Math 419
Conditional Probability
Problem Set 2
1. An insurance salesman sells insurance in Territory X and Territory Y. Three times more people live in
Territory X as Territory Y. If 20% of the people in Territory X buy insurance, and 70% of the people
in Territory Y buy insurance, what is the probability that a randomly chosen person is from Territory
X, given they haven't purchased insurance?
2. A survey shows that 75% of the people claim to donate at least 1000 and 50% of the people
exaggerate. 45% do both. Assuming this survey accurately portrays the population, find the
probability that a person donated less than 1000, given they exaggerate.
3. A broker offers three types of bonds : low risk, average risk, and high risk. She notices that younger
people are three times more likely to choose average risk bonds than low risk bonds, and that older
people are just as likely to choose average risk as low risk. Twenty percent of younger clients hold
high risk bonds and 4% of older clients hold high risk bonds. Fifty percent of her clients are older.
What is the probability that a randomly chosen client is young, given that they hold average risk
bond?
4. A truth serum has the property that 90% of the guilty suspects are properly judged while, of course,
10% of the guilty suspects are improperly found innocent. On the other hand, innocent suspects are
misjudged 1% of the time. If the suspect was selected from a group of suspects of which only 5%
have ever committed a crime, and a serum indicated that he is guilty, what is the probability that he is
innocent?
5. An employer offers three plans: life, health, and retirement. An employee can either have exactly two
of these plans or none of these plans. The probability that an employee has life is 0.66. The
probability that an employee has health is 0.50. The probability that an employee has retirement is
0.33. What is the probability that an employee has both the life and retirement plans, given she has
the retirement plan?
6.Ten people work at a small firm. Six people are from department A and others are from department
B. The probability that an employee having a workplace accident in a year is 0.2, where each
employee is independent of any other. Each employee does not have more than one accident in a
year. What is the probability of at least one accident from B given that there were 3 accidents in a
year?
7. In a sample, all people are either left-handed or right-handed and have brown or green eyes. Assume
50% are both right-handed and have brown eyes, and twice as many people have green eyes as are
left-handed. Also, three-fifths of right handers have brown eyes. What is the proportion of the people
who are both left-handed and have green eyes?
8.We send four packages, two fragile and two non fragile. The probability of a fragile package
breaking is 0.2. The probability of a non-fragile breaking is 0.1. Given exactly two break, what is
the probability that both broken ones are fragile?
9.Consider two insurance plans, A and B. 58% of the people have at least one plan. 40% of the people
who do not have plan B, have plan A. 30% of the people who do not have plan A, have plan B.
What is the probability of having plan A for a person who owns exactly one plan?
Math 419
Conditional Probability
Problem Set 2
10. Suppose that P(A Ç B Ç C) = 0.9, P(B) = 0.4, A and C are mutually exclusive, and B and C are
mutually exclusive. Find the value of r if the probability of B given A is 5r, the probability of A
intersect B is r, and the probability of C is 3r.