Question

Funding for many schools comes from taxes based on assessed values of local properties. People's homes are assessed higher if they have extra features such as garages and swimming pools. Assessment records in a certain school district indicate that \(37 \%\) of the homes have garages and \(3 \%\) have swimming pools. The Addition Rule might suggest, then, that \(40 \%\) of residences have a garage or a pool. What's wrong with that reasoning?

Short answer

The reasoning is flawed because it ignores potential overlap between homes with both features, leading to an overestimate.

Step-by-step solution

Understand the Addition Rule

The Addition Rule in probability states that for any two events A and B, the probability of either event occurring (A or B) is given by \( P(A\cup B) = P(A) + P(B) - P(A\cap B) \). This means we add the probabilities of each event occurring but subtract the probability of both events occurring together.

Define Events and Probabilities

Consider the two events: Event G for homes having garages and Event P for homes having swimming pools. We are given: \( P(G) = 0.37 \) and \( P(P) = 0.03 \). The simple addition of these probabilities is based on the assumption they are mutually exclusive, which may not hold true.

Apply the Addition Rule Correctly

To accurately apply the Addition Rule, we should also consider \( P(G\cap P) \), the probability of homes having both a garage and a swimming pool. Without this probability, simply adding \(0.37\) and \(0.03\) ignores any overlap where homes may have both features.

Identify the Error in Reasoning

The error occurs because the initial calculation assumes there's no overlap between homes with garages and homes with swimming pools (i.e., \( P(G\cap P) = 0 \)). The Calculation of \( P(G\cup P) = 0.37 + 0.03 = 0.40 \) is incorrect unless \( P(G\cap P) = 0 \), which may not be true. If some homes have both garages and pools, then \( P(G\cap P) > 0 \), and we have overestimated the percentage with the union calculation.

Conclusion of the Error

The reasoning is incorrect because it fails to account for the possible existence of homes with both garages and swimming pools. Without data on \( P(G\cap P) \), the assumption that the two events are mutually exclusive leads to an overestimate of \( P(G\cup P) \).

Key concepts

Addition Rule

The Addition Rule is a fundamental principle in probability that helps us determine the probability of either one or both of two events occurring. This rule assists in estimating combined outcomes in scenarios where two distinct events might happen simultaneously. In mathematical terms, the Addition Rule is expressed as:
\[ P(A\cup B) = P(A) + P(B) - P(A\cap B) \] Where

• \(P(A)\) is the probability of event A occurring,
• \(P(B)\) is the probability of event B occurring, and
• \(P(A \cap B)\) is the probability of both events A and B occurring concurrently.

This equation shows that to find the probability of either event A or event B happening, you start by summing the individual probabilities of each event. However, if there's a chance that both could occur simultaneously, you must subtract the probability of both happening to avoid double counting. This adjustment is crucial because the overlap in occurrences is counted twice if we simply sum \(P(A)\) and \(P(B)\) without modification.

Mutually Exclusive Events

In probability, Mutually Exclusive Events are scenarios where two events cannot occur simultaneously. If we have such events, their probability appears straightforward at first glance because there's no overlap to consider. For example, if an event could be either rolling a 3 or a 5 on a standard dice, they are mutually exclusive - the dice cannot show both 3 and 5 at the same time.
This simplicity translates to the equation \[ P(A\cap B) = 0 \] when A and B are mutually exclusive. Here,

• \(P(A\cap B)\) represents the probability of both events happening together, and
• \(0\) denotes that there is no overlap.

However, the error in assuming events are mutually exclusive without verification can lead to inaccurate probability calculations. Analyzing whether events are truly mutually exclusive is key. If they are not, and we presume they are, calculations that seem correct, like simply adding probabilities, might be flawed.

Overlap in Probability

The concept of Overlap in Probability refers to instances where two events can occur at the same time. This intersection, known in probability as the "joint probability", needs careful consideration. When events overlap, it affects how we calculate the likelihood of either event happening.
The probability of the overlap, \(P(A \cap B)\), is critical to the accuracy of our final probability calculation.

• Without it, assumptions might exclude shared outcomes, leading to overestimation.
• The crux of accurate probability assessment relies on acknowledging and adjusting for this overlap.

In our school funding example, ignoring the possibility that some homes have both garages and swimming pools misrepresents the true scenario. By identifying that there might indeed be an overlap, we can refine our calculations using the Addition Rule effectively, accounting for homes counted twice by mistake if not adjusted for joint occurrences. Always remember: proper probability calculations involve including overlaps where they exist - never assume they do not unless otherwise indicated.