Probability
Independent Events
Grade 12
Question:
<p>Let \(A\) and \(B\) be two events. Suppose \(P(A) = 0.4\), \(P(B) = p\), and \(P(A \cup B) = 0.7\). The value of \(p\) for which \(A\) and \(B\) are independent is</p>
<p>1/3</p>
<p>1/4</p>
<p>1/2</p>
<p>1/5</p>
Step-by-Step Solution
Key Concept: For independent events, P(A ∩ B) = P(A)·P(B). Use the union formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B) and substitute the independence condition to solve for p.
<p><strong>Step 1:</strong> Write the union formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)</p><p><strong>Step 2:</strong> For independent events, P(A ∩ B) = P(A) · P(B) = 0.4p</p><p><strong>Step 3:</strong> Substitute into the union formula: 0.7 = 0.4 + p - 0.4p</p><p><strong>Step 4:</strong> Simplify: 0.7 = 0.4 + p(1 - 0.4)</p><p>0.7 = 0.4 + 0.6p</p><p>0.3 = 0.6p</p><p>p = 0.5</p><p>∴ Answer: C (p = 0.5)</p>
Correct Answer: C