An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company’s insured drivers:

Age of Driver‌	Probability of Accident	Portion of Company’s Insured Drivers
16-20	0.06	0.08
21-30	0.03	0.15
31-65	0.02	0.49
66-99	0.04	0.28

A randomly selected driver that the company insures has an accident. Calculate the probability that the driver was age 16-20.

• 0.13
• 0.16
• 0.19
• 0.23
• 0.40

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: B

Apply Bayes’ Formula. Let

A = Event of an accident

B₁ = Event the driver’s age is in the range 16-20

B₂ = Event the driver’s age is in the range 21-30

B₃ = Event the driver’s age is in the range 30-65

B₄ = Event the driver’s age is in the range 66-99

Then

[[math]] \begin{align*} \operatorname{P}(B_1 | A) &= \frac{\operatorname{P}( A | B_1 ) \operatorname{P}( B_1 )}{\operatorname{P}( A | B_1 ) \operatorname{P}( B_1 ) + \operatorname{P}( A |B_2 ) \operatorname{P}( B_2 ) + \operatorname{P}( A | B_3 ) \operatorname{P}( B_3 ) + \operatorname{P}( A | B_4 ) \operatorname{P}( B_4 )} \\ &= \frac{( 0.06 )( 0.08)}{( 0.06 )( 0.08) + ( 0.03)( 0.15) + ( 0.02 )( 0.49 ) + ( 0.04 )( 0.28)} \\ &= 0.1584. \end{align*} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.