An insurance company insures red and green cars. An actuary compiles the following data:

Color of Car	Red	Green
Number insured	300	700
Probability an accident occurs	0.10	0.05
Probability that the claim exceeds the deductible, given an accident occurs from this group	0.9	0.8

The actuary randomly picks a claim from all claims that exceed the deductible. Calculate the probability that the claim is on a red car.

• 0.300
• 0.462
• 0.491
• 0.667
• 0.692

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

Solution: C

Let R be the event the car is red and G be the event the car is green. Let E be the event that the claim exceeds the deductible. Then,

[[math]] \operatorname{P}(R | E ) = \frac{\operatorname{P}( R) \operatorname{P}( E | R)}{\operatorname{P}( R ) \operatorname{P}( E | R ) + \operatorname{P}(G) \operatorname{P}( E | G)} = \frac{0.3(0.09)}{0.3(0.09) + 0.7(0.04)} = \frac{0.027}{0.055} = 0.491. [[/math]]

Note that if A is the probability of an accident,

[[math]] \operatorname{P}( E | R ) = \operatorname{P}( E | R \cap A) \operatorname{P}( A | R ) = 0.1(0.9) = 0.09. [[/math]]

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