Q-2: In a subdivision, 67% of the homes have detached garages, 32% have a patio, and 13% have both. What is the probability that a house has a patio if you know that it has a detached garage?
To find the probability that a house has a patio, given that it has a detached garage, we can use conditional probability. Conditional probability is the probability of an event occurring, given that another event has already occurred.
Let's consider the information given in the problem:
- 67% of the homes have detached garages.
- 32% of the homes have a patio.
- 13% of the homes have both a detached garage and a patio.
We can use the formula for conditional probability:
P(A | B) = P(A and B) / P(B)
Where:
- P(A | B) is the probability of event A occurring, given that event B has already occurred.
- P(A and B) is the probability of both events A and B occurring.
- P(B) is the probability of event B occurring.
In this case, event A is having a patio, and event B is having a detached garage.
Let's substitute the known probabilities into the formula:
P(A | B) = P(A and B) / P(B)
P(patio | detached garage) = P(patio and detached garage) / P(detached garage)
From the given information, we know that 13% of the homes have both a detached garage and a patio (P(patio and detached garage) = 0.13), and 67% of the homes have a detached garage (P(detached garage) = 0.67).
Substituting these values into the formula, we can calculate the probability that a house has a patio, given that it has a detached garage:
P(patio | detached garage) = 0.13 / 0.67
Therefore, the probability that a house has a patio, given that it has a detached garage, is approximately 0.194 or 19.4%.