Multinomial Distribution
A multinomial distribution is the
probability distribution of the outcomes from a multinomial experiment.
Multinomial Experiment
A multinomial experiment is a
statistical experiment that has the following properties:

The trials are independent;
that is, the outcome on one trial does not affect the outcome on other trials.
Consider the following statistical experiment. You toss two dice three times,
and record the outcome on each toss. This is a multinomial experiment because:

The trials are independent; that is, getting a particular outcome on one trial
does not affect the outcome on other trials.
Note: A binomial
experiment is a special case of a multinomial experiment. Here is the
main difference. With a binomial experiment, each trial can result in two  and
only two  possible outcomes. With a multinomial experiment, each trial can
have two or more possible outcomes.
Multinomial Distribution
A multinomial distribution is the
probability distribution of the outcomes from a multinomial experiment.
The multinomial formula defines the probability of any outcome from a
multinomial experiment.
Multinomial Formula. Suppose a
multinomial experiment consists of n trials, and each trial can result
in any of k possible outcomes: E_{1}, E_{2}, . . . , E_{k}.
Suppose, further, that each possible outcome can occur with probabilities p_{1},
p_{2}, . . . , p_{k}. Then, the probability (P) that E_{1}
occurs n_{1} times, E_{2} occurs n_{2} times, . . . ,
and E_{k} occurs n_{k} times is:
P = [ n! / ( n_{1}!
* n_{2}! * ... n_{k}! ) ] * ( p_{1}^{n1}
* p_{2}^{n2} * . . . * p_{k}^{nk}
)
where n = n_{1} + n_{2} + . . . + n_{k}.
The examples below illustrate how to use the multinomial formula to compute the
probability of an outcome from a multinomial experiment.
Test Your Understanding
Problem 1
Suppose a card is drawn randomly from an ordinary deck of playing cards, and
then put back in the deck. This exercise is repeated five times. What is the
probability of drawing 1 spade, 1 heart, 1 diamond, and 2 clubs?
Solution: To solve this problem, we apply the multinomial formula. We
know the following:

On any particular trial, the probability of drawing a spade, heart, diamond, or
club is 0.25, 0.25, 0.25, and 0.25, respectively. Thus, p_{1} = 0.25, p_{2}
= 0.25, p_{3} = 0.25, and p_{4} = 0.25.
We plug these inputs into the multinomial formula, as shown below:
P = [ n! / ( n_{1}! * n_{2}! * ... n_{k}!
) ] * ( p_{1}^{n1} * p_{2}^{n2}
* . . . * p_{k}^{nk} )
P = [ 5! / ( 1! * 1! * 1! * 2! ) ] * [ (0.25)^{1} * (0.25)^{1} *
(0.25)^{1} * (0.25)^{2} ]
P = 0.05859
Thus, if we draw five cards
with replacement from an ordinary deck of playing cards, the
probability of drawing 1 spade, 1 heart, 1 diamond, and 2 clubs is 0.05859.
Multinomial Calculator
The multinomial formula requires many timeconsuming
computations. The Multinomial Calculator can do this work for you 
quickly, easily, and errorfree. Use the Multinomial Calculator to compute the
probability of outcomes from multinomial experiments. The
calculator is free. It can found in the Stat Trek
main menu under the Stat Tools tab. Or you can tap the button below.
Multinomial Calculator
Problem 2
Suppose we have a bowl with 10 marbles  2 red marbles, 3 green marbles, and 5
blue marbles. We randomly select 4 marbles from the bowl,
with replacement. What is the probability of selecting 2 green marbles
and 2 blue marbles?
Solution: To solve this problem, we apply the multinomial formula. We
know the following:

The experiment consists of 4 trials, so n = 4.

On any particular trial, the probability of drawing a red, green, or blue
marble is 0.2, 0.3, and 0.5, respectively. Thus, p_{red} = 0.2, p_{green}
= 0.3, and p_{blue} = 0.5
We plug these inputs into the multinomial formula, as shown below:
P = [ n! / ( n_{1}! * n_{2}! * ... n_{k}!
) ] * ( p_{1}^{n1} * p_{2}^{n2}
* . . . * p_{k}^{nk} )
P = [ 4! / ( 0! * 2! * 2! ) ] * [ (0.2)^{0} * (0.3)^{2} * (0.5)^{2}
]
P = 0.135
Thus, if we draw 4 marbles
with replacement from the bowl, the probability of drawing 0 red
marbles, 2 green marbles, and 2 blue marbles is 0.135.