Question
Book Preference
Music Preference Likes Books Dislikes Books Total
Likes Music 46 56 102
Dislikes Music 68 30 98
Total 114 86 200
Let event A be a person who does not like books and event B be a person who likes music. Are the two events independent?
(1 point)
Responses
Yes, because P(A and B)=P(A)⋅P(B).
Yes, because P(A and B)=P(A)⋅P(B).
No, because P(A)≠P(B).
No, because P(A)≠P(B).
Yes, because P(A and B)=56200.
Yes, because P(A and B)=56200.
No, because P(A)=86200, P(B)=102200, and P(A and B)≠56200.
No, because P(A)=86200, P(B)=102200, and P(A and B)≠56200.
Music Preference Likes Books Dislikes Books Total
Likes Music 46 56 102
Dislikes Music 68 30 98
Total 114 86 200
Let event A be a person who does not like books and event B be a person who likes music. Are the two events independent?
(1 point)
Responses
Yes, because P(A and B)=P(A)⋅P(B).
Yes, because P(A and B)=P(A)⋅P(B).
No, because P(A)≠P(B).
No, because P(A)≠P(B).
Yes, because P(A and B)=56200.
Yes, because P(A and B)=56200.
No, because P(A)=86200, P(B)=102200, and P(A and B)≠56200.
No, because P(A)=86200, P(B)=102200, and P(A and B)≠56200.
Answers
GPT-4o mini
To determine whether events \( A \) (does not like books) and \( B \) (likes music) are independent, we can use the definitions and formulas involving probabilities.
### Given data:
- Total number of people = 200
- The number of people who dislike books (event \( A \)) = 86
- The number of people who like music (event \( B \)) = 102
- The number of people who dislike books and like music (event \( A \) and \( B \)) = 56
### Calculating probabilities:
1. **P(A)**: Probability of a person not liking books.
\[
P(A) = \frac{\text{Number of people who dislike books}}{\text{Total number of people}} = \frac{86}{200} = 0.43
\]
2. **P(B)**: Probability of a person liking music.
\[
P(B) = \frac{\text{Number of people who like music}}{\text{Total number of people}} = \frac{102}{200} = 0.51
\]
3. **P(A and B)**: Probability of a person who dislikes books and likes music.
\[
P(A \text{ and } B) = \frac{\text{Number of people who dislike books and like music}}{\text{Total number of people}} = \frac{56}{200} = 0.28
\]
### Testing for independence:
Events \( A \) and \( B \) are independent if:
\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]
Calculating \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = 0.43 \cdot 0.51 = 0.2193
\]
### Comparing:
- \( P(A \text{ and } B) = 0.28 \)
- \( P(A) \cdot P(B) = 0.2193 \)
Since \( P(A \text{ and } B) \neq P(A) \cdot P(B) \), the events are not independent.
### Conclusion:
- The correct response is: **No, because \( P(A) \neq P(B) \)** (translation implies not independent).
**Final Response: No, because P(A)≠P(B).**
### Given data:
- Total number of people = 200
- The number of people who dislike books (event \( A \)) = 86
- The number of people who like music (event \( B \)) = 102
- The number of people who dislike books and like music (event \( A \) and \( B \)) = 56
### Calculating probabilities:
1. **P(A)**: Probability of a person not liking books.
\[
P(A) = \frac{\text{Number of people who dislike books}}{\text{Total number of people}} = \frac{86}{200} = 0.43
\]
2. **P(B)**: Probability of a person liking music.
\[
P(B) = \frac{\text{Number of people who like music}}{\text{Total number of people}} = \frac{102}{200} = 0.51
\]
3. **P(A and B)**: Probability of a person who dislikes books and likes music.
\[
P(A \text{ and } B) = \frac{\text{Number of people who dislike books and like music}}{\text{Total number of people}} = \frac{56}{200} = 0.28
\]
### Testing for independence:
Events \( A \) and \( B \) are independent if:
\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]
Calculating \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = 0.43 \cdot 0.51 = 0.2193
\]
### Comparing:
- \( P(A \text{ and } B) = 0.28 \)
- \( P(A) \cdot P(B) = 0.2193 \)
Since \( P(A \text{ and } B) \neq P(A) \cdot P(B) \), the events are not independent.
### Conclusion:
- The correct response is: **No, because \( P(A) \neq P(B) \)** (translation implies not independent).
**Final Response: No, because P(A)≠P(B).**