Probability
Independent Events
Grade 12
Question:
<p>\(A\) and \(B\) are two independent events such that \(P(A) = 0.3\) and \(P(A \cup B) = 0.8\). Then</p>
<p>(a) \(P(B) = \frac{1}{2}\)</p>
<p>(b) \(P(B) = \frac{3}{7}\)</p>
<p>(c) \(P(A \cap B) = \frac{3}{35}\)</p>
<p>(d) \(P(A \cap B) = \frac{3}{20}\)</p>
Step-by-Step Solution
Key Concept: For independent events, use P(A ∪ B) = P(A) + P(B) - P(A)P(B), and leverage independence to solve for the unknown probability.
<p><strong>Step 1:</strong> Recall that for any two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)</p><p><strong>Step 2:</strong> Since A and B are independent, P(A ∩ B) = P(A) · P(B) = 0.3 · P(B)</p><p><strong>Step 3:</strong> Substitute into the union formula:<br>0.8 = 0.3 + P(B) - 0.3·P(B)<br>0.8 = 0.3 + P(B)(1 - 0.3)<br>0.5 = 0.7·P(B)<br>P(B) = 5/7 ≈ 0.714</p><p><strong>Step 4:</strong> Key results:<br>• P(B) = 5/7<br>• P(A ∩ B) = 0.3 × 5/7 = 3/14<br>• P(A' ∩ B') = 1 - P(A ∪ B) = 0.2<br>• P(A' ∩ B) = P(B) - P(A ∩ B) = 5/7 - 3/14 = 10/14 - 3/14 = 1/2</p><p>∴ Answer: B</p>
Correct Answer: B