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  1. Introduction
  2. Graphing Distributions
  3. Summarizing Distributions
  4. Describing Bivariate Data

  5. Probability
    1. Contents
      Standard
    2. Introduction to Probability
      Standard
         Video
    3. Basic Concepts
      Standard
         Video
    4. Conditional p Demo
      Standard
    5. Gambler's Fallacy
      Standard
         Video
    6. Permutations and Combinations
      Standard
         Video
    7. Birthday Demo
      Standard
    8. Binomial Distribution
      Standard
         Video
    9. Binomial Demonstration
      Standard
    10. Poisson Distribution
      Standard
         Video
    11. Multinomial Distribution
      Standard  
    12. Hypergeometric Distribution
      Standard
         Video
    13. Base Rates
      Standard
         Video
    14. Bayes Demo
      Standard
    15. Monty Hall Problem
      Standard
    16. Statistical Literacy
      Standard
    17. Exercises
      Standard

  6. Research Design
  7. Normal Distribution
  8. Advanced Graphs
  9. Sampling Distributions
  10. Estimation
  11. Logic of Hypothesis Testing
  12. Tests of Means
  13. Power
  14. Regression
  15. Analysis of Variance
  16. Transformations
  17. Chi Square
  18. Distribution Free Tests
  19. Effect Size
  20. Case Studies
  21. Calculators
  22. Glossary
 

Multinomial Distribution

Author(s)

David M. Lane

Prerequisites

Distributions, Basic Probability, Variability, Binomial Distribution

Learning Objectives
  1. Define multinomial outcomes
  2. Compute probabilities using the multinomial distribution

The binomial distribution allows one to compute the probability of obtaining a given number of binary outcomes. For example, it can be used to compute the probability of getting 6 heads out of 10 coin flips. The flip of a coin is a binary outcome because it has only two possible outcomes: heads and tails. The multinomial distribution can be used to compute the probabilities in situations in which there are more than two possible outcomes. For example, suppose that two chess players had played numerous games and it was determined that the probability that Player A would win is 0.40, the probability that Player B would win is 0.35, and the probability that the game would end in a draw is 0.25. The multinomial distribution can be used to answer questions such as: "If these two chess players played 12 games, what is the probability that Player A would win 7 games, Player B would win 2 games, and the remaining 3 games would be drawn?" The following formula gives the probability of obtaining a specific set of outcomes when there are three possible outcomes for each event:

multinomial 3 outcomes

where

p is the probability,
n is the total number of events
n1 is the number of times Outcome 1 occurs,
n2 is the number of times Outcome 2 occurs,
n3 is the number of times Outcome 3 occurs,
p1 is the probability of Outcome 1
p2 is the probability of Outcome 2, and
p3 is the probability of Outcome 3.


For the chess example,

n = 12 (12 games are played),
n1 = 7 (number won by Player A),
n2 = 2 (number won by Player B),
n3 = 3 (the number drawn),
p1 = 0.40 (probability Player A wins)
p2 = 0.35(probability Player B wins)
p3 = 0.25(probability of a draw)

chess example multinomial


The formula for k outcomes is

k outcomes

Note that the binomial distribution is a special case of the multinomial when k = 2.

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