Problem Statement
An insurance company insured $2000$ scooter drivers, $4000$ car drivers and $6000$ truck drivers. The accident probabilities are $0.01, 0.03$ and $0.15$ respectively. One driver meets with an accident. What is the probability that he is a scooter driver?
Verified Solution & Marking Scheme
Define Prior Probabilities
$P(E_1) = \frac{2000}{12000} = \frac{1}{6}, \, P(E_2) = \frac{1}{3}, \, P(E_3) = \frac{1}{2}$.
Conditional Accident Probabilities
$P(A|E_1) = 0.01 = \frac{1}{100}, \, P(A|E_2) = \frac{3}{100}, \, P(A|E_3) = \frac{15}{100}$.
Bayes Formula
$P(E_1|A) = \frac{\frac{1}{6}\cdot\frac{1}{100}}{\frac{1}{6}\cdot\frac{1}{100} + \frac{1}{3}\cdot\frac{3}{100} + \frac{1}{2}\cdot\frac{15}{100}} = \frac{\frac{1}{600}}{\frac{1 + 6 + 45}{600}} = \frac{1}{52}$