Probability
Basic Probability
Grade 12
Question:
<p>If <em>A</em> and <em>B</em> are events such that \(P(A \cup B) = 3/4\), \(P(A \cap B) = 1/4\), and \(P(A^c) = 2/3\), then find</p><p>(a) \(P(A)\)</p><p>(b) \(P(B)\)</p><p>(c) \(P(A \cap B^c)\)</p><p>(d) \(P(A^c \cap B)\)</p>
<p>\(P(A) = 1/3\)</p>
<p>\(P(B) = 2/3\)</p>
<p>\(P(A \cap B^c) = 1/12\)</p>
<p>\(P(A^c \cap B) = 5/12\)</p>
Step-by-Step Solution
Key Concept: Use the fundamental probability formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B) and the complement rule P(A) = 1 - P(A^c) to systematically solve for all probabilities. The key is recognizing that all four parts are interconnected through these two relationships.
<p><strong>Step 1: Find P(A) using complement</strong></p><p>P(A^c) = 2/3 ⟹ P(A) = 1 - 2/3 = <strong>1/3</strong></p><p><strong>Step 2: Find P(B) using addition rule</strong></p><p>P(A ∪ B) = P(A) + P(B) - P(A ∩ B)</p><p>3/4 = 1/3 + P(B) - 1/4</p><p>P(B) = 3/4 - 1/3 + 1/4 = 9/12 - 4/12 + 3/12 = <strong>8/12 = 2/3</strong></p><p><strong>Step 3: Find P(A ∩ B^c)</strong></p><p>P(A ∩ B^c) = P(A) - P(A ∩ B) = 1/3 - 1/4 = 4/12 - 3/12 = <strong>1/12</strong></p><p><strong>Step 4: Find P(A^c ∩ B)</strong></p><p>P(A^c ∩ B) = P(B) - P(A ∩ B) = 2/3 - 1/4 = 8/12 - 3/12 = <strong>5/12</strong></p><p><strong>Verification:</strong> P(A ∩ B) + P(A ∩ B^c) + P(A^c ∩ B) + P(A^c ∩ B^c) = 1/4 + 1/12 + 5/12 + 2/12 = 1 ✓</p><p>∴ <strong>(a) P(A) = 1/3, (b) P(B) = 2/3, (c) P(A ∩ B^c) = 1/12, (d) P(A^c ∩ B) = 5/12</strong></p>
Correct Answer: A,B,C,D