Probability
Conditional Probability
Grade None
Question:
<p><strong>For Problems 10–12:</strong> Let \(S\) and \(T\) are two events defined on a sample space with probabilities \(P(S) = 0.5\), \(P(T) = 0.69\), \(P(S/T) = 0.5\).</p><p>The value of \(P(S \cap T)\) is</p>
<p>(1) 0.3450</p>
<p>(2) 0.2500</p>
<p>(3) 0.6900</p>
<p>(4) 0.350</p>
Step-by-Step Solution
Key Concept: Use the conditional probability formula P(S|T) = P(S ∩ T)/P(T) to find the intersection probability by rearranging to P(S ∩ T) = P(S|T) × P(T).
<p><strong>Step 1:</strong> Recall the conditional probability formula:</p><p>P(S|T) = P(S ∩ T)/P(T)</p><p><strong>Step 2:</strong> Rearrange to find P(S ∩ T):</p><p>P(S ∩ T) = P(S|T) × P(T)</p><p><strong>Step 3:</strong> Substitute the given values:</p><p>P(S ∩ T) = 0.5 × 0.69 = 0.345</p><p><strong>Step 4:</strong> Verify this makes sense: P(S ∩ T) = 0.345 should be ≤ min(P(S), P(T)) = min(0.5, 0.69) = 0.5 ✓</p><p>∴ Answer: P(S ∩ T) = 0.345 (or 345/1000 = 69/200)</p>
Correct Answer: A