Q.Let $P(A) = \dfrac{7}{13}$, $P(B) = \dfrac{9}{13}$ and $P(A \cap B) = \dfrac{4}{13}$. Then $P(A' \mid B)$ is equal to
(A) $\dfrac{6}{13}$
(B) $\dfrac{4}{13}$
(C) $\dfrac{4}{9}$
(D) $\dfrac{5}{9}$

Concept: Conditional Probability — $P(A' \mid B) = \frac{P(A' \cap B)}{P(B)}$.

Step 1: $A' \cap B$ means "not $A$ but $B$". From the Venn diagram:

$P(A' \cap B) = P(B) - P(A \cap B)$.

Step 2: Substitute the given values:

$P(A' \cap B) = \frac{9}{13} - \frac{4}{13} = \frac{5}{13}$.

Step 3: Apply the conditional formula: …

We are asked for $P(A' \mid B)$ — the probability that $A$ does not happen, given that $B$ has happened. This is a classic conditional probability problem.

Why this approach works:

Conditional probability shrinks the sample space to the event we are conditioning on (here, $B$). Within that restricted space, we want the portion where $A$ is false. That portion is exactly $B$ minus the part where $A$ and $B$ both occur — i.e., $B \setminus (A \cap B)$. So the numerator becomes $P(B) - P(A \cap B)$.

Let’s work through it step by step.

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

Here, $X = A'$ and $Y = B$, so

2. Find $P(A' \cap B)$: The event $A' \cap B$ means “$B$ occurs but $A$ does not.” This is exactly $B$ minus the overlap $A \cap B$. Since $A \cap B$ is a subset of $B$, we have

3. Substitute the given values: $P(B) = \frac{9}{13}$ and $P(A \cap B) = \frac{4}{13}$. So

4. Now compute the conditional probability: …

Method: Conditional probability of a complement, same condition

Use this for $P(A'\mid B)$ — the chance $A$ does not occur given $B$.

Steps

Step 1: Express the numerator as the part of $B$ outside $A$.

Step 2: Divide by the condition's probability.

Step 3: Simplify (and cross-check). …

Common Mistakes

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

Why it's wrong: $A'\cap B$ is the part of $B$ outside $A$, so it equals $P(B)-P(A\cap B)$. Correct approach: subtract the overlap from $P(B)$, keeping everything inside $B$.

Mistake 2: Dividing by $1$ instead of $P(B)$. …