I am having trouble figuring out the answer to the following counting rules problems. No matter what I try, my answer does not match the answer key from my course (and there are no solutions provided, just the final probability). I know that I have to use the combination rule (NCn) as order does not matter in these questions but I just can't seem to crack them. Any help will be appreciated!
These questions come from a document that the professor generated years ago as general practice problems for the course.
Q1. Eight male and eight female Psychology 020 students are going to be assigned to samples of sizes 2, 6, and 8 to participate as subjects in different research experiments. If assignment to the samples is random, what is the probability that no more than 2 female will end up in the sample of size 6?
So N = 16, Males = 8, Females = 8
My initial logic to solve this question would be choosing the number of males and females to make up to group of 6 divided by the total number of ways to pick six people from the group of 16. For this question stating no more than 2 females, this would mean that there are either:
a. 6 males, 0 females b. 5 males, 1 female 6. 4 males, 2 females
(8C6)(8C0)/(16C6) + (8C5)(8C1)/(16C6) + (8C4)(8C2)/(16C6) = .00349 + .05594 + .2447 = .3041
The answer I am given by the professor is a final probability of having no more than 2 females in the group of 6 is .0594.
I am not sure then what logic to follow.
Q2. Twenty people (12 females and 8 males) are applying for jobs at a company that needs to fill 10 Management positions, 6 Sales positions, and 4 Secretarial positions. Eight of the 12 female applicants have a B.A. degree; 5 of the 8 male applicants have a B.A. degree; the remaining applicants have no university degree. If the company hires all 20 of the applicants but then assigns them to the 3 different job categories at random…
a. What is the probability that at least 5 of the Sales positions are filled by people with B.A. degrees?
b. In how many ways can the applicants be assigned such that equal numbers of males and females with B.A.s end up in Management positions and all the Sales positions are filled by females?