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Exercise for October 27, 2007
November 2001 Course 1 Examination, Problem No. 11, also Study Note P-09-07,
Problem No. 43
A company takes out an insurance policy to cover accidents that occur at its
manufacturing plant. The probability that one or more accidents will occur during any
3
given month is . The number of accidents that occur in any given month is independent
5
of the number of accidents that occur in all other months. Calculate the probability that
there will be at least four months in which no accidents occur before the fourth month in
which at least one accident occurs.
A. 0.01
B. 0.12
C. 0.23
D. 0.29
E. 0.41
Solution.
Consider a Bernoulli Trial with success defined as a month with an accident, and a month
3
with no accident being a failure. Then the probability of success is . Now consider a
5
negative binomial random variable, which counts the number of failures (months with no
accident) until 4 successes (months with accidents), call it X. The problems asks us to
find Pr ( X ! 4 ) . But
Pr ( X ! 4 ) = 1 " Pr ( X = 0 ) " Pr ( X = 1) " Pr ( X = 2 ) " Pr ( X = 3) =
4
4
4
2
4
3
# 4 & # 3 & 2 # 5& # 3 & # 2 & # 6& # 3 & # 2 &
# 3& # 3 &
= 1 " % ( ) % ( " % ( ) % ( ) " % ( ) % ( ) % ( " % ( % ( % ( * 0.289792.
$ 1 ' $ 5 ' 5 $ 2 ' $ 5 ' $ 5 ' $ 3' $ 5 ' $ 5 '
$ 0' $ 5 '
Answer D.
© Copyright 2004-2007 by Krzysztof Ostaszewski.
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Exercises from the past actuarial examinations are copyrighted by the Society of
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