Question

Managers of an office selected a committee of five employees to participate in discussions on how to revise the flow of work in the office. The office has 25 employees, of which 10 are men. If the commit- tee is selected at random from the 25 employees, is it likely to have 4 men? (This question and the follow- ing introduce the calculations that define a hypergeo- metric random variable.)

(a) Explain why it would not be appropriate to use a binomial model for the number of men on a randomly selected committee of five.

(b) The binomial coefficient nCx gives the number of ways of picking a subset of x items out of n. Using this fact, how many ways are there to pick a committee of 5 from among all of the office employees?

(c) How many ways are there to pick a subset of 4 male employees from the 10 men? To pick 1 woman from the 15 female employees?

(d) Use your answers to (b) and (c) to find the probability of a randomly selected committee with 4 male and 1 female members.

(e) Would you get a smaller or larger probability by incorrectly using a binomial model? Explain. (You don’t have to do the binomial calculation to answer this question.)

Answer 1

We have given that N=25,M=10,n=5

Let X: Number of Male selected in committee.

then X~HYPERGEOM(25,10,5)

(a) Here random sample of men is selected using without replacement so probability of selecting men changes from trial to trial ,thats why it would not be appropriate to use a binomial model for the number of men on a randomly selected committee of five.

(b) The number of ways to pick a committee of 5 from among all of the office employees:

(c)

The number of ways to pick a subset of 4 male employees from the 10 men

The number of ways to pick 1 woman from the 15 female employees

(d) the probability of a randomly selected committee with 4 male and 1 female members is

(e) We get smaller probability by incorrectly using a binomial model.