Probability
Conditional Probability
Grade 12
Question:
<p>Consider the following two statements:</p><p><strong>Statement-1:</strong> For events \(A\) and \(E\),<br>\[P(E/A) \geq P(A/E) \cdot P(E)\]</p><p><strong>Statement-2:</strong> \(P(A/E) \geq P(A \cap E)\)</p><p>Which of the following is correct?</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1.</p>
<p>Statement-1 is true, Statement-2 is false.</p>
<p>Statement-1 is false, Statement-2 is true.</p>
Step-by-Step Solution
Key Concept: Use conditional probability definition P(E/A) = P(A∩E)/P(A) and Bayes' theorem P(A/E) = P(A∩E)/P(E) to verify each statement algebraically. Statement-1 requires checking if P(A∩E)/P(A) ≥ P(A∩E)·P(E)/P(E), and Statement-2 requires comparing P(A∩E)/P(E) with P(A∩E).
<p><strong>Step 1: Verify Statement-1</strong></p><p>Statement-1: P(E/A) ≥ P(A/E)·P(E)</p><p>LHS: P(E/A) = P(A∩E)/P(A)</p><p>RHS: P(A/E)·P(E) = [P(A∩E)/P(E)]·P(E) = P(A∩E)</p><p>So Statement-1 becomes: P(A∩E)/P(A) ≥ P(A∩E)</p><p>This simplifies to: 1/P(A) ≥ 1, or P(A) ≤ 1 ✓ (Always TRUE)</p><p><strong>Step 2: Verify Statement-2</strong></p><p>Statement-2: P(A/E) ≥ P(A∩E)</p><p>P(A/E) = P(A∩E)/P(E)</p><p>Since P(E) ≤ 1 (and P(E) > 0), dividing P(A∩E) by P(E) gives:</p><p>P(A∩E)/P(E) ≥ P(A∩E) ✓ (Always TRUE)</p><p><strong>Step 3: Conclusion</strong></p><p>Both Statement-1 and Statement-2 are true.</p><p>∴ Answer: B (Both statements are correct)</p>
Correct Answer: B