In a subdivison, 67% of homes have detached garages, 32% have a patio, and 13% have both. What is the probability that a house has a pation if you know that it has a detached garage? Please explain how you did the work, thanks.
To find the probability that a house has a patio given that it has a detached garage, we can use conditional probability.
First, let's define the events:
A = "Having a detached garage"
B = "Having a patio"
We are given three pieces of information:
1. P(A) = 67% = 0.67 (the probability of having a detached garage)
2. P(B) = 32% = 0.32 (the probability of having a patio)
3. P(A ∩ B) = 13% = 0.13 (the probability of having both a detached garage and a patio)
The conditional probability of B given A is denoted as P(B|A), which denotes the probability of B occurring given that A has already occurred.
For our problem, we need to find P(B|A), which is the probability of having a patio given that a house has a detached garage.
We can use the formula for conditional probability:
P(B|A) = P(A ∩ B) / P(A)
Plugging in the given values, we have:
P(B|A) = 0.13 / 0.67 = 0.194 = 19.4%
Therefore, the probability that a house has a patio, given that it has a detached garage, is 19.4%.