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Multinomial Distributions

Probability distributions for experiments with more than 2 outcomes : P= n!/ n1!n2!n3!...nk! x (p1^n1 x p2^n2 x p3^n3...pk^nk

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Multinomial Distributions
Teacher Contributed

Multinomial Distributions


  • Multinomial distributions
  • Factorials
  • Probability

Student Exploration

How do the grades of students at your school measure up? How well does your school prepare its students for college?

What is the likelihood that a student at your high school is on the track to college? Today society expects that almosteveryone goes to college in order to build a strong future for themselves, however it is more difficult and competitive than ever to get into college (partly because so many people are applying). One of the biggest factors that contribute to your college application are your high school grades. So how well is your school preparing its students for this application process? We can investigate this using multinomial distribution. You just learned about multinomial distributions, situations that are made up of more than two events and where each event has a specific probability.

  1. Research a college that you are interested and find the average grade point average (gpa) of their most recent class of accepted students. Then determine all the different grade combinations that make up at that gpa or higher (i.e. a 3.0 is all B’s and a 3.4 is 3 B’s and 2 A’s).These two websites can help you find this information: http://www.education.com/reference/article/why-college-admissions-competitive/ and http://www.collegedata.com/cs/search/college/college_search_tmpl.jhtml.
  2. Next you need to research the grades of the students at your school. You need to determine how many individual A, B, C, D and F grades were earned by all the students at your school last semester. You can find this by looking it up online (if your school or district posts this information) or by asking your school’s administrative assistant.
  3. Lastly set up multinomial distribution equations to calculate the chances that a student picked at random has each of the grade combinations that you determined in #1.
  4. In order to determine the probability that a student at random is on track to being accepted to the college you researched, add up all the probabilities that you calculated in #3.
  5. How did the students at your school measure up? Are they on track to college?

Articles on the college application process today:




Extension Investigation

Multinomial distributions also comes into play when you are looking at blood types or bone marrow types (for transfusions), and you want to determine the likelihood that you find a matching type.

  1. Why might multinomial distributions be more common than binomial distributions in real life? Explain.
  2. Compare the formula for binomial and multinomial distributions. How has the formula for multinomial distributions been modified to accommodate more than two possible outcomes?

Resources Cited






Connections to other CK-12 Subject Areas

  • Binomial Distributions
  • Theoretical and Experimental Probability
  • Probability and Permutations.

Image from: http://pixabay.com/en/three-library-school-stack-36753/

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