# Multinomial Distribution Calculator

The Multinomial Calculator makes it easy to compute multinomial probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.

To learn more, go to Stat Trek's tutorial on the multinomial distribution.

• First, enter the number of outcomes.
• Then, enter the probability and frequency for each outcome.
• Click the Calculate button.
Number of outcomes

 Outcome Probability Frequency 1 2

Multinomial probability:

Instructions: To find the answer to a frequently-asked 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.

### What is a multinomial experiment?

A multinomial experiment is a statistical experiment that has the following characteristics:

• The experiment involves one or more trials.
• Each trial has a discrete number of possible outcomes.
• On any given trial, the probability that a particular outcome will occur is constant.
• All of the trials in the experiment are independent.

Tossing a pair of dice is a perfect example of a multinomial experiment. Suppose we toss a pair of dice three times. Each toss represents a trial, so this experiment would have 3 trials. Each toss also has a discrete number of possible outcomes - 2 through 12. The probability of any particular outcome is constant; for example, the probability of rolling a 12 on any particular toss is always 1/36. And finally, the outcome on any toss is not affected by previous or succeeding tosses; so the trials in the experiment are independent.

### What is a multinomial distribution?

A multinomial distribution is a probability distribution. It refers to the probabilities associated with each of the possible outcomes in a multinomial experiment.

For example, suppose we flip three coins and count the number of coins that land on heads. This multinomial experiment has four possible outcomes: 0 heads, 1 head, 2 heads, and 3 heads. Probabilities associated with each possible outcome are an example of a multinomial distribution, as shown below.

Outcome Probability

The table completely defines the probabilities associated with every possible outcome from this multinomial experiment. It is the multinomial distribution for this experiment.

### What is the number of outcomes?

The number of outcomes refers to the number of different results that could occur from a multinomial experiment. For example, suppose we roll a die. Each roll of the die can have six possible outcomes - 1, 2, 3, 4, 5, or 6. Similarly, the roll of two dice can have eleven possible outcomes - the numbers from 2 to 12.

### What is the probability of an outcome?

Each trial in a multinomial experiment can have a discrete number of outcomes. The likelihood that a particular outcome will occur in a single trial is the probability of the outcome.

For example, suppose we toss two dice. The probability of tossing a 2 is 1/36; the probability of tossing a 3 is 2/36, the probability of tossing 4 is 3/36, etc.

### What is the frequency of an outcome?

In a multinomial experiment, the frequency of an outcome refers to the number of times that an outcome occurs. For example, suppose we toss a single die. This experiment has 6 possible outcomes; the die could land on 1, 2, 3, 4, 5, or 6. Suppose that we roll the die four times and observe the following outcomes: we roll a 1, a 3, and a two 5's? The frequency for each outcome is shown in the table below.

Outcome Frequency
1 1
2 0
3 1
4 0
5 2
6 0

### What is the multinomial probability?

A multinomial probability refers to the probability of obtaining a specified frequency in a multinomial experiment. For example, suppose we toss a single die four times. We might ask: What is the probability that we roll a 1, a 3, and a two 5's?

The probability of getting this particular result would be very small: 0.00154. The easiest way to compute a multinomial probability is to use the Multinomial Calculator. To see how to compute multinomial probabilities by hand, go to Stat Trek's tutorial on the multinomial distribution.

### What is the relation between a multinomial and a binomial experiment?

A binomial experiment is actually a special case of a multinomial experiment. The binomial experiment is a multinomial experiment, in which each trial can have only two possible outcomes. The flip of a coin is a good example of a binomial experiment, since a coin flip can have only two possible outcomes - heads or tails. To learn more about binomial experiments, go to Stat Trek's tutorial on the binomial distribution.

## Multinomial Distribution: Sample Problems

1. 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.

1. 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.