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