Multinomial Distribution Calculator

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.

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
0 heads	0.125
1 head	0.375
2 heads	0.375
3 heads	0.125

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

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.

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.

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

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.

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.