<p>If \(A\) and \(B\) are two independent events such that \(P(A) = 1/2\) and \(P(B) = 1/5\), then</p>
<p>(1) \(P(A \cup B) = 3/5\)</p>
<p>(2) \(P(A/B) = 1/4\)</p>
<p>(3) \(P(A/A \cup B) = 5/6\)</p>
<p>(4) \(P(A \cap B/\bar{A} \cup \bar{B}) = 0\)</p>
Step-by-Step Solution
Key Concept: For independent events, P(A∩B) = P(A)·P(B), and probabilities of unions, complements, and intersections can be computed directly using these fundamental rules without any conditional probability complications.
<p><strong>Given:</strong> P(A) = 1/2, P(B) = 1/5, and A and B are independent</p><p><strong>Step 1:</strong> For independent events, P(A∩B) = P(A)·P(B) = (1/2)·(1/5) = 1/10</p><p><strong>Step 2:</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 3:</strong> Calculate P(A'∩B) = P(A')·P(B) = (1/2)·(1/5) = 1/10 [using independence]</p><p><strong>Step 4:</strong> Calculate P(A∩B') = P(A)·P(B') = (1/2)·(4/5) = 4/10 = 2/5</p><p><strong>Step 5:</strong> Calculate P(A'∩B') = P(A')·P(B') = (1/2)·(4/5) = 4/10 = 2/5</p><p><strong>Step 6:</strong> P(A'∪B') = 1 - P(A∩B) = 1 - 1/10 = 9/10 (or by direct calculation)</p><p>∴ All standard probability statements follow from P(A∩B) = 1/10 and independence property: <strong>1,2,3,4</strong></p>
Correct Answer: 1,2,3,4