<p>If \(P(A \cap B) = \dfrac{1}{2}\), \(P(\bar{A} \cap \bar{B}) = \dfrac{1}{3}\), \(P(A) = p\), \(P(B) = 2p\), then find the value of \(p\).</p>
Step-by-Step Solution
Key Concept: Use the fact that the four mutually exclusive events A∩B, A∩B̄, Ā∩B, and Ā∩B̄ partition the sample space (sum to 1), then apply the definitions P(A) = P(A∩B) + P(A∩B̄) and P(B) = P(A∩B) + P(Ā∩B).
<p><strong>Step 1:</strong> The sample space is partitioned into four disjoint events: A∩B, A∩B̄, Ā∩B, and Ā∩B̄.</p><p>Therefore: P(A∩B) + P(A∩B̄) + P(Ā∩B) + P(Ā∩B̄) = 1</p><p><strong>Step 2:</strong> Substitute known values: 1/2 + P(A∩B̄) + P(Ā∩B) + 1/3 = 1</p><p>This gives: P(A∩B̄) + P(Ā∩B) = 1 - 1/2 - 1/3 = 1/6</p><p><strong>Step 3:</strong> Use the definitions of marginal probabilities:</p><p>P(A) = P(A∩B) + P(A∩B̄) ⟹ p = 1/2 + P(A∩B̄)</p><p>P(B) = P(A∩B) + P(Ā∩B) ⟹ 2p = 1/2 + P(Ā∩B)</p><p><strong>Step 4:</strong> From Step 3: P(A∩B̄) = p - 1/2 and P(Ā∩B) = 2p - 1/2</p><p><strong>Step 5:</strong> Substitute into the equation from Step 2:</p><p>(p - 1/2) + (2p - 1/2) = 1/6</p><p>3p - 1 = 1/6</p><p>3p = 7/6</p><p>∴ p = 7/18</p>
Correct Answer: 7/18