Probability
Independent Events
Grade 12
Question:
<p>Events <i>A</i> and <i>B</i> are such that \(P(A) = 1/2\), \(P(B) = 7/12\) and \(P(\text{not } A \text{ or not } B) = 1/4\). State whether <i>A</i> and <i>B</i> are independent?</p>
<p>A and B are independent</p>
<p>A and B are not independent</p>
<p>Cannot be determined</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Events are independent iff P(A ∩ B) = P(A)·P(B). Use De Morgan's law on the complement: P(A' ∪ B') = 1 - P(A ∩ B), then check the independence condition.
<p><strong>Step 1:</strong> Use De Morgan's law: P(A' ∪ B') = P((A ∩ B)') = 1 - P(A ∩ B)</p><p><strong>Step 2:</strong> Given P(A' ∪ B') = 1/4, so: 1 - P(A ∩ B) = 1/4 → P(A ∩ B) = 3/4</p><p><strong>Step 3:</strong> For independence, check if P(A ∩ B) = P(A)·P(B):</p><p>P(A)·P(B) = (1/2)·(7/12) = 7/24</p><p><strong>Step 4:</strong> Since P(A ∩ B) = 3/4 ≠ 7/24, the events are <strong>NOT independent</strong>.</p><p>∴ Answer: B</p>
Correct Answer: B