JEE Main 2025 Apr 8 Shift 2, Mathematics Q4: Conditional Probability

JEE Main2025Apr 8, Shift 2Mathematics

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

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

Solution

Given $P (A) = 0.7$ , $P (B) = 0.4$ , $P (\overline{B}) = 0.6$ , and $P (A \cap \overline{B}) = 0.5$ .

Compute $P (A \cap B) = P (A) - P (A \cap \overline{B}) = 0.7 - 0.5 = 0.2$ .

Compute $P (A \cup \overline{B}) = P (A) + P (\overline{B}) - P (A \cap \overline{B}) = 0.7 + 0.6 - 0.5 = 0.8$ .

The intersection needed for the conditional probability: $B \cap (A \cup \overline{B}) = (B \cap A) \cup (B \cap \overline{B}) = A \cap B$ , since $B \cap \overline{B} = \emptyset$ .

Therefore $P (B ∣ A \cup \overline{B}) = \frac{P ( A \cap B )}{P ( A \cup B )} = \frac{0.2}{0.8} = \frac{1}{4}$ , matching option (1).

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JEE Main 2025 Apr 8 Shift 2, Mathematics Q4: Conditional Probability

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