Multinomial Distribution Calculator
The Multinomial Calculator makes it easy
to compute multinomial probabilities. For help in using the calculator, read
the FrequentlyAsked Questions or review
the Sample Problems.
To learn more, go to Stat Trek's
tutorial on the multinomial distribution.
FrequentlyAsked Questions
Instructions: To find the answer to a frequentlyasked
question, simply click on the question. If none of the questions addresses your
need, refer to Stat Trek's tutorial
on the multinomial distribution or visit the
Statistics Glossary. Online help is just a mouse click away.
Multinomial Distribution: Sample Problems

Suppose you toss a pair of dice 10 times. What is the probability of getting
the following outcome: two rolls of 7, two rolls of 6, and any other outcome on
the remaining six rolls.
Hint: The probabilities associated with each roll of two dice are shown below.
Outcome 
Probability 
2 
1/36 
3 
2/36 
4 
3/36 
5 
4/36 
6 
5/36 
7 
6/36 
8 
5/36 
9 
4/36 
10 
3/36 
11 
2/36 
12 
1/36 
Solution:
We know the following:

The number of outcomes is 3. (Outcome 1 is a roll of 7, the Outcome 2 is a roll
of 6, and Outcome 3 is any other roll.)

Outcome 1: The probability is 6/36, and the frequency is 2.

Outcome 2: The probability is 5/36, and the frequency is 2.

Outcome 3: The probability is 25/36, and the frequency is 6.
Therefore, we plug those numbers into the Multinomial
Calculator
and hit the Calculate button. The calculator reports that the multinomial
probability is 0.076. Thus, in ten rolls of the dice, the probability of
rolling 7 two times, 6 two times, and something else six times
is 0.076.

A bowl has 2 black marbles, 3 green marbles, and 5 white marbles. A marble is
randomly selected and then put back in the bowl. Suppose this selection process
is repeated five times. What is the probability that 3 white marbles, 1 green
marble, and 1 black marble will be chosen?
Hint: On any given trial, the probability of choosing a black marble is 2/10;
the probability of choosing a green marble is 3/10; and
the probability of choosing a white marble is 5/10.
Solution:
We know the following:

The number of outcomes is 3. (Outcome 1 is a black marble; Outcome 2, a green
marble; and Outcome 3, a white marble.)

Outcome 1: The probability is 0.2, and the frequency is 1.

Outcome 2: The probability is 0.3, and the frequency is 1.

Outcome 3: The probability is 0.5, and the frequency is 3.
Therefore, we plug those numbers into the Multinomial
Calculator and hit the Calculate button. The calculator reports that
the cumulative multinomial probability is 0.150. Thus, the probability of
selecting 1 black marble, 1 green marble, and 3 white marbles is 0.150.