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STAT 408 Spring 2009
Homework
(due Friday, February 13, by 3:00 p.m.)
Be sure to show all your work; your partial credit might depend on it.
No credit will be given without supporting work.
1. A bank classifies borrowers as "high risk" or "low risk," and 16% of its loans are
made to those in the "high risk" category. Of all the bank's loans, 5% are in default. It is also known that 40% of the loans in default are to high-risk borrowers.
a) What is the probability that a randomly selected loan is in default and issued to a high-risk borrower?
b) What is the probability that a loan will default, given that it is issued to a high-risk borrower?
c) What is the probability that a randomly selected loan is either in default or issued to a high-risk borrower, or both?
d) A loan is being issued to a borrower who is not high-risk. What is the probability that this loan will default?
e) Are events {a randomly selected loan is in default} and {a randomly selected loan is issued to a high-risk borrower} independent? Justify your answer.
2. At Initech , 50% of all employees surf the Internet during work hours. 20% of the
employees surf the Internet and play Solitaire during work hours. It is also known that 60% of the employees either surf the Internet or play Solitaire (or both) during work hours.
a) What proportion of the employees play Solitaire during work hours?
b) If it is known that an employee surfs the Internet during work hours, what is the probability that he/she also plays Solitaire?
c) Suppose an employee does not play Solitaire during work hours. What is the probability that he/she surfs the Internet?
d) Are events {an employee surfs the Internet during work hours} and {an employee plays Solitaire during work hours} independent? Justify your answer.
3. An automobile insurance company classifies each driver as a good risk, a medium
risk, or a poor risk. Of those currently insured, 30% are good risks, 50% are medium risks, and 20% are poor risks. In any given year, the probability that a driver will have a traffic accident is 0.1 for a good risk, 0.3 for a medium risk, and 0.5 for a poor risk.
a) What is the probability that a randomly selected driver insured by this company had a traffic accident during 2008?
b) If a randomly selected driver insured by this company had a traffic accident during 2008, what is the probability that the driver is actually a poor risk?
c) If a randomly selected driver insured by this company did not have a traffic accident during 2008, what is the probability that the driver is actually a good risk?
d) Suppose a driver insured by this company is not a poor risk. What is the probability that the driver had a traffic accident during 2008?
e) The company announced that it will raise the insurance premiums for the drivers who either are poor risks or had a traffic accident during 2008, or both. What proportion of customers would have their premiums raised?
f) Are events {a randomly selected driver is a medium risk} and {a randomly selected driver had a traffic accident during 2008} independent? Justify your answer.
g) Are events {a randomly selected driver is a medium risk} and {a randomly selected driver had a traffic accident during 2008} mutually exclusive? Justify your answer.
4. From a group of 16 male and 9 female armadillos, Noah must choose two to
travel on his ark. Unable to distinguish between male and female armadillos, Noah must choose at random.
a) Noah chooses the two armadillos at random. Compute the probability that Noah gets two armadillos of the opposite sex (i.e., one male and one female armadillo).
b) In order to improve his chances of selecting at least one male and one female armadillo, Noah decides to "cheat" and select three armadillos to travel on his ark. Compute the probability that Noah gets at least one male and one female armadillo.
From the textbook:
1.4-6 1.4-
1.3-20 1.1-
