A product manager at Clean & Brite (C&B) wants to determine whether her company

Question

A product manager at Clean & Brite (C&B) wants to determine whether her company should market a new brand of toothpaste. If this new product succeeds in the marketplace, C&B estimates it could earn $1,800,000 in future profits from the sale of the new toothpaste. If this new product fails, however, the company expects that it could lose approximately $750,000. If C&B chooses not to market this new brand, the product manager believes that there would be little, if any, impact on the profits earned through sales of C&B's other products. The manager has estimated that the new toothpaste brand will succeed with probability or 0.50. Before making her decision regarding this toothpaste product, the manager can spend $75,000 on a market research study. Based on similar studies with past products, C&B believes that the study will predict a successful product, given that the product, given that the product would actually be a success, with probability 0.75. It also believes that the study will predict a failure, given that the product would actually be a failure, with probability 0.65.

To maximize the expected profit, what strategy should the C&B product manager follow?

I know that the solution for this problem is on the chegg textbook solutions, but it uses precision tree software, and I have to compute by hand using decision trees, and the properties of probability.

Explanation / Answer

Let M be the event the new product is predicted to be successful and S be the event that the new product is actually successful.

We gather from the question that P(S)=0.5, P(M|S)=0.75 and P(M|S)=0.65 We need to find P(S|M)

P(M/S) * P(S)= P(M intersects S)

=0.75*0.5

P(M intersects S)=P(S intersects M) = 0.375

P(M)= P(S intersects M) + P(S' intersects M)

=0.375 + (0.65)*(0.65)

P(M)=0.7975

P(S/M) = P(S intersects M) / P(M)

= 0.375/0.7975

P(S/M)=0.47=47%

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