10. The probability that automobile sales will increase next month (A) is estimated to be 0.40....

 

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10. The probability that automobile sales will increase next month (A) is estimated to be 0.40. The probability that the sale of replacement parts will increase (R) is estimated to be 0.50. The probability that both industries will experience an increase in sales is estimated to be 0.10. What is the probability that (a) automobile sales have increased during the month given that there is information that replacement parts sales have increased, (b) replacement parts sales have increased given information that automobile sales have increased during the month? Are the two events dependent or independent? 11. A project group of two engineers and three technicians is to be assigned from a departmental group which includes five engineers and nine technicians. How many different project groups can be assigned from the fourteen available personnel? Now suppose the five individuals are assigned randomly from the fourteen personnel in the department, without reference to whether each person is an engineer or a technician. What is the probability that the project group will include (a) exactly two engineers, (b) no engineers. (c) no technicians? 12. A diagnostic test has a probability0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability0.10 of giving a (false) positive when applied to a non-sufferer. It is estimated that 0.5 % of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the following probabilities: (a) that the test result will be positive; (b) that, given a positive result, the person is a sufferer; (c) that, given a negative result, the person is a non-sufferer; (d) that the person will be misclassified. 13. In a TV Game show, a contestant selects one of three doors; behind one of the doors there is a prize, and behind the other two there are no prizes. After the contestant selects a door, the game-show host opens one of the remaining doors, and reveals that there is no prize behind it. The host then asks the contestant whether they want to SWITCH their choice to the other unopened door, or STICK to their original choice. Is it probabilistically advantageous for the contestant to SWITCH doors, or is the probability of winning the prize the same whether they STICK or SWITCH? (Assume that the host selects a door to open, from those available, with equal probability). 10. The probability that automobile sales will increase next month (A) is estimated to be 0.40. The probability that the sale of replacement parts will increase (R) is estimated to be 0.50. The probability that both industries will experience an increase in sales is estimated to be 0.10. What is the probability that (a) automobile sales have increased during the month given that there is information that replacement parts sales have increased, (b) replacement parts sales have increased given information that automobile sales have increased during the month? Are the two events dependent or independent? 11. A project group of two engineers and three technicians is to be assigned from a departmental group which includes five engineers and nine technicians. How many different project groups can be assigned from the fourteen available personnel? Now suppose the five individuals are assigned randomly from the fourteen personnel in the department, without reference to whether each person is an engineer or a technician. What is the probability that the project group will include (a) exactly two engineers, (b) no engineers. (c) no technicians? 12. A diagnostic test has a probability0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability0.10 of giving a (false) positive when applied to a non-sufferer. It is estimated that 0.5 % of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the following probabilities: (a) that the test result will be positive; (b) that, given a positive result, the person is a sufferer; (c) that, given a negative result, the person is a non-sufferer; (d) that the person will be misclassified. 13. In a TV Game show, a contestant selects one of three doors; behind one of the doors there is a prize, and behind the other two there are no prizes. After the contestant selects a door, the game-show host opens one of the remaining doors, and reveals that there is no prize behind it. The host then asks the contestant whether they want to SWITCH their choice to the other unopened door, or STICK to their original choice. Is it probabilistically advantageous for the contestant to SWITCH doors, or is the probability of winning the prize the same whether they STICK or SWITCH? (Assume that the host selects a door to open, from those available, with equal probability).