Q.Two events A and B will be independent, if (A) A and B are mutually exclusive (B) $P(A'B') = [1 - P(A)] [1 - P(B)]$ (C) $P(A) = P(B)$ (D) $P(A) + P(B) = 1$

Concept: Event Independence — Two events are independent if the occurrence of one does not affect the probability of the other. The defining condition is $P(A \cap B) = P(A) P(B)$.

Reasoning:

1. For independent events, $P(A \cap B) = P(A) P(B)$. Using De Morgan’s law, $P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B)$.
2. If $A$ and $B$ are independent, then $P(A \cup B) = P(A) + P(B) - P(A)P(B)$. Substituting gives: $P(A' \cap B') = 1 - [P(A) + P(B) - P(A)P(B)] = [1 - P(A)][1 - P(B)].$ …

We need to recall what it means for two events to be independent. The formal definition is:

That’s the only condition that guarantees independence. Now let’s examine each option carefully — not just by checking formulas, but by understanding why each one does or does not guarantee independence.

1. Option (A): A and B are mutually exclusive

   Mutually exclusive means $A \cap B = \emptyset$, so $P(A \cap B) = 0$.

   For independence, we would need $P(A)P(B) = 0$, which forces at least one of $P(A)$ or $P(B)$ to be zero.

   If both events have non-zero probability, mutual exclusivity actually violates independence. So this is not a general condition — it only works in a trivial case.

   Verdict: Incorrect.

2. Option (B): $P(A'B') = [1 - P(A)][1 - P(B)]$

   Let’s unpack this. $A'B'$ means $A' \cap B'$, the complement of $A \cup B$ (by De Morgan’s law).

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

   The right-hand side is $(1 - P(A))(1 - P(B)) = 1 - P(A) - P(B) + P(A)P(B)$.

   Equating both sides:

Cancel the 1:

Multiply by -1:

But we know the general addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Comparing, we get:

Cancel $P(A) + P(B)$ from both sides:

So $P(A \cap B) = P(A)P(B)$.

That’s exactly the definition of independence.

Verdict: Correct. …

Method: Recognising equivalent forms of the independence condition

For "which statement makes A and B independent?" MCQs, the technique is to reduce each option to the one defining equation and see which is genuinely equivalent to it.

Steps

Step 1: State the definition.

Step 2: Algebraically test each candidate condition.

Rewrite each option and check whether it collapses to the definition. The complement form is the classic disguise — using De Morgan and the addition rule, …

Common Mistakes

Mistake 1: Choosing "$A$ and $B$ are mutually exclusive" as the independence condition.

Why it's wrong: mutual exclusivity gives $P(A\cap B)=0$, which (for positive probabilities) actually forces dependence — the opposite of independence. Correct approach: independence means $P(A\cap B)=P(A)P(B)$.

Mistake 2: Not recognising that the complement statement is the independence condition in disguise. …