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```Krzys’ Ostaszewski: http://www.krzysio.net
Author of the “Been There Done That!” manual for Course P/1 http://smartURL.it/
krzysioP (paper) or http://smartURL.it/krzysioPe (electronic) Instructor for Course P/1
online seminar: http://smartURL.it/onlineactuary
If you find these exercises valuable, please consider buying the manual or attending our
seminar, and if you can’t, please consider making a donation to the Actuarial Program
at Illinois State University: https://www.math.ilstu.edu/actuary/giving/
Donations will be used for scholarships for actuarial students. Donations are taxdeductible to the extent allowed by law.
Questions about these exercises? E-mail: [email protected]
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 '