Q.State whether the following statement is True or False: If $A$ and $B$ are two independent events then $P(A \text{ and } B) = P(A) \cdot P(B)$.

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

Reasoning:

1. By definition, events $A$ and $B$ are independent if and only if $P(A \cap B) = P(A) \cdot P(B)$. …

The core idea here is what it means for two events to be independent. Independence is not about events being unrelated in a vague sense; it has a precise mathematical meaning. When we say $A$ and $B$ are independent, we mean that knowing whether $B$ happened gives you no information about whether $A$ happened, and vice versa.

The formal way to capture this is through conditional probability. If $P(B) > 0$, independence means $P(A \mid B) = P(A)$. That is, the probability of $A$ given $B$ is just the same as the probability of $A$ on its own — $B$'s occurrence doesn't change the odds.

Now, recall the definition of conditional probability:

If $A$ and $B$ are independent, we substitute $P(A \mid B) = P(A)$ into this:

Multiply both sides by $P(B)$:

This is not just a property of independence — it is the definition that most textbooks use. Two events are defined to be independent precisely when $P(A \cap B) = P(A) P(B)$. So the statement given is exactly that definition. …

Method: Recognising the definition of independent events

Some statements are true simply because they are a definition. The skill is to recognise the multiplication rule as the defining condition for independence.

Steps

Step 1: Recall the definition.

Events $A$ and $B$ are independent exactly when

Step 2: Match the notation.

"$P(A\text{ and }B)$" is another way of writing $P(A\cap B)$, so the statement is the definition verbatim.

Step 3: Distinguish from the addition rule. …

Common Mistakes

Mistake 1: Marking this False by confusing independence with mutual exclusivity.

Why it's wrong: $P(A\text{ and }B)=P(A)P(B)$ is precisely the definition of independence; it is mutually exclusive events that give $P(A\cap B)=0$. Correct approac …