What is probability of 12 male and 13 female 25 volunteers vol pool of 30 15 male 15 female

Question

A researcher wants to conduct a genetic study using 25 randomly-selected volunteers. He has a volunteer pool of 30, composed of 15 males and 15 females. What is the probability of 12 male and 13 female volunteers OR 13 male and 12 female volunteers being selected at random for the study?

Answer 1

Volunteer pool of 30: 15M+15F.

Selecting 25 randomly from the pool is limited to the following outcomes:

(1) 15M+10F

(2) 14M+11F

(3) 13M+12F

(4) 12M+13F

(5) 11M+14F

(6) 10M+15F

Out of these 6 possibilities (3) and (4) meet the requirement for 13 males and 12 females, or 12 males and 13 females, the probability is 2/6=1/3.

[(3) gives us 105 ways of selecting 13 males out of 15, and 455 ways of selecting 12 females out of 15. Vice versa for (4).]

Comments

Total ways of selecting 12 boys from the pool of 15 boys = 15C12 =455

Total ways of selecting 13 boys from the pool of 15 boys = 15C13 = 105

Total ways of selecting 12 girls from the pool of 15 girls= 15C12 =455

Total ways of selecting 13 girls from the pool of 15 girls= 15C13 = 105

So total ways = 455*105 + 105 *455 = 95550

Therefore, total probability = 95550/ 30C25 = 0.67

I am not sure, I am very bad at probability.

Answer 2

Total ways of selecting 12 boys from the pool of 15 boys = 15C12 =455

Total ways of selecting 13 boys from the pool of 15 boys = 15C13 = 105

Total ways of selecting 12 girls from the pool of 15 girls= 15C12 =455

Total ways of selecting 13 girls from the pool of 15 girls= 15C13 = 105

So total ways = 455*105 + 105 *455 = 95550

Therefore, total probability = 95550/ 30C25 = 0.67

Comments

I think the difference between our answers is because the two ways of selecting males and females are not mutually exclusive, because there will be a set of males and a set of females in common between the 13 males/12 females group, and the 12 males/13 females group, and this means you cannot simply add the individual probabilities. There's an overlap which reduces the overall probability.