Question 4. Youth Group Officers. A youth group has 8 boys and 6 girls. If a slate of 4 officers is selected, find the probability that exactly
(Hint: sampling is without replacement here)
a) 3 are girls
b) 2 are girls

Step 1 ：The problem is asking for the probability of selecting exactly 3 girls and 1 boy, and exactly 2 girls and 2 boys from a group of 8 boys and 6 girls.

Step 2 ：The total number of ways to select 4 officers from 14 people is given by the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of people and k is the number of people to be selected.

Step 3 ：The number of ways to select exactly 3 girls and 1 boy is given by the product of the number of ways to select 3 girls from 6 and the number of ways to select 1 boy from 8.

Step 4 ：The number of ways to select exactly 2 girls and 2 boys is given by the product of the number of ways to select 2 girls from 6 and the number of ways to select 2 boys from 8.

Step 5 ：The probability is then given by the ratio of the number of favorable outcomes to the total number of outcomes.

Step 6 ：Given that the total number of ways to select 4 officers from 14 people is 1001, the number of ways to select exactly 3 girls and 1 boy is 160, and the number of ways to select exactly 2 girls and 2 boys is 420.

Step 7 ：The probability that exactly 3 officers are girls is \(\frac{160}{1001} = \boxed{0.1598}\)

Step 8 ：The probability that exactly 2 officers are girls is \(\frac{420}{1001} = \boxed{0.4196}\)