Probability
Independent Events
Grade 12
Question:
<p>Events \(A\) and \(C\) are independent. If the probabilities relating \(A\), \(B\), and \(C\) are \(P(A) = 1/5\), \(P(B) = 1/6\); \(P(A \cap C) = 1/20\); \(P(B \cup C) = 3/8\). Then</p>
<p>events \(B\) and \(C\) are independent</p>
<p>events \(B\) and \(C\) are mutually exclusive</p>
<p>events \(B\) and \(C\) are neither independent nor mutually exclusive</p>
<p>events \(B\) and \(C\) are equiprobable</p>
Step-by-Step Solution
Key Concept: Use independence of A and C to find P(C), then apply inclusion-exclusion on B∪C to find P(B∩C), which determines whether B and C are independent.
<p><strong>Step 1: Find P(C) using independence of A and C</strong></p><p>Since A and C are independent: P(A∩C) = P(A)·P(C)</p><p>1/20 = (1/5)·P(C)</p><p>P(C) = 1/4</p><p><strong>Step 2: Find P(B∩C) using inclusion-exclusion on B∪C</strong></p><p>P(B∪C) = P(B) + P(C) - P(B∩C)</p><p>3/8 = 1/6 + 1/4 - P(B∩C)</p><p>3/8 = 2/12 + 3/12 - P(B∩C)</p><p>3/8 = 5/12 - P(B∩C)</p><p>P(B∩C) = 5/12 - 3/8 = 10/24 - 9/24 = 1/24</p><p><strong>Step 3: Check if B and C are independent</strong></p><p>For independence: P(B∩C) should equal P(B)·P(C)</p><p>P(B)·P(C) = (1/6)·(1/4) = 1/24 ✓</p><p>Since P(B∩C) = P(B)·P(C), B and C are independent.</p><p>∴ Answer: C</p>
Correct Answer: C