Q.Fill in the blank: Let $A$ and $B$ be two events. If $P(A \mid B) = P(A)$, then $A$ is __________ of $B$.

Concept: Event Independence

When $P(A \mid B) = P(A)$, knowing $B$ has occurred gives no information about $A$ — the probability of $A$ remains unchanged.

Reasoning:

1. By definition, $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$.
2. Setting this equal to $P(A)$ gives $\frac{P(A \cap B)}{P(B)} = P(A)$. …

When you see $P(A \mid B) = P(A)$, the intuition is simple: the probability of $A$ happening is the same whether or not $B$ has happened. In other words, $B$’s occurrence doesn’t change the odds for $A$ at all. That’s the very meaning of independence between two events.

Let’s walk through the reasoning step by step.

1. Recall the definition of conditional probability. For any two events $A$ and $B$ (with $P(B) > 0$),

2. Plug in the given condition. We are told $P(A \mid B) = P(A)$. So:

3. Multiply both sides by $P(B)$. This gives:

4. Recognise this as the definition of independence. Two events $A$ and $B$ are said to be independent if and only if

That’s exactly what we have here. …

Method: Turning a conditional-probability statement into an independence test

When a question gives an equation involving $P(A\mid B)$ and asks what relationship it forces, translate it into the multiplication form and read off the standard name.

Steps

Step 1: Write the definition of conditional probability.

Step 2: Substitute the given equation and clear the denominator.

Put in whatever $P(A\mid B)$ is said to equal and multiply through by $P(B)$. The condition $P(A\mid B)=P(A)$ yields

Step 3: Match against the definitions. …

Common Mistakes

Mistake 1: Answering "mutually exclusive" for the blank.

Why it's wrong: $P(A\mid B)=P(A)$ clears to $P(A\cap B)=P(A)P(B)$, which is independence; mutual exclusivity is the different condition $P(A\cap B)=0$. …