Probability
Event and Algebra of Events
Grade 12
Question:
<p>If <i>A</i> and <i>B</i> are two events such that \(P(A) = 3/4\) and \(P(B) = 5/8\), then</p>
<p>(1) \(P(A \cup B) \geq 3/4\)</p>
<p>(2) \(P(A' \cap B) \leq 1/4\)</p>
<p>(3) \(1/8 \leq P(A \cap B') \leq 3/8\)</p>
<p>(4) \(3/8 \leq P(A \cap B) \leq 5/8\)</p>
Step-by-Step Solution
Key Concept: Use the fundamental probability bounds: P(A∩B) ≤ min(P(A), P(B)) and P(A∩B) ≥ P(A) + P(B) - 1, combined with P(A∪B) = P(A) + P(B) - P(A∩B) to determine valid ranges for probabilities of intersections and unions.
<p><strong>Step 1:</strong> Find bounds for P(A∩B).</p><p>Lower bound: P(A∩B) ≥ P(A) + P(B) - 1 = 3/4 + 5/8 - 1 = 6/8 + 5/8 - 8/8 = 3/8</p><p>Upper bound: P(A∩B) ≤ min(P(A), P(B)) = min(3/4, 5/8) = 5/8</p><p>Therefore: <strong>3/8 ≤ P(A∩B) ≤ 5/8</strong></p><p><strong>Step 2:</strong> Find bounds for P(A∪B) using P(A∪B) = P(A) + P(B) - P(A∩B).</p><p>When P(A∩B) = 3/8 (minimum): P(A∪B) = 3/4 + 5/8 - 3/8 = 6/8 + 5/8 - 3/8 = 8/8 = 1 (maximum)</p><p>When P(A∩B) = 5/8 (maximum): P(A∪B) = 3/4 + 5/8 - 5/8 = 3/4 (minimum)</p><p>Therefore: <strong>3/4 ≤ P(A∪B) ≤ 1</strong></p><p><strong>Step 3:</strong> Verify which statements are valid:</p><p>1. P(A∩B) = 1/2 ✓ (lies in [3/8, 5/8])</p><p>2. P(A∪B) = 7/8 ✓ (lies in [3/4, 1])</p><p>3. P(A'∩B) = P(B) - P(A∩B) = 5/8 - 1/2 = 1/8 ✓ (valid for P(A∩B) = 1/2)</p><p>4. P(A∩B') = P(A) - P(A∩B) = 3/4 - 1/2 = 1/4 ✓ (valid for P(A∩B) = 1/2)</p><p>∴ Answer: 1, 2, 3, 4 (All statements are possible)</p>
Correct Answer: 1,2,3,4