# N - Department of Mathematics | Illinois State University

```Krzys’ Ostaszewski: http://www.krzysio.net
Author of the ASM Manual (the “Been There Done That!” manual) for Course P/1
http://smartURL.it/krzysioP (paper) or http://smartURL.it/krzysioP (electronic)
Instructor of online P/1 seminar: http://smartURL.it/onlineactuary
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Questions about these exercises? E-mail: [email protected]
Exercise for September 29, 2007
May 2000 Course 1 Examination, Problem No. 36, also Study Note P-09-07, Problem
No. 14
In modeling the number of claims filed by an individual under an automobile policy
during a three-year period, an actuary makes the simplifying assumption that for all
1
integers n ≥ 0, pn +1 = pn , where pn represents the probability that the policyholder
5
files n claims during the period. Under this assumption, what is the probability that a
policyholder files more than one claim during the period?
A. 0.04
B. 0.16
C. 0.20
D. 0.80
E. 0.96
Solution.
∞
In order for this to be a probability distribution we must have
∑p
k=0
k
= 1. Since
1
1 1
1 1
1
pn −1 = ⋅ pn − 2 = ⋅ ⋅…⋅ p0 ,
5
5 5
5 5
5
it follows that
n
∞
∞ ⎛
⎞
1
⎛ 1⎞
1 = ∑ pn = ∑ ⎜ ⎜ ⎟ ⋅ p0 ⎟ = p0 ⋅
= 1.25 p0 .
1
⎝ ⎠
⎠
n=0
n=0 ⎝ 5
1−
5
n
4 ⎛ 1⎞
4
Consequently, p0 = and pn = ⋅ ⎜ ⎟ . Finally,
5 ⎝ 5⎠
5
pn =
Pr ( N &gt; 1) = 1 − Pr ( N = 0 ) − Pr ( N = 1) = 1 − p0 − p1 = 1 −
4 1 4 1
− ⋅ =
.
5 5 5 25