Probability
Conditional Probability
Grade 12
Question:
<p>Let \(A\) and \(B\) be two events such that \(P(A \cap B') = 0.20\), \(P(A' \cap B) = 0.15\), \(P(A' \cap B') = 0.1\), then \(P(A/B)\) is equal to</p>
<p>(1) 11/14</p>
<p>(2) 2/11</p>
<p>(3) 2/7</p>
<p>(4) 1/7</p>
Step-by-Step Solution
Key Concept: First find P(A∩B) using the fact that all four mutually exclusive regions partition the sample space: P(A∩B) + P(A∩B') + P(A'∩B) + P(A'∩B') = 1. Then apply the conditional probability formula P(A/B) = P(A∩B)/P(B).
<p><strong>Step 1:</strong> Find P(A∩B) using the partition property.</p><p>Since the four regions {A∩B, A∩B', A'∩B, A'∩B'} partition the sample space:</p><p>P(A∩B) + P(A∩B') + P(A'∩B) + P(A'∩B') = 1</p><p>P(A∩B) + 0.20 + 0.15 + 0.10 = 1</p><p>P(A∩B) = 1 - 0.45 = 0.55</p><p><strong>Step 2:</strong> Find P(B).</p><p>P(B) = P(A∩B) + P(A'∩B) = 0.55 + 0.15 = 0.70</p><p><strong>Step 3:</strong> Apply conditional probability formula.</p><p>P(A/B) = P(A∩B)/P(B) = 0.55/0.70 = 55/70 = 11/14</p><p>∴ Answer: A</p>
Correct Answer: A