Probability
Independent Events
Grade 12
Question:
<p>Let \(A\) and \(B\) be two events such that \(P(\overline{A \cup B}) = 1/6\), \(P(A \cap B) = 1/4\) and \(P(\bar{A}) = 1/4\), where \(\bar{A}\) stands for complement of event \(A\). Then events \(A\) and \(B\) are</p>
<p>equally likely but not independent</p>
<p>equally likely and mutually exclusive</p>
<p>mutually exclusive and independent</p>
<p>independent but not equally likely</p>
Step-by-Step Solution
Key Concept: Use the complement rule P(A∪B) = 1 - P(A̅∪B̅) and the inclusion-exclusion principle to find P(B), then check independence by verifying if P(A∩B) = P(A)·P(B).
<p><strong>Step 1:</strong> Find P(A∪B) using complement.</p><p>P(A̅∪B̅) = 1/6, so P(A∪B) = 1 - P(A̅∪B̅) = 1 - 1/6 = 5/6</p><p><strong>Step 2:</strong> Find P(A) from complement.</p><p>P(A̅) = 1/4, therefore P(A) = 1 - 1/4 = 3/4</p><p><strong>Step 3:</strong> Use inclusion-exclusion to find P(B).</p><p>P(A∪B) = P(A) + P(B) - P(A∩B)</p><p>5/6 = 3/4 + P(B) - 1/4</p><p>5/6 = 1/2 + P(B)</p><p>P(B) = 5/6 - 1/2 = 5/6 - 3/6 = 2/6 = 1/3</p><p><strong>Step 4:</strong> Check independence: P(A)·P(B) = 3/4 × 1/3 = 1/4 = P(A∩B) ✓</p><p>∴ Events A and B are <strong>independent</strong> (Answer: D)</p>
Correct Answer: D