To find \( P(A \text{ and } B) \), we first identify the number of people who are both in the age group of 31 to 40 years old (event A) and prefer fiction (event B).
From the table provided:
- The number of people aged 31 to 40 who prefer fiction is 24.
- The total number of people surveyed is 45.
Thus, we can calculate \( P(A \text{ and } B) \) as follows:
\[ P(A \text{ and } B) = \frac{\text{Number of people who are 31 to 40 and prefer fiction}}{\text{Total number of people}} = \frac{24}{45} \]
Next, we need to determine if events A and B are independent. Two events A and B are independent if:
\[ P(A \text{ and } B) = P(A) \times P(B) \]
First, we calculate \( P(A) \) and \( P(B) \):
-
Calculate \( P(A) \):
- Total people aged 31 to 40 = 30.
- Therefore, \( P(A) = \frac{30}{45} \).
-
Calculate \( P(B) \):
- Total people who prefer fiction = 36.
- Therefore, \( P(B) = \frac{36}{45} \).
Now we can compute \( P(A) \times P(B) \):
\[ P(A) \times P(B) = \left(\frac{30}{45}\right) \times \left(\frac{36}{45}\right) = \frac{30 \times 36}{45 \times 45} = \frac{1080}{2025} \]
Next, we simplify the fractions where necessary:
- Compare \( P(A \text{ and } B) \) with \( P(A) \times P(B) \):
\[ P(A \text{ and } B) = \frac{24}{45} \] \[ P(A) \times P(B) = \frac{1080}{2025} \]
Since \( \frac{24}{45} \) does not equal \( \frac{1080}{2025} \), events A and B are not independent.
So the choice from the options is:
\[ P(A \text{ and } B) = \frac{24}{45} \text{ but does not indicate that the two events are independent.} \]
Final Choice:
"P(A and B) = 24/45 but does not indicate that the two events are independent."