Question

Real estate ads suggest that \(64 \%\) of homes for sale have garages, \(21 \%\) have swimming pools, and \(17 \%\) have both features. What is the probability that a home for sale has a) a pool or a garage? b) neither a pool nor a garage? c) a pool but no garage?

Short answer

a) 68%, b) 32%, c) 4%

Step-by-step solution

Understand the Given Percentages

We are given that 64% of homes have garages (let's denote this as event G), 21% have swimming pools (denote as event P), and 17% have both features (denote as event P \(\cap\) G).

Calculate Probability of Pool or Garage

We need to find the probability of a pool or a garage, denoted as \(P(P \cup G)\). Using the formula for the union of two events: \[ P(P \cup G) = P(P) + P(G) - P(P \cap G) \] Substituting in the given probabilities: \[ P(P \cup G) = 0.21 + 0.64 - 0.17 = 0.68 \] Thus, the probability that a home has either a pool or a garage is 68%.

Calculate Probability of Neither Pool nor Garage

To find the probability of neither a pool nor a garage, we need the complement of \(P(P \cup G)\). The complement rule states that \(P(A^c) = 1 - P(A)\): \[ P((P \cup G)^c) = 1 - P(P \cup G) = 1 - 0.68 = 0.32 \] Thus, the probability that a home has neither a pool nor a garage is 32%.

Calculate Probability of Pool but No Garage

To find the probability of a pool but no garage, we use: \[ P(P \cap G^c) = P(P) - P(P \cap G) \] Substituting in the given probabilities: \[ P(P \cap G^c) = 0.21 - 0.17 = 0.04 \] Thus, the probability that a home has a pool but no garage is 4%.

Key concepts

Union of Events

In probability theory, when we talk about the union of events, we're discussing the scenario in which either one event occurs, the other occurs, or both occur. It's like asking, "What's the chance of at least one of these happening?" For instance, in our exercise, we're interested in the probability that a home has either a pool or a garage, or both. To calculate this probability, we use the formula:

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

Here, \(A\) represents one event (like having a pool), and \(B\) is another event (such as having a garage). The term \(P(A \cap B)\) accounts for the overlap where both events occur, ensuring we don't double-count it. Thus, the union gives us a comprehensive view of all homes with either or both features.

Complement Rule

The complement rule is a fundamental concept in probability. It helps us understand the probability of something not happening. If you know the chance of an event occurring, you can easily find out the chance of it not occurring. The rule is simple: if \(A\) is an event, then its complement \(A^c\) is everything not in \(A\).

• Mathematically, this relation is given by:\[ P(A^c) = 1 - P(A) \]

In our real estate example, once we calculated the probability of a home having a pool or a garage, the complement rule allowed us to deduce the likelihood of a home having neither feature. This helped us complete the picture to understand all possibilities.

Set Theory

Set theory is a branch of mathematical logic that deals with collections of objects, known as sets. In probability, events are often treated as sets of outcomes. Set theory provides a framework to understand and express relationships between different events.

• The basic operations include:
• Union (\(A \cup B\)): Combines all elements from sets \(A\) and \(B\).
• Intersection (\(A \cap B\)): Includes only the elements that are common to both sets \(A\) and \(B\).
• Complement (\(A^c\)): Contains everything not in set \(A\).

In our exercise, set theory is applied as we evaluate homes with certain features as different sets and use operations like union and intersection to determine probabilities.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It's an essential concept for understanding how the probability of one event affects another.

• The formula used is:\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]where \(P(A|B)\) is the probability of \(A\) given \(B\), and \(P(A \cap B)\) is the probability of both \(A\) and \(B\) happening.

Although not directly utilized in our exercise solution, understanding conditional probability can help in deeper analyses of more complex real estate scenarios, like finding out the probability of having a pool given there's already a garage.