Probability Of Selecting Two Democrats And Two Republicans Committee

Jul 12, 2025 by ADMIN 69 views

Probability of Selecting a Committee with Two Democrats and Two Republicans

Introduction

In the realm of probability, we often encounter scenarios where we need to determine the likelihood of specific outcomes when selecting a subset from a larger group. This article delves into such a problem, focusing on the probability of selecting a committee with a specific composition from a city council. Specifically, we will explore the probability of forming a four-person committee consisting of two Democrats and two Republicans from a city council comprising seven Democrats and eight Republicans. This problem exemplifies the application of combinatorial principles and probability calculations in real-world scenarios. Understanding these concepts is crucial for various fields, including statistics, data analysis, and decision-making.

Problem Statement

Consider a city council consisting of seven Democrats and eight Republicans. Our task is to determine the probability of selecting a committee of four people that comprises exactly two Democrats and two Republicans. This problem requires us to apply our understanding of combinations and probability. We need to figure out the total number of ways to form a four-person committee and the number of ways to form a committee with the specified composition (two Democrats and two Republicans). The ratio of these two numbers will give us the desired probability. To solve this, we'll use the combination formula, which helps us calculate the number of ways to choose a subset of items from a larger set without considering the order of selection. This is perfect for our problem since the order in which the committee members are chosen doesn't matter.

Understanding Combinations

Before we dive into the calculations, let's clarify the concept of combinations. In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter. The number of combinations of n items taken k at a time is denoted as C(n, k) or "n choose k," and it is calculated using the formula:

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

Where "!" denotes the factorial function (e.g., 5! = 5 * 4 * 3 * 2 * 1). This formula tells us how many different ways we can choose a group of k items from a larger group of n items, where the order of selection is not important. In our problem, we'll use this formula to calculate the number of ways to choose Democrats and Republicans for the committee. For instance, we'll need to know how many ways we can choose two Democrats from the seven available and how many ways we can choose two Republicans from the eight available. These calculations will be crucial in determining the probability of forming the desired committee.

Calculating the Total Number of Ways to Form a Committee

To begin, we need to calculate the total number of ways to form a four-person committee from the entire city council. The city council consists of 7 Democrats and 8 Republicans, making a total of 15 members. We want to choose 4 people out of these 15, so we use the combination formula:

Total ways = C(15, 4) = 15! / (4! * 11!) = (15 * 14 * 13 * 12) / (4 * 3 * 2 * 1) = 1365

This means there are 1365 different ways to select a committee of four people from the 15 members of the city council. This number represents the total possible outcomes when forming a committee. Now, we need to figure out how many of these committees have the specific composition we're interested in: two Democrats and two Republicans. This will involve calculating the number of ways to choose two Democrats from the available Democrats and the number of ways to choose two Republicans from the available Republicans. These calculations will be combined to find the total number of committees with the desired composition.

Calculating the Number of Ways to Select Two Democrats

Next, we calculate the number of ways to select two Democrats from the seven Democrats available. Using the combination formula again:

Ways to select 2 Democrats = C(7, 2) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21

This tells us that there are 21 different ways to choose two Democrats from the seven Democrats in the city council. Each of these combinations of Democrats could potentially be part of a committee with the desired composition. Now, we need to perform a similar calculation for the Republicans. We need to figure out how many ways we can choose two Republicans from the eight Republicans in the city council. This calculation will be combined with the number of ways to choose the Democrats to find the total number of committees with two Democrats and two Republicans.

Calculating the Number of Ways to Select Two Republicans

Similarly, we calculate the number of ways to select two Republicans from the eight Republicans available:

Ways to select 2 Republicans = C(8, 2) = 8! / (2! * 6!) = (8 * 7) / (2 * 1) = 28

So, there are 28 different ways to choose two Republicans from the eight Republicans in the city council. Now that we know the number of ways to choose two Democrats and the number of ways to choose two Republicans, we can combine these numbers to find the total number of committees with the desired composition. This will involve multiplying the two numbers together, as each combination of Democrats can be paired with each combination of Republicans to form a unique committee.

Calculating the Number of Committees with Two Democrats and Two Republicans

To find the number of committees with two Democrats and two Republicans, we multiply the number of ways to select two Democrats by the number of ways to select two Republicans:

Number of committees with 2 Democrats and 2 Republicans = 21 * 28 = 588

This means that there are 588 different committees that can be formed with exactly two Democrats and two Republicans. Now that we know this number and the total number of possible committees, we can calculate the probability of selecting a committee with this specific composition. This involves dividing the number of favorable outcomes (committees with two Democrats and two Republicans) by the total number of possible outcomes (all possible committees).

Calculating the Probability

Finally, we calculate the probability of selecting a committee with two Democrats and two Republicans by dividing the number of favorable outcomes (588) by the total number of possible outcomes (1365):

Probability = 588 / 1365 = 0.4308 (approximately)

Therefore, the probability of selecting a committee of four people with two Democrats and two Republicans is approximately 0.4308, or 43.08%. This means that if we were to randomly select committees of four people from the city council, we would expect about 43% of those committees to have two Democrats and two Republicans. This result highlights the likelihood of achieving this specific composition given the makeup of the city council.

Conclusion

In conclusion, the probability of selecting a committee of four people with two Democrats and two Republicans from a city council of seven Democrats and eight Republicans is approximately 43.08%. This problem demonstrates the application of combinatorial principles and probability calculations. By understanding how to calculate combinations and probabilities, we can analyze and predict the likelihood of various outcomes in real-world scenarios. This skill is valuable in many fields, from statistics and data analysis to decision-making and risk assessment. The ability to quantify the likelihood of events allows for more informed decisions and a deeper understanding of the world around us.