1) In an experiment, the probability that event \( A \) occurs is \( \frac{2}{9} \), the probability that event \( B \) occurs is \( \frac{2}{7} \), and the probability that event \( A \) occurs given that event \( B \) occurs is \( \frac{2}{9} \). 4) Are \( A \) and \( B \) independent events? 4) yes

Yes, events \( A \) and \( B \) are **independent**. ### Explanation: Two events \( A \) and \( B \) are independent if and only if the occurrence of one event does not affect the probability of the other. Mathematically, this is expressed as: \[ P(A \mid B) = P(A) \] Let's verify this with the given probabilities: - **Given:** - \( P(A) = \frac{2}{9} \) - \( P(B) = \frac{2}{7} \) - \( P(A \mid B) = \frac{2}{9} \) - **Check for Independence:** Compare \( P(A \mid B) \) with \( P(A) \): \[ P(A \mid B) = \frac{2}{9} = P(A) \] Since \( P(A \mid B) = P(A) \), this confirms that the occurrence of event \( B \) does not affect the probability of event \( A \). Therefore, events \( A \) and \( B \) are independent. - **Alternative Verification:** Another way to confirm independence is by checking if: \[ P(A \cap B) = P(A) \times P(B) \] Calculate \( P(A \cap B) \) using the conditional probability: \[ P(A \cap B) = P(A \mid B) \times P(B) = \frac{2}{9} \times \frac{2}{7} = \frac{4}{63} \] Now, calculate \( P(A) \times P(B) \): \[ P(A) \times P(B) = \frac{2}{9} \times \frac{2}{7} = \frac{4}{63} \] Since both calculations yield \( \frac{4}{63} \), this further confirms that \( A \) and \( B \) are independent events. ### Conclusion: Your answer is correct. Events \( A \) and \( B \) are independent.