Calculating Probability for Forming a Committee with Exactly 3 Boys from a Group of 15

In this article, we will walk through the process of calculating the probability that a committee of 5 students selected from a group consisting of 10 boys and 5 girls contains exactly 3 boys. We will use the concept of combinations to solve this problem and ensure the steps resonate well with Google's search engine optimization (SEO) standards.

Problem Statement

A committee of 5 students is selected at random from a group of 15 students (10 boys and 5 girls). What is the probability that the committee has exactly 3 boys?

Step-by-Step Solution

Step 1: Calculate the Total Number of Ways to Select a Committee

First, we need to determine the total number of ways to select 5 students from a group of 15. We use the combination formula to do this:

[ binom{15}{5} frac{15!}{5! (15 - 5)!} frac{15!}{5!10!} 3003 ]

Step 2: Calculate the Number of Ways to Select 3 Boys and 2 Girls

We need to find the number of ways to select 3 boys out of 10 and 2 girls out of 5. We use the combination formula for this calculation as well:

Ways to choose 3 boys from 10:

[ binom{10}{3} frac{10!}{3! (10 - 3)!} frac{10!}{3!7!} 120 ]

Ways to choose 2 girls from 5:

[ binom{5}{2} frac{5!}{2! (5 - 2)!} frac{5!}{2!3!} 10 ]

Step 3: Calculate the Total Number of Ways to Select 3 Boys and 2 Girls

Multiplying the number of ways to choose the boys and the girls gives us the total number of favorable outcomes:

[ binom{10}{3} times binom{5}{2} 120 times 10 1200 ]

Step 4: Calculate the Probability

Now, we can find the probability of selecting a committee with exactly 3 boys:

[ P(text{exactly 3 boys}) frac{binom{10}{3} times binom{5}{2}}{binom{15}{5}} frac{1200}{3003} approx 0.399 ]

Thus, the probability that the committee has exactly 3 boys is:

[ boxed{frac{400}{1001} text{ or approximately } 0.399} ]

Related Keywords and Articles

This article is relevant for those interested in probability, specifically the application of combinations to solve real-world problems. Understanding this concept can aid in solving similar probability questions involving committee selection or random selections from larger groups.

Conclusion

By following the steps outlined in this article, one can accurately calculate the probability of forming a committee with a specific number of boys (or girls) from a larger group. This method is not only useful for academic purposes but also practical for real-life scenarios requiring the application of probability and combinatorics.