EXAM P SAMPLE SOLUTIONS

31. Solution: D
Let X denote the number of employees that achieve the high performance level. Then X
follows a binomial distribution with parameters n= 20 and p = 0.02 . Now we want to
determine x such that
Pr X > x ≤ 0.01
[ ]
or, equivalently,
[ ] ( )( ) ( ) 20
x 20
k −k
0.99 ≤Pr X ≤ x =∑ 0.02 0.98
k = 0 k
The following table summarizes the selection process for x:
x Pr X = x Pr X ≤ x
[ ] [ ]
(
20
) =
0 0.98 0.668 0.668
( )( )
1 20 0.02 0.98 = 0.272 0.940
19
( ) ( )
2 18
2 190 0.02 0.98 = 0.053
0.993
Consequently, there is less than a 1% chance that more than two employees will achieve
the high performance level. We conclude that we should choose the payment amount C
such that
2C = 120,000
or
C = 60,000
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32. 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
Pr [ z ≥ x+ 2] = f ( 0,0, 4) + f ( 1,0,3) + f ( 0,1,3 ) + f ( 0, 2, 2)
=
= 0.0488
+ +
4!
+
2!2!
4 3 3 2
( 0.20) 4( 0.50)( 0.20) 4( 0.30)( 0.20) ( 0.30) ( 0.20)
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