If A and B are two independent events, then which of the following is…

The correct answer is Option (1) → (B) and (D) only

For independent events A and B, \(P(A\cap B) = P(A)P(B)\), and the complements A′ and B′ are also independent.

Checking the statements

(A) \(P(A\cap B) = 0\) — Incorrect. \(P(A\cap B) = P(A)P(B)\), which is 0 only if one of the events has probability 0. A zero intersection describes mutually exclusive events, not independent ones.

(B) \(P(A\cup B) = 1-P(A') P(B')\) — Correct. By De Morgan’s law and the independence of A′ and B′:

    \( \displaystyle P(A\cup B) = 1 - P\big((A\cup B)'\big) = 1 - P(A'\cap B') \)

    \( \displaystyle = 1 - P(A')P(B') \)

(C) \(P(A\cup B) = P(A) P(B)\) — Incorrect. The product is the probability of the intersection. For example, with \(P(A) = P(B) = \frac{1}{2}\):

    \( \displaystyle P(A\cup B) = \frac{1}{2} + \frac{1}{2} - \frac{1}{4} = \frac{3}{4} \)

    \( \displaystyle P(A)P(B) = \frac{1}{4} \)

(D) \(P(A\cap B) = P(A) P(B)\) — Correct. This is the definition of independent events.