Page 143 - Probability Demystified

132 CHAPTER 8 Other Probability Distributions

The Multinomial Distribution

Recall that for a probability experiment to be binomial, two outcomes are
necessary. But if each trial of a probability experiment has more than two
outcomes, a distribution that can be used to describe the experiment is called
a multinomial distribution. In addition, there must be a fixed number of
independent trials, and the probability for each success must remain the same
for each trial.
A short version of the multinomial formula for three outcomes is given
next. If X consists of events E 1 , E 2 , and E 3 , which have corresponding
probabilities of p 1 , p 2 , and p 3 of occurring, where x 1 is the number of times E 1
will occur, x 2 is the number of times E 2 will occur, and x 3 is the number of
times E 3 will occur, then the probability of X is
n!
x 1
p p p x 3 where x þ x þ x ¼ n and p þ p þ p ¼ 1:
x 2
3
2
1
3
2
1
x !x !x ! 1 2 3
3
1
2
EXAMPLE: In a large city, 60% of the workers drive to work, 30% take the
bus, and 10% take the train. If 5 workers are selected at random, find the
probability that 2 will drive, 2 will take the bus, and 1 will take the train.
SOLUTION:
n ¼ 5, x 1 ¼ 2, x 2 ¼ 2, x 3 ¼ 1 and p 1 ¼ 0.6, p 2 ¼ 0.3, and p 3 ¼ 0.1
Hence, the probability that 2 workers will drive, 2 will take the bus, and
one will take the train is
5! 2 2 1
ð0:6Þ ð0:3Þ ð0:1Þ ¼ 30 ð0:36Þð0:09Þð0:1Þ¼ 0:0972
2!2!1!
EXAMPLE: A box contains 5 red balls, 3 blue balls, and 2 white balls. If
4 balls are selected with replacement, find the probability of getting 2 red
balls, one blue ball, and one white ball.

SOLUTION:
5 3 2
n ¼ 4, x 1 ¼ 2, x 2 ¼ 1, x 3 ¼ 1, and p ¼ , p ¼ , and p ¼ : Hence, the
3
2
1
10 10 10
probability of getting 2 red balls, one blue ball, and one white ball is

4! 5 2 3 1 2 1 3 9
¼ 12 ¼ ¼ 0:18
2!1!1! 10 10 10 200 50