Mathematics · Probability

JEE Main 2025 — 8 April, Evening Shift — Question 33

If $A$ and $B$ are two events such that $P(A)=0.7, P(B)$ $=0.4$ and $P(A \cap \bar{B})=0.5$ , where $\bar{B}$ denotes the complement of $B$ , then $P(B \mid(A \cup \bar{B}))$ is equal to

Option A:
$\frac{1}{2}$
Option B:
$\frac{1}{4}$
Correct
Option C:
$\frac{1}{3}$
Option D:
$\frac{1}{6}$

Answer: B

Step-by-step solution

$P(A)=0.7$

$P(B)=0.4$

$P\left(A \cap B^{C}\right)=0.5$

$P\left(B / A \cup B^{C}\right)=\frac{P\left(B \cap\left(A \cup B^{C}\right)\right)}{P\left(A \cup B^{C}\right)}$

$=\frac{P(A \cap B)}{P\left(A \cup B^{C}\right)}$

Using Venn diagram

$\Rightarrow \frac{P(A \cap B)}{P\left(A \cup B^{C}\right)}=\frac{0.2}{0.5+0.2+0.1}=\frac{2}{8}=\frac{1}{4}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 8 April, Evening Shift
Subject: Mathematics
Chapter: Probability
Topic: Conditional Probability and Multiplication Theorem