Probability Of An All-Boys Committee: A Step-by-Step Guide

Let's dive into this probability problem! We're tasked with figuring out the likelihood of selecting a committee consisting entirely of boys from a larger group of boys and girls. This involves understanding combinations and how to calculate probabilities in scenarios like this. So, let's break it down step by step.

Understanding the Problem

We have a group of eight boys and six girls, making a total of 14 students. We need to form a committee of four, and we want to know the probability that all four members are boys. This means we'll be dealing with combinations, as the order in which the students are chosen doesn't matter. Combinations are a fundamental concept in probability and combinatorics, and mastering them is crucial for solving a wide range of problems. In this particular scenario, we need to figure out how many ways we can select four boys from the group of eight, and then compare that to the total number of ways we can select any four students from the entire group.

Calculating Total Possible Committees

First, we need to determine the total number of possible committees of four that can be formed from the 14 students (8 boys and 6 girls). This is a combination problem, represented as "14 choose 4", or C(14, 4). The formula for combinations is:

C(n, k) = n! / (k!(n-k)!)

Where:

• n is the total number of items
• k is the number of items to choose
• ! denotes the factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1)

So, for our problem, we have:

C(14, 4) = 14! / (4!(14-4)!) = 14! / (4!10!)

Let's break this down:

• 14! = 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
• 4! = 4 × 3 × 2 × 1 = 24
• 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

We can simplify the expression by canceling out the 10! in the numerator and denominator:

C(14, 4) = (14 × 13 × 12 × 11) / (4 × 3 × 2 × 1)

Now, let's do some more simplification:

C(14, 4) = (14 × 13 × 12 × 11) / 24

C(14, 4) = (14 × 13 × 11) / 2 (since 12 / 24 = 1/2)

C(14, 4) = 7 × 13 × 11 (since 14/2 = 7)

C(14, 4) = 1001

So, there are 1001 possible committees of four that can be formed from the 14 students. This number represents the total possible outcomes when we are forming our committee, and it's the denominator in our probability calculation. This initial calculation is crucial, as it sets the stage for determining the likelihood of our specific event: selecting an all-boys committee. Understanding combinations and factorials is essential not just for this problem but for many areas of mathematics and statistics.

Calculating the Number of All-Boys Committees

Next, we need to figure out how many committees can be formed with only boys. We have eight boys, and we want to choose four of them. This is another combination problem, represented as "8 choose 4", or C(8, 4). Using the same formula:

C(8, 4) = 8! / (4!(8-4)!) = 8! / (4!4!)

Let's break this down:

• 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
• 4! = 4 × 3 × 2 × 1 = 24

So, we have:

C(8, 4) = (8 × 7 × 6 × 5) / (4 × 3 × 2 × 1)

Now, let's simplify:

C(8, 4) = (8 × 7 × 6 × 5) / 24

C(8, 4) = (2 × 7 × 5) (since 8/24 simplifies to 1/3, and 6/3 = 2)

C(8, 4) = 70

Therefore, there are 70 possible committees consisting of only boys. This number represents the favorable outcomes for our specific event. The ability to calculate combinations like this is a cornerstone of probability theory, and it allows us to quantify the likelihood of different events occurring. In this case, we've determined the number of ways to form an all-boys committee, which is a critical piece of information for calculating the final probability.

Calculating the Probability

Now that we know the total number of possible committees (1001) and the number of all-boys committees (70), we can calculate the probability of selecting an all-boys committee. Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.

Probability (All-Boys Committee) = (Number of All-Boys Committees) / (Total Number of Committees)

Probability (All-Boys Committee) = 70 / 1001

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 7:

Probability (All-Boys Committee) = (70 ÷ 7) / (1001 ÷ 7)

Probability (All-Boys Committee) = 10 / 143

So, the probability of selecting a committee consisting of all boys is 10/143. This result gives us a precise measure of the likelihood of this specific event occurring. Understanding how to calculate probabilities like this is essential for making informed decisions in various fields, from science and engineering to finance and everyday life.

Conclusion

Therefore, the probability that the committee consists of all boys is 10/143. This problem demonstrates the importance of understanding combinations and how they are used to calculate probabilities. By breaking down the problem into smaller steps, we were able to calculate the total possible outcomes and the favorable outcomes, ultimately leading us to the final probability. Mastering these concepts will not only help you solve similar problems but also enhance your overall understanding of mathematical reasoning and problem-solving.

For further exploration of probability and combinations, you can visit Khan Academy's Probability and Combinations Section.