Probability
Independent Events
Grade None
Question:
<p>If \(A\) and \(B\) are two independent events such that \(P(A) = 1/2\), \(P(B) = 1/5\), then</p>
<p>\(P(A/B) = 1/2\)</p>
<p>\(P\!\left(\dfrac{A}{A \cup B}\right) = \dfrac{5}{6}\)</p>
<p>\(P\!\left(\dfrac{A \cap B}{A' \cup B'}\right) = 0\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: For independent events, P(A∩B) = P(A)·P(B), and P(A∪B) = P(A) + P(B) - P(A∩B). Also recognize that P(A'∩B) = P(A')·P(B) when events are independent.
<p><strong>Step 1:</strong> Identify given information: P(A) = 1/2, P(B) = 1/5, A and B are independent events.</p><p><strong>Step 2:</strong> For independent events, P(A∩B) = P(A)·P(B) = (1/2)·(1/5) = 1/10</p><p><strong>Step 3:</strong> Calculate P(A∪B) = P(A) + P(B) - P(A∩B) = 1/2 + 1/5 - 1/10 = 5/10 + 2/10 - 1/10 = 6/10 = 3/5</p><p><strong>Step 4:</strong> For complement and intersection: P(A'∩B) = P(A')·P(B) = (1/2)·(1/5) = 1/10, where P(A') = 1 - P(A) = 1/2</p><p><strong>Step 5:</strong> P(A∪B)' = 1 - P(A∪B) = 1 - 3/5 = 2/5 (probability that neither occurs)</p><p>∴ Answer: A</p>
Correct Answer: A