Probability
Addition Theorem of Probability
Grade 12
Question:
<p><strong>For Problems 10–12:</strong> Let \(S\) and \(T\) are two events defined on a sample space with probabilities \(P(S) = 0.5\), \(P(T) = 0.69\), \(P(S/T) = 0.5\).</p><p>The value of \(P(S \text{ or } T)\) is</p>
<p>(1) 0.6900</p>
<p>(2) 1.19</p>
<p>(3) 0.8450</p>
<p>(4) 0</p>
Step-by-Step Solution
Key Concept: Use the conditional probability formula P(S/T) = P(S∩T)/P(T) to find P(S∩T), then apply the addition rule P(S∪T) = P(S) + P(T) - P(S∩T).
<p><strong>Step 1:</strong> Find P(S∩T) using conditional probability formula.</p><p>Given: P(S/T) = P(S∩T)/P(T)</p><p>0.5 = P(S∩T)/0.69</p><p>P(S∩T) = 0.5 × 0.69 = 0.345</p><p><strong>Step 2:</strong> Apply the addition rule for probability of union of two events.</p><p>P(S∪T) = P(S) + P(T) - P(S∩T)</p><p>P(S∪T) = 0.5 + 0.69 - 0.345</p><p>P(S∪T) = 1.19 - 0.345 = 0.845</p><p><strong>Step 3:</strong> Verify the result is between 0 and 1, and that P(S∪T) ≥ max{P(S), P(T)}.</p><p>0.845 ≥ 0.69 ✓</p><p>∴ Answer: P(S or T) = 0.845</p>
Correct Answer: C