Question

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.

• A. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

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.

Answer 1

The Correct Option is C

- Assertion : If $P(A \cap B) = 0$, then A and B are independent events. This assertion is true. Independent events imply that the occurrence of one does not influence the probability of the other. A condition of $P(A \cap B) = 0$ signifies that A and B cannot occur concurrently, a characteristic of independence. Consequently, the assertion is accurate. - Reason (R): Two events are independent if the occurrence of one does not affect the occurrence of the other. This definition is insufficient. The precise definition of independent events is: two events A and B are independent if and only if: \[ P(A \cap B) = P(A) \times P(B) \] The provided reason is false as it omits the correct condition for independence, which relies on the multiplication rule. Therefore, Reason (R) is incorrect. Thus, Assertion is true, and Reason (R) is false. The correct option is (C).