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Exercise for May 10, 2008
May 2001 Course 1 Examination, Problem No. 29, also Study Note P-09-05, Problem
No. 32
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 4 new policies for adults
earning their first driver’s license. What is the probability that these 4 will contain at least
two more high-risk drivers than low-risk drivers?
A. 0.006
B. 0.012
C. 0.018
D. 0.049
E. 0.0073
Solution.
Let X be the number of low-risk drivers insured among the four, Y be the number of
moderate-risk drivers insured among the four, Z be the number of high-risk drivers
insured among the four, and f X ,Y , Z ( x, y, z ) is the joint probability function of X, Y, and Z.
The joint distribution of X, Y, and Z is trinomial, and
Pr ( Z ! X + 2 ) = f ( 0, 0, 4 ) + f (1, 0, 3) + f ( 0,1, 3) + f ( 0, 2, 2 ) =
4!
4!
4!
4!
" 0.20 4 +
" 0.50 " 0.20 3 +
" 0.30 " 0.20 3 +
" 0.30 2 " 0.20 2 =
1!0!3!
0!1!3!
0!2!2!
0!0!4!
= 0.0488.
=
Answer D.
© Copyright 2004-2008 by Krzysztof Ostaszewski.
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Exercises from the past actuarial examinations are copyrighted by the Society of
Actuaries and/or Casualty Actuarial Society and are used here with permission.