Let A and B be two independent events. Then which of following statem…

The correct answer is Option (4) → \( \bar{\mathrm{A}} \) and \( \bar{\mathrm{B}} \)  are not independent.

If A and B are independent, then P(A ∩ B) = P(A) P(B), and so are the pairs \( \bar{A} \) and B, A and \( \bar{B} \), and \( \bar{A} \) and \( \bar{B} \). For the last pair:

    \( P(\bar{A}\cap\bar{B})=1-P(A\cup B) \)

    \( =1-P(A)-P(B)+P(A)P(B) \)

    \( =(1-P(A))(1-P(B)) \)

    \( =P(\bar{A})P(\bar{B}) \)

So \( \bar{A} \) and \( \bar{B} \) are independent, and Option (4), which says they are not, is the statement that is not true.

NCERT (2026-27 edition):
Independent Events