Q.If two events are independent, then
(A) they must be mutually exclusive
(B) the sum of their probabilities must be equal to $1$
(C) (A) and (B) both are correct
(D) None of the above is correct

The key idea is Event Independence: two events $A$ and $B$ are independent if $P(A \cap B) = P(A) \cdot P(B)$. This has no inherent connection to mutual exclusivity or to probabilities summing to 1.

1. Mutual exclusivity means $P(A \cap B) = 0$. For independent events with non-zero probabilities, $P(A \cap B) = P(A)P(B) > 0$, so they cannot be mutually exclusive. …

The Core Idea: What Independence Really Means

When two events are independent, the occurrence of one gives no information about the other. Mathematically, this means:

This is the defining property. It says the probability that both happen is simply the product of their individual probabilities — no adjustment needed.

Now contrast this with mutual exclusivity. Two events are mutually exclusive if they cannot happen at the same time:

That is a completely different condition. One says "they can happen together, and the chance is the product of their chances." The other says "they can never happen together."

Step-by-Step Reasoning

1. Check option (A): "they must be mutually exclusive"

   For two events to be mutually exclusive, we need $P(A \cap B) = 0$. But for independent events with non-zero probabilities, $P(A \cap B) = P(A)P(B) > 0$. So independence contradicts mutual exclusivity (except in the trivial case where one event has probability zero). Option (A) is false.

2. Check option (B): "the sum of their probabilities must be equal to 1" …

Method: Independence vs. mutual exclusivity (reasoning pattern)

This is a conceptual question — the technique is to test each claim against precise definitions, not to compute.

Steps

Step 1: State both definitions side by side.

• Independent: $P(A\cap B)=P(A)\,P(B)$ — the events can co-occur.
• Mutually exclusive: $P(A\cap B)=0$ — the events cannot co-occur.

Step 2: See that they conflict for positive-probability events.

If $P(A),P(B)>0$ and the events are independent, then $P(A\cap B)=P(A)P(B)>0$, so they cannot also be mutually exclusive. …

Common Mistakes

Mistake 1: Thinking independent events must be mutually exclusive.

Why it's wrong: independence gives $P(A\cap B)=P(A)P(B)>0$ (for positive-probability events), while mutual exclusivity demands $P(A\cap B)=0$ — the two are opposites, not the same. Correct approach: treat them as distinct, unrelated properties. …