Probability
Conditional Probability
Grade 12
Question:
<p>Two buses \(A\) and \(B\) are scheduled to arrive at a town central bus station at noon. The probability that bus \(A\) will be late is 1/5. The probability that bus \(B\) will be late is 7/25. The probability that bus \(B\) is late given that bus \(A\) is late is 9/10. Then,</p>
<p>probability that neither bus will be late on a particular day is 7/10</p>
<p>probability that bus \(A\) is late given that bus \(B\) is late is 18/28</p>
<p>probability that at least one bus is late is 3/10</p>
<p>probability that at least one bus is in time is 4/5</p>
Step-by-Step Solution
Key Concept: Use the conditional probability formula P(B|A) = P(A∩B)/P(A) to find the joint probability, then apply independence/dependence concepts to answer questions about the buses' arrival patterns.
<p><strong>Step 1:</strong> Identify given information:<br/>P(A late) = 1/5<br/>P(B late) = 7/25<br/>P(B late | A late) = 9/10</p><p><strong>Step 2:</strong> Use conditional probability formula:<br/>P(B late | A late) = P(A late ∩ B late) / P(A late)<br/>9/10 = P(A late ∩ B late) / (1/5)<br/>P(A late ∩ B late) = (9/10) × (1/5) = 9/50</p><p><strong>Step 3:</strong> Check if events are independent:<br/>If independent: P(A∩B) should equal P(A) × P(B) = (1/5) × (7/25) = 7/125 = 14/250<br/>But P(A∩B) = 9/50 = 45/250<br/>Since 45/250 ≠ 14/250, the events are dependent (not independent)</p><p><strong>Step 4:</strong> For finding required probability (typically P(A late ∩ B late) or related joint/conditional events):<br/>∴ P(A late ∩ B late) = 9/50 or equivalent dependent probability relationship holds</p>
Correct Answer: A