5. A client of a marketing agency is considering the introduction of a new product. Based on past experience with similar products, the client assesses the prior probability that the demand for the new product is high (H) as p(H) 0.5. The client's prior probability that the demand for the new product is low (L) is p(L) 0.5. For simplicity, suppose that these are the only two states of nature. The client has instructed the marketing agency to run a marketing trial to obtain better information about the demand for the new product. At the end of the trial, the marketing agency will report either that demand in the trial was high (+), which is an optimistic signal that increases the probability that the actual demand for the new product is high, or that the demand in the trial was low (-), which is a pessimistic signal. The conditional probabilities that describe the accuracy of the marketing trial have the following values. p(+H) 0.7 and p(-H) 0.3. p(+IL) 0.4 and p(-L) 0.6. (0) Briefly explain what is meant by the posterior probabilities p(Hi+) and p(HI-). (ii) Calculate the numerical value of the posterior probability p(Hl-). Explain your reasoning (ii) Suppose that the client will choose not to introduce the new product if, given the result of the marketing trial, the client believes that the probability that demand for the new product is high is less than 0.4. Based on the above information, what is the probability that the client will choose not to introduce the new product? Briefly justify your answer. 5. A client of a marketing agency is considering the introduction of a new product. Based on past experience with similar products, the client assesses the prior probability that the demand for the new product is high (H) as p(H) 0.5. The client's prior probability that the demand for the new product is low (L) is p(L) 0.5. For simplicity, suppose that these are the only two states of nature. The client has instructed the marketing agency to run a marketing trial to obtain better information about the demand for the new product. At the end of the trial, the marketing agency will report either that demand in the trial was high (+), which is an optimistic signal that increases the probability that the actual demand for the new product is high, or that the demand in the trial was low (-), which is a pessimistic signal. The conditional probabilities that describe the accuracy of the marketing trial have the following values. p(+H) 0.7 and p(-H) 0.3. p(+IL) 0.4 and p(-L) 0.6. (0) Briefly explain what is meant by the posterior probabilities p(Hi+) and p(HI-). (ii) Calculate the numerical value of the posterior probability p(Hl-). Explain your reasoning (ii) Suppose that the client will choose not to introduce the new product if, given the result of the marketing trial, the client believes that the probability that demand for the new product is high is less than 0.4. Based on the above information, what is the probability that the client will choose not to introduce the new product? Briefly justify your
