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Chapter 11: Problem 14

A parent-teacher committee consisting of four people is to be selected from fifteen parents and five teachers. Find the probability of selecting two parents and two teachers.

Short Answer

Expert verified
The probability is approximately 1.083 (or 108.3%), indicating that there's something wrong with our calculations. In step 2, a mistake has been made while counting the combinations. The correct calculation would yield \( C(15, 2) \times C(5, 2) = 45 \times 10 = 450 \), not 5250. Therefore, the desired probability is \( \frac{450}{4845} \approx 0.093 \) or 9.3%.

Step by step solution

01

Calculate Total Combinations

We need to find the total number of ways to select a committee of 4 people from total 20 people (15 parents and 5 teachers). The combination formula is used here: \( C(n, k) = \frac{n!}{k!(n-k)!} \). Substituting n=20 and k=4 gives \( C(20, 4) = \frac{20!}{4!(20-4)!} \). After calculation, the total combinations are 4845.
02

Calculate Desired Combinations

Now we need to find the number of ways to select exactly 2 parents from 15 and 2 teachers from 5. This will be the product of combination of selecting 2 parents from 15 and combination of selecting 2 teachers from 5. So we calculate \( C(15, 2) \times C(5, 2) = \frac{15!}{2!(15-2)!} \times \frac{5!}{2!(5-2)!} \), which gives 5250.
03

Calculate Probability

Finally, the probability that we seek can be found by dividing the number of desired combinations by total combinations. So, the desired probability is \( \frac{5250}{4845} \)

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