Problem 14 - Algebra for College Students

Question

A committee consists of 1 Democrat, 5 Republicans, and 6 independents. If one person is randomly selected from the committee to be the chairperson, then what is the probability that a) the person is a Democrat? b) the person is either a Democrat or a Republican? c) the person is not a Republican?

Step-by-Step Solution

Answer

a) \(\frac{1}{12}\) b) \(\frac{1}{2}\) c) \(\frac{7}{12}\)

Step 1: Identify the Total Number of Committee Members

Count the total number of committee members by adding the number of Democrats, Republicans, and Independents: Number of Democrats = 1 Number of Republicans = 5 Number of Independents = 6 Total number of committee members = 1 + 5 + 6 = 12

Step 2: Calculate the Probability of Selecting a Democrat (Part a)

The probability of selecting a Democrat is calculated by dividing the number of Democrats by the total number of committee members: Number of Democrats = 1 Total number of committee members = 12 Probability of selecting a Democrat = \ \ \( P(D) = \frac{1}{12} \)

Step 3: Calculate the Probability of Selecting a Democrat or a Republican (Part b)

The probability of selecting either a Democrat or a Republican is calculated by adding the number of Democrats and the number of Republicans, then dividing by the total number of committee members: Number of Democrats = 1 Number of Republicans = 5 Total number of Democrats and Republicans = 1 + 5 = 6 Probability of selecting a Democrat or a Republican = \ \ \( P(D \cup R) = \frac{6}{12} = \frac{1}{2} \)

Step 4: Calculate the Probability of Not Selecting a Republican (Part c)

The probability of not selecting a Republican is calculated by subtracting the number of Republicans from the total number of committee members and then dividing by the total number of committee members: Number of Republicans = 5 Total number of committee members = 12 Number of non-Republicans = 12 - 5 = 7 Probability of not selecting a Republican = \ \ \( P(eg R) = \frac{7}{12} \)

Key Concepts

Calculating Probability

When dealing with probability in statistics, one of the most fundamental steps is calculating the probability of an event. To do this, you need to identify the total number of possible outcomes and the number of favorable outcomes.

In this exercise, you needed to find the probability of selecting a Democrat, a Democrat or a Republican, and not selecting a Republican. Here are the steps:

• First, identify the total number of possible outcomes. In this case, it's the total number of committee members.
• Next, identify the number of favorable outcomes for each scenario (e.g., the number of Democrats, the combined number of Democrats and Republicans, and the number of non-Republicans).
• Finally, divide the number of favorable outcomes by the total number of outcomes.

For example, the probability of selecting a Democrat is calculated as follows:

\( P(D) = \frac{1}{12} \)

This formula states that the number of outcomes favorable to selecting a Democrat is 1 and the total number of committee members is 12.

Complementary Events

Complementary events are pairs of events that together make up the entire sample space.

For instance, in this problem, selecting a Republican or not selecting a Republican are complementary events. Every possible outcome in the sample space is either selecting a Republican or not selecting a Republican.

To understand complementary events, consider these points:

• The sum of the probabilities of complementary events equals 1.
• If you know the probability of an event occurring, you can find the probability of the complementary event by subtracting the probability of the event from 1.

In the given exercise, the probability of not selecting a Republican is calculated by taking the total number of members and subtracting the number of Republicans, and then dividing by the total number of members.

\( P(eg R) = \frac{7}{12} \)

It's vital to understand that complementary events, when added together, cover all possible outcomes without any overlap.

Probability Rules

There are several key rules in probability that help simplify calculations and ensure accurate results. Here are some of the most essential ones used in this example:

• Rule of Total Probability: The total probability of all possible outcomes in a sample space is always 1.
• Addition Rule: To find the probability of either of two mutually exclusive events occurring, you add their individual probabilities.
• Complement Rule: The probability of an event occurring is 1 minus the probability of the event not occurring.

Applying these rules helps solve complex probability problems more easily. For instance, in part (b) of the exercise, the probability of selecting a Democrat or a Republican was calculated using the addition rule:

\( P(D \cup R) = \frac{6}{12} = \frac{1}{2} \)

Understanding these foundational rules allows you to efficiently and correctly calculate the probability of various events, such as finding the probability of selecting different types of members from a committee.