Probability
Independence of Events
Grade 12
Question:
<p>If \(P(A) = \frac{2}{3}\), \(P(B) = \frac{1}{2}\) and \(P(A \cup B) = \frac{5}{6}\) then the events \(A\) and \(B\) are</p>
<p>(a) mutually exclusive</p>
<p>(b) independent</p>
<p>(c) independent as well as mutually exclusive</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Use the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B) to find P(A ∩ B), then check if P(A ∩ B) = P(A)·P(B) for independence or if one event is contained in another for mutual exclusivity.
<p><strong>Step 1:</strong> Use the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B)</p><p>5/6 = 2/3 + 1/2 - P(A ∩ B)</p><p>5/6 = 4/6 + 3/6 - P(A ∩ B)</p><p>5/6 = 7/6 - P(A ∩ B)</p><p>P(A ∩ B) = 7/6 - 5/6 = 2/6 = 1/3</p><p><strong>Step 2:</strong> Check for independence: P(A) × P(B) = (2/3) × (1/2) = 2/6 = 1/3</p><p>Since P(A ∩ B) = P(A) × P(B) = 1/3, the events are <strong>independent</strong>.</p><p><strong>Step 3:</strong> Verify they are not mutually exclusive: P(A ∩ B) = 1/3 ≠ 0, so they can occur together.</p><p>∴ Answer: B (Independent events)</p>
Correct Answer: B