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The Multinomial Distribution is a probability distribution used to calculate the probability of counts for a given number of outcomes. In R, the function dmultinom() is used to calculate the probability of multiple outcomes. The function takes a vector of probabilities, the number of trials and the number of successes as parameters and returns the probability of the specified outcome. The function can also be used in conjunction with other functions such as pnorm() and qnorm() to calculate more complex probabilities.
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! * (p1x1 * p2x2 * … * pkxk) / (x1! * x2! … * xk!)
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.
The following tutorials provide additional information about the multinomial distribution: