To determine \( P(A \text{ and } B) \) and check if the events are independent, let's first look at the definitions:
- Event A: Someone is 31 to 40 years old.
- Event B: Someone prefers fiction.
From the table, we can find the number of people who are both in the age group of 31 to 40 and who prefer fiction:
- The number of people who are 31 to 40 years old and prefer fiction is 24.
Now, we can compute \( P(A \text{ and } B) \):
\[ P(A \text{ and } B) = \frac{\text{Number of people who are 31 to 40 and prefer fiction}}{\text{Total number of people}} = \frac{24}{45} \]
Next, we need to check if events A and B are independent. Two events A and B are independent if:
\[ P(A \text{ and } B) = P(A) \cdot P(B) \]
Calculating \( P(A) \):
\[ P(A) = \frac{\text{Number of people who are 31 to 40 years old}}{\text{Total number of people}} = \frac{30}{45} = \frac{2}{3} \]
Calculating \( P(B) \):
\[ P(B) = \frac{\text{Number of people who prefer fiction}}{\text{Total number of people}} = \frac{36}{45} = \frac{4}{5} \]
Now, we compute \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = \left(\frac{2}{3}\right) \cdot \left(\frac{4}{5}\right) = \frac{8}{15} \]
Now, we compare \( P(A \text{ and } B) \) and \( P(A) \cdot P(B) \):
- \( P(A \text{ and } B) = \frac{24}{45} \) which simplifies to \( \frac{8}{15} \) (matching the calculated \( P(A) \cdot P(B) \)).
Since \( P(A \text{ and } B) = P(A) \cdot P(B) \), the two events are independent.
The correct answer is:
\[ P(A \text{ and } B) = \frac{24}{45} \text{ and shows that the two events are independent.} \]
Thus, the response would be:
P(A and B)=2445 and shows that the two events are independent.