Probability Of Forming An All-Boys Committee Detailed Solution
Calculating probability is a fundamental concept in mathematics and statistics, with applications spanning across various fields, from simple games of chance to complex real-world scenarios. In this article, we will delve into a specific probability problem involving committee selection. We'll explore the step-by-step solution to determine the probability of forming an all-boys committee from a group of boys and girls. Understanding the underlying principles of combinations and probability will empower you to tackle similar problems with confidence. This article aims to provide a comprehensive explanation and insightful analysis for both students and enthusiasts interested in probability calculations. We'll break down the problem into manageable parts, ensuring a clear understanding of each step involved.
Problem Statement: Forming an All-Boys Committee
The probability problem presented to us involves selecting a four-person committee from a group of eight boys and six girls. The question we aim to answer is: If students are chosen at random, what is the probability that the committee consists entirely of boys? This is a classic combinatorial probability problem that requires us to understand the principles of combinations and how they apply to probability calculations. To solve this, we must first determine the total number of ways to form a four-person committee from the entire group of students. Then, we'll calculate the number of ways to form a committee consisting only of boys. Finally, we'll divide the number of favorable outcomes (all-boys committee) by the total number of possible outcomes (any four-person committee) to find the desired probability. This approach will allow us to systematically arrive at the correct answer.
Step 1: Calculating the Total Number of Possible Committees
To begin solving the committee selection problem, our initial step is to determine the total number of ways to form a four-person committee from the entire group of students. We have a total of eight boys and six girls, which gives us a combined pool of 14 students. We need to choose four students out of these 14 to form our committee. This is a combination problem because the order in which the students are chosen does not matter. We use the combination formula, which is denoted as C(n, k) or "n choose k," where n is the total number of items, and k is the number of items to choose. The formula is given by:
C(n, k) = n! / (k!(n-k)!)
Where "!" denotes the factorial function. Applying this to our problem, we want to find C(14, 4), which is the number of ways to choose four students from 14. Plugging the values into the formula, we get:
C(14, 4) = 14! / (4!(14-4)!) = 14! / (4!10!)
Calculating the factorials and simplifying, we have:
C(14, 4) = (14 × 13 × 12 × 11) / (4 × 3 × 2 × 1) = 1001
Thus, there are 1001 different ways to form a four-person committee from the group of 14 students. This number represents the total possible outcomes when selecting a committee of four from the combined group of boys and girls. Understanding this total is crucial as it forms the denominator in our probability calculation. We now move on to calculating the number of committees that consist only of boys.
Step 2: Determining the Number of All-Boys Committees
Having calculated the total number of possible committees, our next crucial step in solving this probability problem is to determine the number of committees that consist entirely of boys. Since we have eight boys in total, we need to figure out how many ways we can choose four boys from this group of eight. This is another combination problem, where we are choosing four items (boys) from a set of eight. We again use the combination formula, C(n, k) = n! / (k!(n-k)!), but this time with n = 8 and k = 4. So we want to calculate C(8, 4), which represents the number of ways to select four boys from a group of eight. Plugging these values into the formula, we get:
C(8, 4) = 8! / (4!(8-4)!) = 8! / (4!4!)
Calculating the factorials and simplifying, we have:
C(8, 4) = (8 × 7 × 6 × 5) / (4 × 3 × 2 × 1) = 70
This result tells us that there are 70 different ways to form a committee consisting of only boys. This is a key piece of information because it represents the number of favorable outcomes—the outcomes we are interested in—when calculating the probability of forming an all-boys committee. Understanding the number of all-boys committees allows us to compare it to the total number of possible committees, which we calculated in the previous step. With both the number of favorable outcomes and the total number of possible outcomes, we are now ready to calculate the probability. The next step will involve dividing the number of all-boys committees by the total number of possible committees to arrive at our final answer.
Step 3: Calculating the Probability of an All-Boys Committee
With the total possible committees and the number of all-boys committees calculated, we can now proceed to the final step in solving our probability problem: determining the probability of forming an all-boys committee. Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. In our case, the favorable outcomes are the committees consisting entirely of boys, and the total possible outcomes are all the possible four-person committees that can be formed from the group of 14 students. We found that there are 70 ways to form an all-boys committee (C(8, 4) = 70) and 1001 total possible committees (C(14, 4) = 1001). Therefore, the probability of forming an all-boys committee is:
Probability = (Number of all-boys committees) / (Total number of possible committees) = 70 / 1001
This fraction can be simplified by finding the greatest common divisor (GCD) of 70 and 1001. Both numbers are divisible by 7, so we can simplify the fraction as follows:
70 / 1001 = (70 ÷ 7) / (1001 ÷ 7) = 10 / 143
Thus, the probability that the committee consists of all boys is 10/143. This means that out of every 143 randomly selected four-person committees, we would expect about 10 of them to consist entirely of boys. This calculation completes our solution to the problem. We have successfully navigated the steps from calculating combinations to determining probability, providing a clear and concise answer to the initial question. The final answer encapsulates the likelihood of a specific event occurring, given a set of conditions, and demonstrates the power of combinatorial principles in probability calculations.
Conclusion: Key Takeaways on Committee Selection Probability
In conclusion, the probability analysis of forming a four-person committee consisting entirely of boys from a group of eight boys and six girls highlights several key concepts in combinatorics and probability. We've systematically worked through the problem, emphasizing the importance of understanding combinations and their role in calculating probabilities. First, we determined the total number of possible committees by calculating C(14, 4), which represents the number of ways to choose four students from a group of 14. This calculation gave us a total of 1001 possible committees. Next, we focused on finding the number of favorable outcomes, which in this case was the number of committees consisting only of boys. We calculated C(8, 4), representing the number of ways to choose four boys from a group of eight, which resulted in 70 all-boys committees. Finally, we calculated the probability by dividing the number of favorable outcomes by the total number of possible outcomes, giving us a probability of 70/1001, which simplifies to 10/143. This final result provides a clear and concise answer to the original question: the probability that a randomly selected four-person committee consists entirely of boys is 10/143. This problem illustrates the application of combinatorial principles in probability calculations and underscores the importance of breaking down complex problems into manageable steps. Understanding these concepts is crucial for tackling similar probability challenges in various mathematical and real-world contexts. By mastering these techniques, one can confidently approach probability problems involving selections, arrangements, and other combinatorial scenarios, enhancing problem-solving skills and analytical thinking.
Problem
A four-person committee is chosen from a group of eight boys and six girls. If students are chosen at random, what is the probability that the committee consists of all boys?
Answer
