Cooper Realty is a small real estate company located in Albany, New York, specializing primarily in residential listings. They are interested in determining the likelihood of one of their listings being sold within a certain number of days. An analysis of company sales homes in previous years produced the following data:

Days Listed Until Sold

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline & & Under 30 & $31-90$ & Over 90 & Total \\
\hline & Under $\$150,000$ & 60 & 30 & 20 & 110 \\
\hline Initial Asking & $\$150,000-\$199,999$ & 40 & 160 & 80 & 280 \\
\hline \multirow[t]{3}{*}{ Price } & $\$200,000-\$250,000$ & 20 & 290 & 90 & 400 \\
\hline & Over $\$250,000$ & 10 & 40 & 10 & 60 \\
\hline & Total & 130 & 520 & 200 & 850 \\
\hline
\end{tabular}
\][/tex]

a. If [tex]$A$[/tex] is defined as the event that a home is listed for more than 90 days before being sold, estimate the probability of [tex]$A$[/tex] (to 2 decimals):
[tex]\[ 0.24 \][/tex]

b. If [tex]$B$[/tex] is defined as the event that the initial asking price is under [tex]$\$[/tex]150,000[tex]$, estimate the probability of $[/tex]B[tex]$ (to 3 decimals):
\[ 0.129 \]

c. What is the probability of $[/tex]A \cap B[tex]$ (to 4 decimals)?
\[ 0.0235 \]

d. Assuming that a contract was just signed to list a home with an initial asking price of less than $[/tex]\[tex]$150,000$[/tex], what is the probability that the home will take Cooper Realty more than 90 days to sell (to 2 decimals)?
[tex]\[ 0.18 \][/tex]



Answer :

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]