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STT 441: Fall 2014
Midterm II
• You may use your calculator and one page of notes.
• Exam time: 13:50 am
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14:40 am, November 5, 2014
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1. A recent college graduate is planning to take the first three actuarial examinations in the
coming summer. She will take the first actuarial exam in June. If she passes t.hat exam,
then she will take the second exam in July, and if she also passes that one, then she will
take the third exam in September. If she fails an exam, then she is not allowed to take any
others. The probability that she passes the first. exam is .9. If she passes the first exam,
then the conditional probability that she passes the second one is .8, and if she passes both
the first and the second exams, then the conditional probability that she passes the third
exam is .7.
(a) What is the probability that she passes all three exams? (2pt)
(b) Given that she did not pass all three exams. what is the conditional probability that
she failed the second exam? (2pt)
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2. Suppose that an insurance company classifies people into one of three classes: good risks,
average risks, and bad risks. The companys records indicate that the probabilities that
good-, average-, and bad-risk persons will be involved in an accident over a 1-year span
are, respectively, .05, .15, and .30. If 20 percent of the population is a good risk, 50 percent
an average risk, and 30 percent a bad risk,
(a) What proportion of people have accidents in a fixed year ? (2pt)
(b) If policyholder A had no accidents in 1997, what is the probability that he or she is
a good or average risk? (2pt)
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3. To determine whether they have a certain disease, 100 people are to have their blood
tested. However, rather than testing each individual separately, it has been decided first
to place the people into groups of 10. The blood samples of the 10 people in each group
will be pooled and analyzed together. If the test is negative, one test will suffice for the
10 people, whereas if the test is positive, each of the 10 people will also be individually
tested and, in all, 11 tests will be made on this group. Assume that the probability that a
person has the disease is 0.1 for all people, independently of each other, and compute the
expected number of tests necessary for each group. (Note that we are assuming that the
pooled test will be positive if at least one person in the pool has the disease.) (2pt)
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5. A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly.
If they
are the same color, then you win $10; if they are different colors, then you win
-$9.00.
(That is, you lose $9.00.) Calculate
(a) the expected value of the amount you win; (2pt)
(b) the standard deviation of the amount you win. (2pt)
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