# The Multinomial Distribution

## Multinomial Experiment

A multinomial experiment is an experiment that has the following properties:

• The experiment consists of repeated trials
• Each trial has a discrete number of possible outcomes
• The probability that a particular outcome will occur on any given trial is constant
• Each trial is independent; the outcome of one trial does not affect the outcome of another

Suppose we toss a dice three times and record the number that the dice lands on each time. This is an example of a multinomial experiment because:

• The experiment consists of 3 repeated tosses
• Each toss has a discrete number of possible outcomes (it can land on one of six numbers each time)
• The probability of any outcome is constant – the probabilities do not change from one toss to the next
• Each toss is independent; the outcome of one toss does not affect the outcome of another

Note: A binomial experiment is a special case of a multinomial experiment in which each trial has exactly two possible outcomes.

## Multinomial Distribution

A multinomial distribution tells us the probability of getting a certain set of outcomes in a multinomial experiment.

Suppose we have a multinomial experiment with:

• n trials
• possible outcomes in each trial (O1, O2, … , Ok)
• each possible outcome can occur with probability p1, p2, … , pk

Then the probability (P) that Ooccurs ntimes, Ooccurs ntimes, … , and Ooccurs ntimes is:

P = [n! / (n1! * n1! * … * nk! )] * (p1n1 * p2n2 * … * pknk)

Let’s walk through some examples to gain a better understanding of the multinomial distribution.

## Multinomial Distribution Example Problems

### Example 1

We have 10 marbles in a bag – 2 green marbles, 3 blue marbles, and 5 yellow marbles. We randomly select 3 marbles from the bag, with replacement. What is the probability of selecting exactly 1 green, 1 blue, and 1 yellow marble?

Step 1: Identify the total number of trials (n), the number of times we’re interested in selecting each possible color (n1, n2, … , nk), and the probability of choosing each color on a given trial (p1, p2, … , pk).

= 3 trials

ni = we’re interested in the probability of selecting a green marble once, a blue marble once, and a yellow marble once, so n1 = 1, n2 = 1, n3 = 1.

pi = On any given trial, the probability of selecting a green marble is 2/10 = 0.2, the probability of selecting a blue marble is 3/10 = 0.3, and the probability of selecting a yellow marble is 5/10 = 0.5. Thus, p1 = 0.2, p2 = 0.3, p3 = 0.5.

Step 2: Plug these numbers into the multinomial formula or a multinomial calculator

Using the formula:

P = [n! / (n1! * n1! * … * nk! )] * (p1n1 * p2n2 * … * pknk)

P = [3! / (1! * 1! * 1! )] * (0.21 * 0.31  * 0.51)

P = 0.18

If we randomly select three marbles from the bag with replacement, the probability that we select exactly one green marble, one blue marble, and one yellow marble is 0.18

Using the calculator:

Plug the following numbers into the Multinomial Distribution Calculator:

This matches the result that we got using the formula.

### Example 2

In a certain city, 20% of voters prefer candidate A, 40% prefer candidate B, and 40% prefer candidate C. If you randomly sample 10 voters, what is the probability that 2 will prefer candidate A, 4 will prefer candidate B, and 4 will prefer candidate C?

Step 1: Identify the total number of trials (n), the number of voters we’re interested in preferring each possible candidate (n1, n2, … , nk), and the probability of selecting a voter who prefers a certain candidate on a given trial (p1, p2, … , pk).

= 10 trials

ni = we’re interested in the probability of selecting 2 voters who prefer candidate A, 4 voters who prefe candidate B, and 4 voters who prefer candidate C, so n1 = 2, n2 = 4, n3 = 4.

pi = On any given trial, the probability of selecting a voter who prefers candidate A is 2/10 = 0.2, the probability of selecting a voter who prefers candidate B is 4/10 = 0.4, and the probability of selecting a voter who prefers candidate C is 4/10 = 0.4. Thus, p1 = 0.2, p2 = 0.4, p3 = 0.4.

Step 2: Plug these numbers into the multinomial formula or a multinomial calculator

Using the formula:

P = [n! / (n1! * n1! * … * nk! )] * (p1n1 * p2n2 * … * pknk)

P = [10! / (2! * 4! * 4! )] * (0.22 * 0.44  * 0.44)

P = 0.08258

If we randomly select three voters, the probability that two of the voters prefer candidate A, four of the voters prefer candidate B, and four of the voters prefer candidate C is 0.08258.

Using the calculator:

Plug the following numbers into the Multinomial Distribution Calculator:  This matches the result that we got using the formula.

### Example 3

We randomly select a card from a deck of cards 4 times without replacement. What is the probability that we select one spade, one club, one heart, and one diamond?

Step 1: Identify the total number of trials (n), the number of times we’re interested in selecting each possible suit (n1, n2, … , nk), and the probability of choosing each suit on a given trial (p1, p2, … , pk).

= 4 trials

ni = we’re interested in the probability of selecting each suit exactly one time, so n1 = 1, n2 = 1, n3 = 1, and , n4 = 1.

pi = On any given trial, the probability of selecting a certain suit is 13/52 = 0.25. So, p1 = 0.25, p2 = 0.25, p3 = 0.25, and p4 = 0.25.

Step 2: Plug these numbers into the multinomial formula or a multinomial calculator

Using the formula:

P = [n! / (n1! * n1! * … * nk! )] * (p1n1 * p2n2 * … * pknk)

P = [4! / (1! * 1! * 1! )] * (0.251 * 0.251  * 0.251  * 0.251)

P = 0.09375

If we randomly select four cards from the deck with replacement, the probability that we select exactly one spade, one club, one heart, and one diamond is 0.09375

Using the calculator:

Plug the following numbers into the Multinomial Distribution Calculator:  This matches the result that we got using the formula.

### Example 4

A certain game consists of spinning a spinner, which can land on three different outcomes: a square, a circle, and a triangle. For any given spin, the probability of landing on a square is 0.7, a circle is 0.2, and a triangle is 0.1. If you randomly spin the spinner 10 times, what is the probability that it lands on a square 7 times, a circle 3 times, and a triangle 0 times?

Step 1: Identify the total number of trials (n), the number of times we’re interested in landing on each possible outcome (n1, n2, … , nk), and the probability of landing on each outcome on a given trial (p1, p2, … , pk).

= 10 trials

ni = we’re interested in the probability of landing on the square 7 times, the circle 3 times, and the triangle 0 times, so n1 = 7, n2 = 3, and n3 = 0.

pi = On any given trial, the probability of landing on a square is 0.7, the probability of landing on a circle is 0.2, and the probability of landing on a triangle is 0.1. So, p1 = 0.7, p2 = 0.2, and p3 = 0.1.

Step 2: Plug these numbers into the multinomial formula or a multinomial calculator

Using the formula:

P = [n! / (n1! * n1! * … * nk! )] * (p1n1 * p2n2 * … * pknk)

P = [10! / (7! * 3! * 0! )] * (0.77 * 0.23  * 0.1)

P = 0.07906

If we spin the spinner ten times, the probability that it lands on a square 7 times, a circle 3 times, and a triangle 0 times is 0.07906

Using the calculator:

Plug the following numbers into the Multinomial Distribution Calculator:  This matches the result that we got using the formula.