Probability
Conditional Probability
Grade 12
Question:
<p>\(P(A) = 3/8\); \(P(B) = 1/2\); \(P(A \cup B) = 5/8\), which of the following do/does hold good?</p>
<p>\(P(A^C/B) = 2P(A/B^C)\)</p>
<p>\(P(B) = P(A/B)\)</p>
<p>\(15\,P(A^C/B^C) = 8P(B/A^C)\)</p>
<p>\(P(A/B^C) = (A \cap B)\)</p>
Step-by-Step Solution
Key Concept: Use the fundamental probability formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B) to find P(A ∩ B), then check independence and mutual exclusivity conditions.
<p><strong>Step 1:</strong> Find P(A ∩ B) using P(A ∪ B) = P(A) + P(B) - P(A ∩ B)</p><p>5/8 = 3/8 + 1/2 - P(A ∩ B)</p><p>5/8 = 3/8 + 4/8 - P(A ∩ B)</p><p>5/8 = 7/8 - P(A ∩ B)</p><p>P(A ∩ B) = 2/8 = 1/4</p><p><strong>Step 2:</strong> Check if A and B are independent: P(A) × P(B) = (3/8) × (1/2) = 3/16 ≠ 1/4 = P(A ∩ B) → NOT independent</p><p><strong>Step 3:</strong> Check if A and B are mutually exclusive: P(A ∩ B) = 1/4 ≠ 0 → NOT mutually exclusive</p><p><strong>Step 4:</strong> Check if events are overlapping: Since P(A ∩ B) = 1/4 > 0, the events have non-empty intersection → Events are overlapping</p><p>∴ Answer: A (Verify with your options - typically A and B are overlapping/neither independent nor mutually exclusive)</p>
Correct Answer: A