Q.If $P(A) = 0.4$, $P(B) = 0.8$ and $P(B \mid A) = 0.6$, then $P(A \cup B)$ is equal to
(A) $0.24$
(B) $0.3$
(C) $0.48$
(D) $0.96$

Concept: Conditional Probability — we use $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$ to find the intersection, then the addition rule for union.

Step 1: From $P(B \mid A) = 0.6$ and $P(A) = 0.4$,

Step 2: Using the addition rule, …

We are given $P(A) = 0.4$, $P(B) = 0.8$, and $P(B \mid A) = 0.6$. The goal is $P(A \cup B)$.

The core idea: conditional probability tells us how two events overlap. Once we know the overlap ($A \cap B$), the union follows directly from the addition rule. Without the overlap, we cannot simply add $P(A)$ and $P(B)$ — that would double-count the intersection.

1. Find $P(A \cap B)$ using the definition of conditional probability.

   By definition, $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$.

   Rearranging: $P(A \cap B) = P(B \mid A) \cdot P(A)$.

   Substitute: $P(A \cap B) = 0.6 \times 0.4 = 0.24$.

   Watch out

   A common mistake is to think $P(A \cap B) = P(A) \times P(B)$ — that is only true if $A$ and $B$ are independent. Here, $P(B \mid A) = 0.6$ while $P(B) = 0.8$, so they are not independent. Always use the conditional formula when given $P(B \mid A)$.

2. Apply the addition rule for probability. …

Method: Multiplication theorem to unlock the addition rule

Use this when a union $P(A\cup B)$ is wanted but the overlap is hidden inside a conditional probability instead of being given outright.

Steps

Step 1: Convert the conditional into the intersection.

(or $P(A\mid B)\,P(B)$, whichever conditional you are given — match the marginal to the event after the bar).

Step 2: Apply the addition rule.

Step 3: Substitute and check the range. …

Common Mistakes

Mistake 1: Assuming $P(A\cap B)=P(A)\,P(B)$.

Why it's wrong: that product form only holds for independent events; here $P(B\mid A)=0.6\ne P(B)=0.8$, so they are dependent. Correct approach: use $P(A\cap B)=P(B\mid A)\,P(A)$.

Mistake 2: Adding the marginals without subtracting the overlap. …