Question

An insurance company insured 4000 doctors, 8000 teachers and 12000 engineers. The probabilities of a doctor, a teacher and an engineer dying before the age of 58 years are 0.01, 0.03 and 0.05 respectively. If one of the insured persons dies before the age of 58 years, find the probability that he is a doctor.

Answer 1

Let E₁ , E₂ and E₃ be the events that the person is a doctor, teacher and engineer respectively and A be the event that person dies before the age of 58 years.

\(P(E_1)=\frac{4000}{4000+8000+12000}=\frac{4000}{24000}=\frac{1}{6}\)

Similarly, 

\(P(E_2)=\frac{1}{3},P(E_3)=\frac{1}{2}\)

and P(A/E₁) = 0.01, P(A/E₂) = 0.03 and P(A/E₃) = 0.05

By Bayes' theorem,

\(P(\frac{E_1}{A})=\frac{P(E_1).P(A/E_1)}{P(E_1).P(\frac{A}{E_1})+P(E_2).P(\frac{A}{E_2})+P(E_3).P(\frac{A}{E_3})}\)