<p>If \(A\) and \(B\) are two independent events such that \(P(\bar{A} \cap B) = 2/15\) and \(P(A \cap \bar{B}) = 1/6\), then \(P(B)\) is</p>
Step-by-Step Solution
Key Concept: Use independence to express conditional probabilities in terms of individual probabilities: P(Ā ∩ B) = P(Ā)·P(B) and P(A ∩ B̄) = P(A)·P(B̄), then solve the system of equations.
<p><strong>Step 1:</strong> Use independence property. Since A and B are independent:</p><p>P(Ā ∩ B) = P(Ā) · P(B) = (1 - P(A)) · P(B) = 2/15</p><p>P(A ∩ B̄) = P(A) · P(B̄) = P(A) · (1 - P(B)) = 1/6</p><p><strong>Step 2:</strong> Let P(A) = a and P(B) = b. Then:</p><p>(1 - a)b = 2/15 ... (i)</p><p>a(1 - b) = 1/6 ... (ii)</p><p><strong>Step 3:</strong> From (i): b - ab = 2/15</p><p>From (ii): a - ab = 1/6</p><p><strong>Step 4:</strong> Subtract equation (ii) from (i):</p><p>b - a = 2/15 - 1/6 = 4/30 - 5/30 = -1/30</p><p>So: b = a - 1/30 ... (iii)</p><p><strong>Step 5:</strong> Substitute (iii) into (ii):</p><p>a(1 - a + 1/30) = 1/6</p><p>a(31/30 - a) = 1/6</p><p>31a/30 - a² = 1/6</p><p>180a² - 186a + 5 = 0</p><p><strong>Step 6:</strong> Using quadratic formula or factoring:</p><p>(6a - 5)(30a - 1) = 0</p><p>So a = 5/6 or a = 1/30</p><p><strong>Step 7:</strong> If a = 5/6, then b = 5/6 - 1/30 = 25/30 - 1/30 = 24/30 = 4/5</p><p>If a = 1/30, then b = 1/30 - 1/30 = 0 (invalid since P(Ā ∩ B) ≠ 0)</p><p>∴ <strong>P(B) = 4/5</strong></p>
Correct Answer: A