Probability
Conditional Probability
Grade 12
Question:
<p>If \(P(A) = 0.8\), \(P(B) = 0.5\), and \(P(B/A) = 0.4\), find \(P(A \cap B)\).</p>
<p>\(0.20\)</p>
<p>\(0.32\)</p>
<p>\(0.40\)</p>
<p>\(0.64\)</p>
Step-by-Step Solution
Key Concept: Use the conditional probability formula P(B/A) = P(A ∩ B)/P(A) to directly find the intersection by rearranging: P(A ∩ B) = P(B/A) × P(A).
<p><strong>Step 1:</strong> Recall the definition of conditional probability: P(B/A) = P(A ∩ B)/P(A)</p><p><strong>Step 2:</strong> Rearrange to solve for P(A ∩ B): P(A ∩ B) = P(B/A) × P(A)</p><p><strong>Step 3:</strong> Substitute the given values: P(A ∩ B) = 0.4 × 0.8 = 0.32</p><p><strong>Step 4:</strong> Verify this makes sense: P(A ∩ B) = 0.32 < P(B) = 0.5 ✓ (the intersection cannot exceed either event's probability)</p><p>∴ Answer: <strong>P(A ∩ B) = 0.32</strong></p>
Correct Answer: B