A company is considering whether to market a new product. Assume, for simplicity, that if this product is marketed, there are only two possible outcomes: success or failure.

The company assesses that the probabilities of these two outcomes are $p$ and $1-p,$ respectively. If the product is marketed and it proves to be a failure, the company will have

a net loss of $$450,000.$ If the product is marketed and it proves to be a success, the company will have a net gain of $$620,000.$ If the company decides not to market the

product, there is no gain or loss. The company can first survey prospective buyers of this new product. The results of the consumer survey can be classified as favorable,

neutral, or unfavorable. Based on similar surveys for previous products, the company assesses the probabilities of favorable, neutral, and unfavorable survey results to be $0\mathrm{.}6,$

$0\mathrm{.}3,$ and $0\mathrm{.}1$ for a product that will eventually be a success, and it assesses these probabilities to be $0\mathrm{.}1,0\mathrm{.}2,$ and $0\mathrm{.}7$ for a product that will eventually be a failure. The total

cost of administering this survey is $C$ dollars.

a$.$ Let $p=0\mathrm{.}4.$ For which values of $C,$ if any, would this company choose to conduct the survey?

The company should choose to conduct the survey for total survey costs that are less than $

b$.$ Let $p=0\mathrm{.}4.$ What is the largest amount this company would be willing to pay for perfect information about the potential success or failure of the new product?

The company should pay up to $

for perfect information about the potential success or failure of the new product.

c$.$ Let $p=0\mathrm{.}5$ and $C=$$$15,000.$ Find the strategy that maximizes the company's expected net earnings. What is the company's optimal strategy?