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The multinomial distribution is a probability distribution used to describe the outcomes of a categorical variable with more than two categories. It is often used in probability and statistics to model and analyze data involving multiple independent variables and their corresponding outcome probabilities. This distribution can be applied in various fields such as economics, biology, and social sciences to understand the likelihood of different outcomes occurring simultaneously. In essence, the multinomial distribution provides a framework for understanding and predicting the occurrence of multiple outcomes in a given scenario, making it a valuable tool in statistical analysis.
An Introduction to the Multinomial Distribution
The multinomial distribution 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, outcome 3 occurs exactly x3 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
For example, suppose there are 5 red marbles, 3 green marbles, and 2 blue marbles in an urn. If we randomly select 5 marbles from the urn, with replacement, what is the probability of obtaining exactly 2 red marbles, 2 green marbles, and 1 blue marble?
To answer this, we can use the multinomial distribution with the following parameters:
- n: 5
- x1 (# red marbles) = 2, x2 (# green marbles) = 2, x3 (# blue marbles) = 1
- p1 (prob. red) = 0.5, p2 (prob. green) = 0.3, p3 (prob. blue) = 0.2
Plugging these numbers in the formula, we find the probability to be:
Probability = 5! * (.52 * .32 * .21) / (2! * 2! * 1!) = 0.135.
Multinomial Distribution Practice Problems
Use the following practice problems to test your knowledge of the multinomial distribution.
Note: We will use the to calculate the answers to these questions.
Problem 1
Question: 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?
Answer: Using the Multinomial Distribution Calculator with the following inputs, we find that the probability is 0.0504:
Problem 2
Question: Suppose an urn contains 6 yellow marbles, 2 red marbles, and 2 pink marbles. If we randomly select 4 balls from the urn, with replacement, what is the probability that all 4 balls are yellow?
Answer: Using the Multinomial Distribution Calculator with the following inputs, we find that the probability is 0.1296:
Problem 3
Question: 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?
Answer: Using the Multinomial Distribution Calculator with the following inputs, we find that the probability is 0.038272:
Additional Resources
The following tutorials provide an introduction to other common distributions in statistics: