Q.State whether the following statement is True or False: Let $P(A) > 0$ and $P(B) > 0$. Then $A$ and $B$ can be both mutually exclusive and independent.

Concept: Event Independence — Two events are independent if $P(A \cap B) = P(A)P(B)$. They are mutually exclusive if $P(A \cap B) = 0$.

Step 1: For mutual exclusivity, $P(A \cap B) = 0$.

Step 2: For independence, $P(A \cap B) = P(A)P(B)$.

Step 3: Since $P(A) > 0$ and $P(B) > 0$, their product $P(A)P(B) > 0$. …

The statement is False.

Why this question matters

This is a classic trap in probability. It tests whether you truly understand the definitions of mutual exclusivity and independence — not just the formulas, but what they mean in terms of real events.

Let’s unpack both ideas.

Mutually exclusive means the two events cannot happen at the same time. If $A$ occurs, $B$ cannot — and vice versa. In set terms, $A \cap B = \varnothing$, so $P(A \cap B) = 0$.

Independent means the occurrence of one event gives you no information about whether the other will occur. Formally, $P(A \cap B) = P(A) P(B)$.

Now here’s the tension: if $A$ and $B$ are mutually exclusive and both have positive probability, then knowing $A$ happened tells you for sure that $B$ did not happen — that’s a huge amount of information. That directly contradicts the idea of independence.

Step-by-step reasoning

1. Write down what we are given.

   $P(A) > 0$, $P(B) > 0$. We are asked whether $A$ and $B$ can be both mutually exclusive and independent.

2. Assume they are mutually exclusive.

   Then $A \cap B = \varnothing$, so

3. Now check the independence condition. For independence, we need

Substituting from step 2, this would require

4. But $P(A) > 0$ and $P(B) > 0$. Their product is strictly positive:

So $0 = P(A) P(B)$ is impossible.

5. Conclusion. …

Method: Deciding whether two probability properties can coexist

For a "can $A$ and $B$ be both X and Y?" question, write each property as an equation on the same quantity $P(A\cap B)$ and check whether they can hold together.

Steps

Step 1: Express each property as an equation.

• Mutually exclusive: $P(A\cap B)=0$.
• Independent: $P(A\cap B)=P(A)P(B)$.

Step 2: Set the two expressions for $P(A\cap B)$ equal.

Step 3: Test against the stated conditions. …

Common Mistakes

Mistake 1: Thinking "cannot occur together" and "do not affect each other" are compatible.

Why it's wrong: for positive-probability events, mutual exclusivity ($P(A\cap B)=0$) forces $P(A)P(B)=0$, which is impossible — so they cannot also be independ …