Q.State whether the following statement is True or False: If $A$ and $B$ are independent events, then $A'$ and $B'$ are also independent.

Concept: Event Independence — Two events are independent if the occurrence of one does not affect the probability of the other.

Reasoning:

1. If $A$ and $B$ are independent, then $P(A \cap B) = P(A) P(B)$.
2. We need to check if $P(A' \cap B') = P(A') P(B')$.
3. Using De Morgan’s law, $A' \cap B' = (A \cup B)'$, so $P(A' \cap B') = 1 - P(A \cup B)$.
4. For independent events, $P(A \cup B) = P(A) + P(B) - P(A)P(B)$. Then: …

Why This Works: The Intuition

Independence means that knowing whether $A$ happened gives you no information about whether $B$ happened. If that's true, then knowing whether $A$ didn't happen should also give you no information about whether $B$ didn't happen. The logic is symmetric — the relationship of "no information" survives complementation.

The formal proof is clean and short, and it's a classic result that often appears in board exams (CBSE, ISC, etc.) as a one-mark true/false or a short-answer question.

Step-by-Step Proof

1. Recall the definition of independence. Two events $A$ and $B$ are independent if and only if

2. What we need to show. We want to prove that $A'$ and $B'$ are independent, i.e.,

3. Express $A' \cap B'$ using De Morgan's law.

So the probability we need is

4. Expand $P(A \cup B)$ using the addition rule. For any two events,

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

5. Substitute into the complement expression.

6. Factor the right-hand side. Notice that $1 - P(A) = P(A')$ and $1 - P(B) = P(B')$. So …

Method: Proving a complement-independence result

To show that some pair of complements is independent, convert to the product test and use De Morgan plus the addition rule.

Steps

Step 1: State what "independent" means for the target pair.

You must show $P(A'\cap B')=P(A')\,P(B')$.

Step 2: Rewrite the intersection of complements with De Morgan.

Step 3: Expand the union and factor. …

Common Mistakes

Mistake 1: Assuming complements of independent events lose independence.

Why it's wrong: independence is preserved under complementation. Using $A'\cap B'=(A\cup B)'$ and $P(A\cup B)=P(A)+P(B)-P(A)P(B)$ gives $P(A'\cap B')=P(A')P(B')$. Correct approach: verify via De Morgan and factor $1-P(A)-P(B)+P(A)P(B)=(1-P(A))(1-P(B))$. …