Practice problems – Set I

1. A bank classifies borrowers as high-risk or low-risk. Only 15% of its loans are made to those in high-risk category. Of all its loans, 5% are in default, and 40% of those in default were made to high-risk borrowers. What is the probability that a high-risk borrower will default?

2. A fund manager is considering investing in the stock of a healthcare provider. The manager’s assessment of probabilities of the rates of return on this stock over the next year is summarized in the table below.
Rate of return Less than -10% -10% to 0% 0% to 10% 10% to 20% More than 20%
Probability 0.04 0.14 0.28 0.33 0.21
Let A be the event “Rate of return will be more than 10%,” and B be the event “Rate of return will be negative.”
• Find the probability of events A and B.
• Describe the even that is complement of event A.
• Find the probability of the complement of A.
• Describe the even that is complement of event B.
• Find the probability of the complement of B.
• Describe and find the probability of event that is intersection of A and B.
• Describe and find the probability of event that is union of A and B.
• Are events A and B mutually exclusive? Collectively exhaustive?

3. A random sample of 52 personal property insurance policies showed the following number of claims over the last year:
# of claims 0 1 2 3 4 5 6
# of policies 21 15 5 4 2 3 2
• Find the mean number of claims per day.
• Find the sample variance and standard deviation.

4. A survey carried out for a supermarket classified customers according to whether their visits to the store are frequent or infrequent and whether they often, sometimes, or never purchase generic products. The accompanying table gives the proportions of people surveyed in each of the six joint classifications.
Frequency of visit Purchase of generic product Often Sometimes Never