show all work. 14 points each + 2 bonus points 1. Suppose a jar contains 3 red, 2 white, and 3 blue marbles. If three marbles are drawn without replacement, find the following probabilities. (Use this and next page for this problem) a. b. c. d. P(Two red and one white) P(one of each color) P(none blue) P(at least one blue) 2. If P(A) = .7, P(B) = .4, and P( a. P( A | B) b. A B P( B| A) ) = .20 , find the following probabilities. 3. Forty percent of the students in an economics class major in business and 70% are from St. Louis Missouri. Also, 20% are neither business majors nor from St Louis. What is the probability that a student selected at random from the economics class is a business major from St Louis? 4. Four hundred randomly selected automobile owners were asked whether they selected the particular make and model of their present car mainly because of its appearance or because of its performance. The results were as follows: Owner Appearance, A Male, M 95 Performance, R 55 Total 150 Female, F 85 165 250 Total 180 220 400 a. What is the probability that an automobile owner purchases the car because of its performance or the automobile owner is a male? b. Given that a female automobile owner purchases the car what is the probability that it was due to its appearance? c. Let M be the event that a car owner is a male. Let A be the event that a car owner purchased it because of its appearance. Answer the two parts below. Are the events M and A mutually exclusive? Are the events M and A independent? 5. An insurance policy costs $100, and will pay policy holders $10,000 if they suffer a major injury (resulting in hospitalization) or $3,000 if they suffer a minor injury (resulting in lost time from work). The company estimates that each year one in every 2000 policy holders may have a major injury, and 1 in 500 a minor injury. a. Create a probability model (a table) for the profit on a policy. (Use this table to answer b.) b. What is the expected company's profit on this policy? 6. The sales manager of an insurance company knows that the company's best salesperson can sell an insurance policy 60% of the time. If the sales person were to make 15 calls to sell insurance, a. what is the probability that 5 insurance policies would be sold? b. what is the probability that at least 10 insurance policies would be sold? 7. A large manufacturing company is in the process of training its personnel in quality control procedures. At present, 40% of the assembly lines use control charts, 40% use inspection techniques, and 20% do not use any method for controlling quality. The assembly lines that use control charts have a 1% defective rate. The assembly lines that use inspection techniques have a 5% defective rate. The assembly lines that do not use any quality control techniques have a 12% defective rate. Given that a defective component is found, what is the probability that it was produced by the assembly lines that do not use any quality control techniques? (Bayes)