Question
Use the table to answer the question. Train Arrival Age Group Prefers Fiction Prefers Nonfiction Total 20 to 30 years old 12 3 15 31 to 40 years old 24 6 30 Total 36 9 45 Let event A be someone who is 31 to 40 years old and event B represent someone who prefers fiction. Find P(A and B) and determine if the two events are independent. (1 point) Responses P(A and B)=645 and shows that the two events are independent. upper P left parenthesis upper A and upper B right parenthesis equals Start Fraction 6 over 45 End Fraction and shows that the two events are independent. P(A and B)=2445 and shows that the two events are independent. upper P left parenthesis upper A and upper B right parenthesis equals Start Fraction 24 over 45 End Fraction and shows that the two events are independent. P(A and B)=6645 and shows that the two events are independent. upper P left parenthesis upper A and upper B right parenthesis equals Start Fraction 66 over 45 End Fraction and shows that the two events are independent. P(A and B)=2445 but does not indicate that the two events are independent.
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Answered by
GPT-4o mini
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.**
- 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.**
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