Multinomial Distribution The binomial distribution applies only to cases involving two types of outcomes, whereas the multinomial distribution involves more than two categories.

Suppose we have three types of mutually exclusive outcomes denoted by A, B, and C. Let P1A2 = p1, P1B2 = p2, and P1C2 = p3. In n independent trials, the probability of x1 outcomes of type A, x2 outcomes of type B, and x3 outcomes of type C is given by

Data Set 8 “IQ and Lead” in Appendix B includes 78 subjects from a low lead exposure group, 22 subjects from a medium lead exposure group, and 21 subjects from a high lead exposure group. Find the probability of randomly selecting 10 subjects for a follow-up study and getting 5 from the low lead group, 2 from the medium lead group, and 3 from the high lead group.
Assume that the selections are made with replacement. Can we use the above expression for finding the probability if the sampling is done without replacement?

n! PPP (x1)!(x2)!(x3)!