Question

1. (May, 1986) A company is considering the introduction of a new product. The marketing vice president believes that the probability of success for the product is .5. It will either be a complete success or a complete failure. The product will be test marketed regionally before being introduced nationally. In the past, the test market has been 75% reliable in making favorable predictions - that is, if a product is destined to be successful nationally, the test market will produce favorable results 75% of the time. The test market however has been 70% reliable in making an unfavorable prediction. That is, a product destined to be a failure nationally will produce an unfavorable result 70% of the time.

If the product is successfully nationally, the net profit will be $30,000,000. If it fails, the net loss will be $10,000,000.

If the test market indicates a favorable result, what is the expected net profit?

(a) $2 million (b) $10 million (c) $15.5 million (d) $18.5 million (e) $20 million

need the full solution not just the multiple choice answer please and thank you.

Answer 1

P(A) = Probability of success nationally = 0.5

P(B) = Test market produce favorable result = (0.75* 0.5) + (0.3 *0.5 )

P(B|A) = Probability of favorable result given product successfully nation = 0.75

Bayes' theorem:

P(A|B) = P(B|A) * P(A) / P(B)

=0.75 * 0.5 /((0.75* 0.5) + (0.3 *0.5 )

=0.714

Net profit = Probability of success * Profit - (1-Probability of success ) * Loss

Net profit = 0.714 * 30 - 0.286 * 10 = 18.5

(d) $18.5 million

Please let me know in case of any doubt.