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:
-
P(A): Probability of a person not liking books. \[ P(A) = \frac{\text{Number of people who dislike books}}{\text{Total number of people}} = \frac{86}{200} = 0.43 \]
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P(B): Probability of a person liking music. \[ P(B) = \frac{\text{Number of people who like music}}{\text{Total number of people}} = \frac{102}{200} = 0.51 \]
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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).