<p>Let \(A\) and \(B\) are events of an experiment and \(P(A) = 1/4\), \(P(A \cup B) = 1/2\), then value of \(P(B/A^c)\) is</p>
Step-by-Step Solution
Key Concept: Use the formula P(B|A^c) = P(B ∩ A^c)/P(A^c), which requires finding P(B ∩ A^c) using the relationship P(A ∪ B) = P(A) + P(B) - P(A ∩ B) and the partition B = (B ∩ A) ∪ (B ∩ A^c).
<p><strong>Step 1:</strong> From P(A ∪ B) = P(A) + P(B) - P(A ∩ B), we have:</p><p>1/2 = 1/4 + P(B) - P(A ∩ B)</p><p>∴ P(B) - P(A ∩ B) = 1/4 ... (i)</p><p><strong>Step 2:</strong> Note that B = (B ∩ A) ∪ (B ∩ A^c) (disjoint union), so:</p><p>P(B) = P(A ∩ B) + P(B ∩ A^c)</p><p>∴ P(B ∩ A^c) = P(B) - P(A ∩ B) = 1/4 [from equation (i)]</p><p><strong>Step 3:</strong> Calculate P(A^c) = 1 - P(A) = 1 - 1/4 = 3/4</p><p><strong>Step 4:</strong> Apply conditional probability formula:</p><p>P(B|A^c) = P(B ∩ A^c)/P(A^c) = (1/4)/(3/4) = <strong>1/3</strong></p><p>∴ Answer: A</p>
Correct Answer: A