Definition 1: Given an experiment with the following characteristics:
- the experiment consists of n independent trials, each with k mutually exclusive outcomes Ei
- for each trial the probability of outcome Ei is pi
Let x1 …, xk be discrete random variables whose values are the number of times outcome Ei occurs in n trials. Then the probability distribution function for x1 …, xk is called the multinomial distribution and is defined as follows:
The case where k = 2 is equivalent to the binomial distribution.
Example 1: Suppose that a bag contains 8 balls: 3 red, 1 green and 4 blue. You reach in the bag pull out a ball at random and then put the ball back in the bag and pull out another ball. This experiment is repeated a total of 10 times. What is the probability that the outcome will result in exactly 4 reds and 6 blues?
The possible outcomes for each trial in this experiment are E1 = a red ball is drawn, E2 = a green ball is drawn and E3 = a blue ball is drawn. Thus p1 = 3/8, p2 = 1/8 and p3 = 4/8, x1 = 4, x2 = 0 and x3 = 6.
Excel Function: While Excel does not provide a function for the multinomial distribution, it does provide the following function:
MULTINOMIAL(x1 …, xk) = n! / (x1!∙…∙xk!)
Thus we could also calculate the answer to Example 9.10 by using the formula
MULTINOMIAL(4,0,6)*(3/8)^4*(1/8)^0*(4/8)^6 = .064888
We can also use a range as the argument of MULTINOMIAL as in Figure 1.
Figure 1 – Multinomial distribution
We can use the following Excel array formula to calculate the same result
Alternatively, we can use the following more complicated non-array formula
Real Statistics Excel Function: The following supplemental function in the Real Statistics Resource Pack can be used to calculate the multinomial distribution.
MULTINOMDIST(R1, R2) = the value of the multinomial pdf where R1 is a range containing the values x1, …, xk and R2 is a range containing the values p1, …, pk
Referring to Figure 1, we have MULTINOMDIST(B3:B5,B6:B8) = 0.064888.