A large pool of adults earning their first driver’s license includes 50% low-risk drivers, 30% moderate-risk drivers, and 20% high-risk drivers. Because these drivers have no prior driving record, an insurance company considers each driver to be randomly selected from the pool.

This month, the insurance company writes four new policies for adults earning their first driver’s license.

Calculate the probability that these four will contain at least two more high-risk drivers than low-risk drivers.

• 0.006
• 0.012
• 0.018
• 0.049
• 0.073

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

Solution: D

Let

X = number of low-risk drivers insured

Y = number of moderate-risk drivers insured

Z = number of high-risk drivers insured

f(x, y, z) = probability function of X, Y, and Z.

Then f is a trinomial probability function, so

[[math]] \begin{align*} \operatorname{P}[ z ≥ x + 2] &= f ( 0, 0, 4 ) + f (1, 0,3) + f ( 0,1,3) + f ( 0, 2, 2 ) \\ &= 0.2^4 + 4 ( 0.50 )( 0.20 )^3 + 4 ( 0.30 )( 0.20 )^3 + \frac{4!}{2!2!}(0.3)^2(0.2)^2 \\ &= 0.0488. \end{align*} [[/math]]

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