Problem 39 - Algebra for College Students

Question

In a recent election there were 1000 eligible voters. They were asked to vote on two issues, \(A\) and \(B\). The results were as follows: 300 people voted for \(A, 400\) people voted for \(B\), and 175 voted for both \(A\) and \(B\). If one person is chosen at random from the 1000 eligible voters, find the probability that the person voted for A or B. \(0.525\)

Step-by-Step Solution

Answer

The probability is 0.525.

Step 1: Define the Problem

We need to find the probability that a randomly chosen voter voted for either Issue A or Issue B. This involves using the principle of Inclusion-Exclusion to calculate the number of voters who voted for at least one of the issues, then calculate the probability.

Step 2: Apply Inclusion-Exclusion Principle

The formula for the union of two sets is \( |A \cup B| = |A| + |B| - |A \cap B| \). In this problem, \(|A| = 300\), \(|B| = 400\), and \(|A \cap B| = 175\). Calculating the union, we find:\[|A \cup B| = 300 + 400 - 175 = 525\]

Step 3: Calculate Probability

The probability that a randomly chosen voter voted for either Issue A or Issue B is calculated by dividing the number of voters who voted for at least one of the issues by the total number of eligible voters. The formula is:\[P(A \cup B) = \frac{|A \cup B|}{\text{Total eligible voters}}\]Plugging in the numbers, we get:\[P(A \cup B) = \frac{525}{1000} = 0.525\]

Key Concepts

Inclusion-Exclusion Principle

In probability and set theory, the Inclusion-Exclusion Principle is a powerful tool used to calculate the size of the union of multiple sets. When dealing with two sets, it helps account for those elements that appear in both sets, which could otherwise be counted twice if simply adding the sizes of the sets. In our election example, we are interested in the set of voters who voted for either Issue A or Issue B.

To apply the Inclusion-Exclusion Principle, we use the formula:

• \[ |A \cup B| = |A| + |B| - |A \cap B| \]

This formula ensures we do not double-count the 175 people who voted for both issues. By subtracting \(|A \cap B|\), we adjust for this overlap, providing an accurate count of voters supporting at least one issue.

This principle is especially useful in elections, where overlap in preferences is common, and helps us understand the true extent of voter engagement across issues.

Union of Sets

The concept of a union in set theory refers to combining all elements from two or more sets. In the context of our voting example, the union of the sets of voters who supported Issues A and B is the collection of all voters who voted for at least one of the issues.

The union, denoted by \(A \cup B\), includes:

• Those who voted only for Issue A (300 - 175)
• Those who voted only for Issue B (400 - 175)
• Those who voted for both issues (175)

By considering these components, we determine \(|A \cup B| = 525\) voters voted for either issue. This computation is essential for calculating probabilities as it directly relates to the share of participants involved, reflecting the broad support or preference for the issues presented.

Voting Statistics

Understanding voting statistics involves more than just counting votes; it involves categorizing voter behavior and preferences. In any election scenario, knowing how many people voted for single or multiple issues informs various analyses.

In our example:

• 300 people voted for Issue A
• 400 people voted for Issue B
• 175 people voted for both A and B

These figures help understand the engagement level of voters and the overlap between issues, which can be crucial for strategizing future campaigns or understanding voter interests. Calculating how many voted for at least one of the issues via the Inclusion-Exclusion Principle gives an accurate picture of overall voter involvement (525, in this case).

Voting statistics also play a role in determining possible correlations between issues and predicting future voting behavior.

Probability Calculation

Probability calculation allows us to quantify the likelihood of a specific event happening in a well-defined sample space. In our election scenario, we calculate the probability of a random voter having voted for at least one of the issues, which is a straightforward application of set theory to real-life contexts.

The total count of eligible voters forms the basis of our probability calculation. With 1000 eligible voters and 525 having voted for either Issue A or B, the probability is given by the formula:

• \[ P(A \cup B) = \frac{|A \cup B|}{\text{Total eligible voters}} = \frac{525}{1000} = 0.525 \]

This calculation gives 0.525, meaning there is a 52.5% chance that a randomly selected voter from this group supported at least one issue. Finding such probabilities is vital for understanding and interpreting voting trends and outcomes, making it easier for candidates and policymakers to gauge public interest and plan accordingly.