Assertion : If A and B are two events such that $P(A \cap B) = 0$, then A and B are independent events.
Reason (R): Two events are independent if the occurrence of one does not affect the occurrence of the other.
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For two events to be independent, the condition $P(A \cap B) = P \times P$ must hold true. A probability of 0 for the intersection of two events indicates that the events cannot occur together, which can be one interpretation of independence, but it is not the full definition.
Both Assertion (A) and Reason (R) are true and the Reason (R) is the
correct explanation of the Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the
correct explanation of the Assertion (A).
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.
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The Correct Option isC
Solution and Explanation
- Assertion : If $P(A \cap B) = 0$, then A and B are independent events. This assertion is true. If two events are independent, it means that the occurrence of one event does not affect the probability of the other event. The condition $P(A \cap B) = 0$ implies that A and B cannot occur together, which is a characteristic of independent events. Therefore, the assertion is correct.
- Reason (R): Two events are independent if the occurrence of one does not affect the occurrence of the other. This definition is incomplete. The correct definition of independent events is: two events A and B are independent if and only if:
\[
P(A \cap B) = P \times P
\]
The reason provided in the question is false because it does not account for the correct condition for independence, which is based on the multiplication rule. Thus, Reason (R) is incorrect.
Therefore, while Assertion is true, Reason (R) is false. Thus, the correct answer is option (C).
