Probability
Independent Events
Grade 12
Question:
<p>Let \(A\), \(B\), \(C\) be three mutually independent events. Consider the two statements \(S_1\) and \(S_2\).<br>\(S_1\): \(A\) and \(B \cup C\) are independent.<br>\(S_2\): \(A\) and \(B \cap C\) are independent.<br>Then</p>
<p>(1) both \(S_1\) and \(S_2\) are true</p>
<p>(2) only \(S_1\) is true</p>
<p>(3) only \(S_2\) is true</p>
<p>(4) neither \(S_1\) nor \(S_2\) is true</p>
Step-by-Step Solution
Key Concept: For mutually independent events A, B, C: use P(A∩(B∪C)) = P(A)·P(B∪C) and P(A∩(B∩C)) = P(A)·P(B∩C) by expanding with independence properties. Both statements follow from the multiplicative rule for independent events applied to composite events.
<p><strong>Step 1:</strong> For mutually independent events A, B, C: P(A∩B) = P(A)P(B), P(A∩C) = P(A)P(C), P(B∩C) = P(B)P(C), and P(A∩B∩C) = P(A)P(B)P(C).</p><p><strong>Step 2 (Check S₁ - A and B∪C independent):</strong> P(A∩(B∪C)) = P((A∩B)∪(A∩C)) = P(A∩B) + P(A∩C) - P(A∩B∩C) = P(A)P(B) + P(A)P(C) - P(A)P(B)P(C) = P(A)[P(B) + P(C) - P(B)P(C)] = P(A)·P(B∪C). ✓ S₁ is TRUE.</p><p><strong>Step 3 (Check S₂ - A and B∩C independent):</strong> P(A∩(B∩C)) = P(A∩B∩C) = P(A)P(B)P(C) = P(A)·P(B∩C). ✓ S₂ is TRUE.</p><p><strong>Step 4:</strong> Both statements are true because mutual independence allows us to factorize probabilities through all combinations of the events.</p><p>∴ Both S₁ and S₂ are true.</p>
Correct Answer: A