Q.State whether the following statement is True or False: If $A$ and $B$ are mutually exclusive events, then they will be independent also.

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

Reasoning:

1. For mutually exclusive events $A$ and $B$, $A \cap B = \emptyset$, so $P(A \cap B) = 0$.
2. For independence, we require $P(A \cap B) = P(A) P(B)$. …

Why this question trips students up

The confusion here comes from mixing up two different kinds of "relationship" between events. Mutually exclusive events cannot happen together — they share no outcomes. Independent events can happen together, and the occurrence of one tells you nothing about the other. These are fundamentally opposite ideas.

Let's make this concrete. Suppose you roll a fair die. Let $A$ be "roll a 1" and $B$ be "roll a 2". These are mutually exclusive — you can't get both on a single roll. But are they independent? If you know $A$ happened, does that change the probability of $B$? Absolutely — it drops it to zero. So they are dependent.

The only exception is when one event has probability zero — then it's both mutually exclusive and independent, but that's a degenerate case, not the general rule.

Step-by-step reasoning

1. Recall the definitions precisely.

Two events $A$ and $B$ are:

• Mutually exclusive if $A \cap B = \emptyset$, so $P(A \cap B) = 0$.
• Independent if $P(A \cap B) = P(A) \cdot P(B)$.

2. Assume both conditions hold.

If $A$ and $B$ are both mutually exclusive and independent, then we must have:

3. Interpret the equation.

The product $P(A) \cdot P(B) = 0$ means at least one of $P(A)$ or $P(B)$ is zero. So the only way mutual exclusivity and independence can coexist is if one of the events has zero probability.

4. Apply to the general case. …

Method: Testing an implication between event properties

To judge "if $A,B$ are mutually exclusive then they are independent," treat it as: does the first equation force the second?

Steps

Step 1: Assume the hypothesis as an equation.

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

Step 2: Write the target property as an equation.

Independence would require $P(A\cap B)=P(A)P(B)$.

Step 3: Combine and check whether it is forced. …

Common Mistakes

Mistake 1: Believing mutual exclusivity implies independence.

Why it's wrong: if $A\cap B=\varnothing$ then knowing $A$ occurred makes $B$ impossible — that is maximal dependence, not independence. Formally $P(A\cap B)=0\ne P(A)P(B)$ when both are positive. Correct ap …