## Multinomial Distribution

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# Multinomial Distribution

It is mainly used to get probabilities in statistical experiments which have multiple outcomes i.e. outcomes > 2.When you limit the number of outcomes to two you get a binomial distribution. The difference between a binomial and a multinomial distribution is the number of outcomes.

In simple terms, the probability density function of a multinomial experiment is referred to as a multinomial distribution just as we say that a binomial experiment has a binomial distribution.

*Properties of a Multinomial Experiment*

- The statistical experiment has a fixed number of repeated trials
- Each trial can only have a definite number of outcomes
- The probability of each outcome is constant on each trial
- Each trial is independent from the others

*Definition*

Assume a multinomial experiment has n trials. Further assume that each trial has k independent possible outcomes i.e. E1, E2, …, Ek. Let each of these possible outcomes have constant probabilities in each trial i.e. p1, p2, …, pk. The multinomial distribution gives us the probability of E1occurringn1 times, E2 occurring n2 times, … and Ek occurring nk times and is expressed as;

**P = [ n! / ( n _{1}! * n_{2}! * ... n_{k}! ) ] * ( p_{1}^{n1} * p_{2}^{n2} * . . . * p_{k}^{nk} ) **

where n = n1 + n2 +... + nk

*Example*

Assume you have a standard deck of 52 playing cards. You go ahead and randomly draw five cards from the deck with replacement. Find the probability of drawing 2 clubs, 1 heart, 1 spade and 1 diamond.

*Answer*

n=5

n1=2

n2=n3=n4=1

p1=p2=p3=p4=0.25

P = [ 5! / ( 1! * 1! * 1! * 2! ) ] * [ (0.25)^{1} * (0.25)^{1} * (0.25)^{1} * (0.25)^{2} ]

P = 0.05859