MAT150 Statistics Prof. Miller
Due: Tuesday, November 8th
- Probability and Probability Distributions
Directions: Answer any five of the six questions. Show all work. You may consult with other students in our class and with me, but not with any one outside of our class except for the most general help (not working on these specific questions.)
1. Suppose 30% of the population has type A blood.
(a) If 8 people are selected at random, what is the probability that at least three of them have
type A blood.
(b) If 80 donors come to give blood one day, what is the probability that at least thirty of them have Type A blood? Explain why this is higher or lower than the answer in part (a).
(c) If 20 people come to give blood, what is the probability that at least one of the donors is of Type A?
3. A website has determined that the length of time that users spend on its site is normally distributed with a mean of 5.83 minutes and a standard deviation of 0.89 minutes.
Find: (a) the probability that a user will stay on the site for more than 6 minutes.
(b) the probability that a user will stay on the site t between 2 and 5 minutes.
(c) the 41st percentile for the length of time spent on the site (a time such that 41% of users stay less than that time).
(d ) the probability that a random sample of 30 users has a mean of more than 5 minutes on the site.
4. A health insurance company charges policyholders a $2500 annual premium for health insurance for hospitalization. The company estimates that each time a patient is hospitalized costs the company $2850. Furthermore, they have estimated that 85% of patients will not be hospitalized, 10% will be hospitalized once a year, and no one will be hospitilized more than twice.
(a) Find the insurance company’s expected profit per policyholder.
(b) What is the expected profit if they enroll 80,000 policyholders?
5. Suppose the weight of eggs produced by Henly Farms has a mean of 59.45 g. with a standard deviation of 3.17 g. Find the probability that a carton of a dozen eggs will weigh less than 724 g.
5. Consider the experiment of rolling two dice and the following events:
A: ‘The sum of the dice is 8’ and B: ‘The first die is an even number’ and C: “The difference (absolute value) of the dice is 2”
Find (a) p(A and B) (HINT: You cannot assume these are independent events.)
(b) p(A or B)
(c) Are A and B mutually exclusive events? Explain.
(d) Are A and B independent events? Explain. (no explanationsno points)
(e) Are B and C independent events? Explain.
6. Describe in a short paragraph the relationship between independent events and mutually exclusive events. Your answer should include (but should not be limited to) an answer to the following questions:
· Give an example of two events A and B that are mutually exclusive, and explain why
they are dependent events.
· Give an example of two events that are dependent, but are not mutually exclusive.
Both examples and explanations are required.
Also, your answer should not be exactly the same as any other student’s.