Two events A and B will be independent, then

The correct answer is Option (1) → \(P(\overline{A} \cap \overline{B}) = [1 - P(A)][1 - P(B)]\)

A and B are independent when \(P(A \cap B) = P(A)\,P(B)\). By De Morgan’s law, \(\overline{A} \cap \overline{B} = \overline{A \cup B}\), so

    \(P(\overline{A} \cap \overline{B}) = 1 - P(A \cup B)\)
\(= 1 - P(A) - P(B) + P(A \cap B)\)
\(= 1 - P(A) - P(B) + P(A)P(B)\)
\(= [1 - P(A)][1 - P(B)]\)

So Option (1) always holds for independent events. The other three options put conditions on the sizes of P(A) and P(B), which independence does not; for example, two independent events can have \(P(A) = \frac{1}{2}\) and \(P(B) = \frac{1}{4}\).

NCERT (2026-27 edition):
Independent Events