6. Ch4,Q39: The prior probabilities for events A1 and A2 are P(A1) = 0.40 and P(A2) = 0.60. It is also known that P(A1 0A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. (a) Are A1 and A2 mutually exclusive? Explain. (b) Compute P(A1 ['1 B) and P(A2 D B). (c) Compute P(B). ((1) Apply Bayes' theorem to compute P(A1 | B) and P(A2 | B). 7. Ch4,Q41: A consulting firm submitted a bid for a large research project. The rm's management initially felt they had a 50-50 chance of getting the project. However, the agency to which the bid was submitted subsequently requested additional in- formation on the bid. past experience indicates that for 75% of the successful bids and 40% of the unsuccessful bids the agency requested additional information. (a) What is the prior probability of the bid being successful (that is, prior to the request for additional information)? (b) What is the conditional probability of a request for additional information given that the bid will ultimately be successful? )item Compute the poste rior probability that the bid will be successful given a request for additional information. 8. Ch4,Q55: A large consumer goods company ran a television advertisement for one of its soap products. On the basis of a survey that was conducted, probabilities were assigned to the following events. B = individual purchased the product 3 = individual recalls seeing the advertise- ment 303 = individual purchased the product and recalls seeing the advertisement. The probabilities assigned were P(B) = 0.20, 13(5) = 0.40, and P(B ['1 S) = 0.12. (a) What is the probability of an individual's purchasing the product given that the individual recalls seeing the advertisement? Does seeing the advertisement increase the probability that the individual will purchase the product? As a decision maker, would you recommend continuing the advertisement (assuming that the cost is reasonable)? (b) Assume that individuals who do not purchase the company's soap product buy from its competitors. What would be your estimate of the company's market share? Would you expect that continuing the advertisement will increase the company's market share? Why or why not? (c) The company also tested another advertisement and assigned it values of 13(3) = 0.30 and P(B3) = 0.10. What is P(B | S) for this other ad- vertisement? Which advertisement seems to have had the bigger effect on customer purchases