CBSE 2025, Set 3

Question

**Assertion (A):** 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 effect the occurrence of the other. A). Both Assertion (A) and Reason (R) are true and 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.

Edvaya · Accepted Answer

The correct answer is Assertion (A) is false, but Reason (R) is true.. **Solution:**

**Assertion (A):** Two events $A$ and $B$ are defined as independent if their joint probability equals the product of their individual probabilities, i.e., $P(A \cap B) = P(A)P(B)$. The condition $P(A \cap B) = 0$ actually defines **mutually exclusive** events. Mutually exclusive events are generally not independent (unless $P(A)=0$ or $P(B)=0$). Therefore, the assertion is **false**.

**Reason (R):** The statement "Two events are independent if the occurrence of one does not affect the occurrence of the other" is the correct theoretical definition of independent events. Therefore, the reason is **true**.

Conclusion: **Assertion (A) is false, but Reason (R) is true.**

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