Probability
Bayes' Theorem
Grade 12
Question:
<p>An insurance company insured 2000 scooter drivers, 4000 car drivers, and 6000 truck drivers. The probability of accidents are 0.01, 0.03, and 0.15, respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?</p>
<p>\(\dfrac{1}{52}\)</p>
<p>\(\dfrac{3}{52}\)</p>
<p>\(\dfrac{15}{52}\)</p>
<p>\(\dfrac{1}{26}\)</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem: P(scooter|accident) = P(accident|scooter) × P(scooter) / P(accident). The total probability of accident is found by summing over all driver types weighted by their probabilities.
<p><strong>Step 1:</strong> Find the prior probabilities of each driver type.</p><p>Total insured persons = 2000 + 4000 + 6000 = 12000</p><p>P(Scooter) = 2000/12000 = 1/6</p><p>P(Car) = 4000/12000 = 1/3</p><p>P(Truck) = 6000/12000 = 1/2</p><p><strong>Step 2:</strong> Use the law of total probability to find P(Accident).</p><p>P(Accident) = P(A|S)·P(S) + P(A|C)·P(C) + P(A|T)·P(T)</p><p>P(Accident) = (0.01)(1/6) + (0.03)(1/3) + (0.15)(1/2)</p><p>P(Accident) = 1/600 + 1/100 + 3/20</p><p>P(Accident) = 1/600 + 6/600 + 90/600 = 97/600</p><p><strong>Step 3:</strong> Apply Bayes' theorem to find P(Scooter|Accident).</p><p>P(S|A) = [P(A|S)·P(S)] / P(A)</p><p>P(S|A) = [(0.01)(1/6)] / (97/600)</p><p>P(S|A) = (1/600) / (97/600)</p><p>P(S|A) = 1/97</p><p>∴ Answer: A</p>
Correct Answer: A