If A and B are independent events, then which of the following is not…

The correct answer is Option (4) → A and B are mutually exclusive

Since A and B are independent, \(P(A \cap B) = P(A)P(B)\). Then

    \(P(A \cap B') = P(A) - P(A \cap B)\)

    \(= P(A) - P(A)P(B)\)

    \(= P(A)P(B')\)

so A and B' are independent. In the same way, with A and B interchanged, A' and B are independent. Also

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

    \(= 1 - P(A) - P(B) + P(A)P(B)\)

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

so A' and B' are independent. Options (1), (2) and (3) are true.

Independent events with non-zero probabilities are never mutually exclusive:

    \(P(A \cap B) = P(A)P(B) \gt 0\)

so A and B occur together with positive probability, and Option (4) is not true.