Question

Suppose that \(46 \%\) of families living in a certain county own a car and \(18 \%\) own an SUV. The Addition Rule might suggest, then, that \(64 \%\) of families own either a car or an SUV. What's wrong with that reasoning?

Short answer

The reasoning mistakenly ignores families who own both, leading to double counting.

Step-by-step solution

Understand Given Probabilities

We are given that 46% of families own a car, so we denote this probability as \( P(A) = 0.46 \), where \( A \) is the event of owning a car. Similarly, 18% own an SUV, denoted as \( P(B) = 0.18 \), where \( B \) represents owning an SUV.

Recall the Addition Rule with Overlap

The Addition Rule for probabilities of two events \( A \) and \( B \) is given by: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]where \( P(A \cup B) \) represents the probability of owning either a car or an SUV, and \( P(A \cap B) \) represents the probability of owning both a car and an SUV.

Identify Missing Probability

The 64% ownership figure was calculated by simply adding 46% and 18% (\( 0.46 + 0.18 = 0.64 \)), without considering that some families might own both a car and an SUV. This represents double-counting in the given scenario.

Explain Double-Counting

By ignoring \( P(A \cap B) \), we assume that no families own both a car and an SUV when determining the ownership percentages using the Addition Rule. This oversight adds the probability of owning both (potentially nonzero), resulting in an erroneously high total of 64%.

Key concepts

Understanding the Addition Rule in Probability

Probability is a fundamental concept in mathematics, especially when dealing with events and their likelihood of occurring. One important aspect is the Addition Rule, which helps us calculate the probability of either one event or another occurring. This rule is crucial when dealing with two events, such as owning a car or an SUV in our initial exercise.

The **Addition Rule** for two events \(A\) and \(B\) states that the probability of either event \(A\) or event \(B\) happening is given by:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

This formula accounts for the possibility that some individuals or cases might belong to both events. By subtracting \(P(A \cap B)\), or the probability that both events happen, from the combined probability, we ensure accurate results. In simpler terms, this subtraction prevents any overlap between events from being double-counted, which leads us to our next concept.

Avoiding Double-Counting in Probability Calculations

When calculating probabilities for events that may overlap, double-counting becomes a significant concern. Double-counting occurs when we mistakenly add the probabilities of overlapping events together without accounting for the overlap itself.

Consider the scenario from the exercise: owning either a car or an SUV. By merely adding 46% and 18%, we arrive at an incorrect logical conclusion of 64%. This mistake arises from the inadvertent presumptions that no family owns both a car and an SUV. In such cases, it's critical to:

• Identify overlapping parties (families owning both a car and an SUV)
• Use the addition formula \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
• Subtract \(P(A \cap B)\) from the total to rectify the issue

This approach ensures the accuracy of probability predictions by addressing any double-counting occurrences beforehand.

Addressing Event Overlap in Probability

Event overlap is a common scenario in probability where two events can occur simultaneously. In probability terms, this is represented as \(P(A \cap B)\), which reads as the probability that both events \(A\) and \(B\) occur together.

In our car-SUV exercise, event overlap plays a crucial role. Families owning both a car and an SUV are part of this overlap. Ignoring these families in the initial calculation resulted in an overestimated probability. Integrating the overlap requires us to subtract \(P(A \cap B)\) when applying the Addition Rule.

• First, identify overlapping areas (like families with both items)
• Calculate \(P(A \cap B)\), if data allows
• Apply \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) to obtain accurate results

Recognizing event overlap prevents miscalculations and ensures correct application of probability concepts.