Q.$A$ and $B$ are events such that $P(A) = 0.4$, $P(B) = 0.3$ and $P(A \cup B) = 0.5$. Then $P(B' \cap A)$ equals
(A) $\dfrac{2}{3}$
(B) $\dfrac{1}{2}$
(C) $\dfrac{3}{10}$
(D) $\dfrac{1}{5}$

Concept: Probability Complement Rule and set decomposition.

We need $P(B' \cap A)$ — the part of $A$ that lies outside $B$.

Step 1: Use the addition rule:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$0.5 = 0.4 + 0.3 - P(A \cap B)$

So $P(A \cap B) = 0.2$.

Step 2: Decompose $A$ into the part inside $B$ and the part outside $B$:

$P(A) = P(A \cap B) + P(A \cap B')$ …

We are given three probabilities: $P(A) = 0.4$, $P(B) = 0.3$, and $P(A \cup B) = 0.5$. The question asks for $P(B' \cap A)$ — the probability that event $A$ occurs and event $B$ does not occur. This is the part of $A$ that lies outside $B$.

Think of a Venn diagram. The event $A$ is split into two disjoint parts: the part inside $B$ (i.e., $A \cap B$) and the part outside $B$ (i.e., $A \cap B'$). So:

If we can find $P(A \cap B)$, we can subtract it from $P(A)$ to get the desired $P(A \cap B')$.

How do we find $P(A \cap B)$? Use the inclusion-exclusion principle, which relates the union of two events to their individual probabilities and their intersection:

This is a central formula for any two events. Rearranging it gives:

Now plug in the given numbers:

1. Find $P(A \cap B)$

2. Use the partition of $A$ Since $A = (A \cap B) \cup (A \cap B')$ and these two sets are disjoint, we have:

Substitute $P(A) = 0.4$ and $P(A \cap B) = 0.2$: …

Method: Splitting an event into "inside" and "outside" another

Use this to find $P(A\cap B')$ — the part of $A$ lying outside $B$ — from marginal and union data.

Steps

Step 1: Recover the overlap from the addition rule.

Step 2: Partition $A$ by whether $B$ occurs.

Since $A=(A\cap B)\cup(A\cap B')$ and these two pieces are disjoint,

Step 3: Solve for the piece you want. …

Common Mistakes

Mistake 1: Estimating $P(B'\cap A)$ as $P(A)-P(B)$.

Why it's wrong: it must be $P(A)$ minus the part of $A$ inside $B$, i.e. $P(A)-P(A\cap B)$, not $P(A)-P(B)$. Correct approach: find $P(A\cap B)$ from the addition rule first, then subtract.

Mistake 2: Skipping the intersection step. …