 voted for candidate A, 4 voted for candidate B, and 4 voted for candidate C?

We can use the following code in R to answer this question:

#calculate multinomial probability
dmultinom(x=c(2, 4, 4), prob=c(.1, .4, .5))

[1] 0.0504

The probability that exactly 2 people voted for A, 4 voted for B, and 4 voted for C is 0.0504.

Example 2

If we randomly select 4 balls from the urn, with replacement, what is the probability that all 4 balls are yellow?

We can use the following code in R to answer this question:

#calculate multinomial probability
dmultinom(x=c(4, 0, 0), prob=c(.6, .2, .2))

[1] 0.1296

The probability that all 4 balls are yellow is 0.1296.

Example 3

Suppose two students play chess against each other. The probability that student A wins a given game is 0.5, the probability that student B wins a given game is 0.3, and the probability that they tie in a given game is 0.2.

If they play 10 games, what is the probability that player A wins 4 times, player B wins 5 times, and they tie 1 time?

We can use the following code in R to answer this question:

#calculate multinomial probability
dmultinom(x=c(4, 5, 1), prob=c(.5, .3, .2))

[1] 0.0382725

The probability that player A wins 4 times, player B wins 5 times, and they tie 1 time is about 0.038.

Additional Resources

The following tutorials provide additional information about the multinomial distribution: