Question

In the real-estate ads described in Exercise 1 , \(64 \%\) of homes for sale have garages, \(21 \%\) have swimming pools, and \(17 \%\) have both features. a) If a home for sale has a garage, what's the probability that it has a pool too? b) Are having a garage and a pool independent events? Explain. c) Are having a garage and a pool mutually exclusive? Explain.

Short answer

a) ~26.56%. b) No, events are not independent. c) No, events are not mutually exclusive.

Step-by-step solution

Understand Given Probabilities

From the exercise, we know the following probabilities: - Probability of a home having a garage, \(P(G) = 0.64\) - Probability of a home having a pool, \(P(P) = 0.21\) - Probability of a home having both a garage and a pool, \(P(G \cap P) = 0.17\).

Solving Part a

To find the probability that a home has a pool given that it already has a garage, use the formula for conditional probability: \[ P(P|G) = \frac{P(G \cap P)}{P(G)} \]Plug in the known values: \[ P(P|G) = \frac{0.17}{0.64} \approx 0.2656 \] Thus, the probability is approximately \(26.56\%\).

Check for Independence in Part b

Two events are independent if \(P(G \cap P) = P(G) \times P(P)\). Calculate this product:\[ P(G) \times P(P) = 0.64 \times 0.21 = 0.1344 \]Since \(0.1344 eq 0.17\), the events are not independent.

Assess Mutual Exclusivity in Part c

Events are mutually exclusive if \(P(G \cap P) = 0\). Here, \(P(G \cap P) = 0.17\), so having both a garage and a pool is not mutually exclusive.

Key concepts

Conditional Probability

Conditional probability is a measure of the probability of an event occurring, given that another event has already occurred. It's a way to update the likelihood of an event based on new information. In formulaic terms, if we want to find the probability of event A happening given that B has happened, we use the formula:

• \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)

For instance, in the context of our real-estate example, if you know a home has a garage, you're interested in finding out how likely it is to also have a pool. This is symbolized as \( P(P|G) \) where G stands for a garage and P for a pool. Given that the probability of both a garage and a pool \( P(G \cap P) \) is 0.17, and the probability of having just a garage \( P(G) \) is 0.64, we calculate:

• \( P(P|G) = \frac{0.17}{0.64} \approx 0.2656 \)

This tells us there's about a 26.56% chance a home has a pool if it already has a garage.

Independence

Events are independent when the occurrence of one event doesn't influence the probability of the other. Mathematically, two events A and B are independent if the probability that both occur is equal to the product of their individual probabilities:

• \( P(A \cap B) = P(A) \times P(B) \)

In simpler terms, for two events to be independent, knowing that one event has happened should not change the probability of the other event.

Looking at our real-estate problem, we calculate whether having a garage (G) and a pool (P) are independent:

• \( P(G \cap P) = 0.17 \)
• \( P(G) \times P(P) = 0.64 \times 0.21 = 0.1344 \)

Since \( 0.1344 eq 0.17 \), the events are not independent. Having a garage seems to affect the likelihood of having a pool and vice versa.

Mutual Exclusivity

Mutual exclusivity in probability refers to events that cannot occur at the same time. If two events are mutually exclusive, the probability of both events occurring simultaneously is zero:

• \( P(A \cap B) = 0 \)

For example, a coin cannot land heads and tails simultaneously on a single flip—they are mutually exclusive outcomes.

In our real-estate scenario, we're examining whether a home having a garage and a pool at the same time is mutually exclusive. With \( P(G \cap P) = 0.17 \), there's actually a 17% chance of both features occurring together, indicating that these events are not mutually exclusive. Both features can be present in the same home, so they can happen simultaneously, unlike heads and tails on a coin flip.