Sure! Let's go through the detailed step-by-step solution for each part of the question.
### Given Data:
- Total number of homes, [tex]\( N \)[/tex] = 850
- Number of homes listed for more than 90 days before being sold ([tex]\( A \)[/tex]) = 200
- Number of homes with an initial asking price under [tex]$150,000 (\( B \)) = 110
- Number of homes with an initial asking price under $[/tex]150,000 and listed for more than 90 days ([tex]\( A \cap B \)[/tex]) = 20
### Part (a):
Event [tex]\( A \)[/tex]: A home is listed for more than 90 days before being sold.
The probability of event [tex]\( A \)[/tex] can be estimated as the ratio of homes listed for more than 90 days to the total number of homes.
[tex]\[ P(A) = \frac{\text{Number of homes listed for more than 90 days}}{\text{Total number of homes}} \][/tex]
[tex]\[ P(A) = \frac{200}{850} \][/tex]
[tex]\[ P(A) = 0.24 \][/tex]
So, the probability of [tex]\( A \)[/tex] is [tex]\( 0.24 \)[/tex].
### Part (b):
Event [tex]\( B \)[/tex]: The initial asking price of the home is under [tex]$150,000.
The probability of event \( B \) can be estimated as the ratio of homes with an initial asking price under $[/tex]150,000 to the total number of homes.
[tex]\[ P(B) = \frac{\text{Number of homes with an initial asking price under $150,000}}{\text{Total number of homes}} \][/tex]
[tex]\[ P(B) = \frac{110}{850} \][/tex]
[tex]\[ P(B) \approx 0.129 \][/tex]
So, the probability of [tex]\( B \)[/tex] is [tex]\( 0.129 \)[/tex].
### Part (c):
Event [tex]\( A \cap B \)[/tex]: The home is listed for more than 90 days and has an initial asking price under [tex]$150,000.
The probability of event \( A \cap B \) can be estimated as the ratio of homes listed for more than 90 days with an initial asking price under $[/tex]150,000 to the total number of homes.
[tex]\[ P(A \cap B) = \frac{\text{Number of homes listed for more than 90 days and under $150,000}}{\text{Total number of homes}} \][/tex]
[tex]\[ P(A \cap B) = \frac{20}{850} \][/tex]
[tex]\[ P(A \cap B) \approx 0.0235 \][/tex]
So, the probability of [tex]\( A \cap B \)[/tex] is [tex]\( 0.0235 \)[/tex].
### Part (d):
Conditional Probability [tex]\( P(A|B) \)[/tex]: The probability that a home will take more than 90 days to sell given that the initial asking price is under [tex]$150,000.
This can be estimated using the formula for conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
From parts (b) and (c):
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.0235}{0.129} \]
\[ P(A|B) \approx 0.18 \]
So, the probability that a home will take more than 90 days to sell given that the initial asking price is under $[/tex]150,000 is [tex]\( 0.18 \)[/tex].
Therefore, the detailed probabilities for each part are:
- [tex]\( P(A) = 0.24 \)[/tex]
- [tex]\( P(B) = 0.129 \)[/tex]
- [tex]\( P(A \cap B) = 0.0235 \)[/tex]
- [tex]\( P(A|B) = 0.18 \)[/tex]
