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 **

Given: A and B are independent events.

For independent events: $P(A \cap B) = P(A)P(B)$

Checking each statement:

1. $A'$ and $B'$:

$P(A' \cap B') = 1 - P(A \cup B) = 1 - [P(A) + P(B) - P(A)P(B)] = (1 - P(A))(1 - P(B)) = P(A')P(B')$

⇒ Independent ✔

2. $A$ and $B'$:

$P(A \cap B') = P(A) - P(A \cap B) = P(A) - P(A)P(B) = P(A)[1 - P(B)] = P(A)P(B')$

⇒ Independent ✔

3. $A$ and $B$:

Given independent ✔

4. $A$ and $B$ are mutually exclusive:

For mutually exclusive events, $P(A \cap B) = 0$, which contradicts $P(A)P(B) ≠ 0$ (if both have nonzero probability).

⇒ Not true ❌

Answer: A and B are mutually exclusive.