A company is considering producing some new Gameboy electronic games.Based on past records,management believes that there is a 70 percent chance that each of these will be successful,and a 30 percent chance of failure.Market research may be used to revise these probabilities.In the past,the successful products were predicted to be successful based on market research 90 percent of the time.However,for products that failed,the market research predicted these would be successes 20 percent of the time.If market research is performed for a new product,what is the probability that the product will be successful if the market research indicates a success?
A)0.10
B)0.90
C)0.91
D)0.63
E)0.09

Question

Quizplus · Accepted Answer

Let S be the event that a new product is successful, and M be the event that the market research indicates a success. We are interested in finding P(S | M).

Using Bayes' theorem, we have:

P(S | M) = P(M | S) P(S) / [P(M | S) P(S) + P(M | ~S) P(~S)]

where ~S denotes the complement of S (i.e., failure).

From the problem statement, we have:

- P(S) = 0.7
- P(~S) = 0.3
- P(M | S) = 0.9
- P(M | ~S) = 0.2

Plugging these values into Bayes' theorem, we get:

P(S | M) = (0.9)(0.7) / [(0.9)(0.7) + (0.2)(0.3)] = 0.91

Therefore, if the market research indicates a success, the probability of the new product being successful is 0.91, which is closest to choice C (0.91).

A Company Is Considering Producing Some New Gameboy Electronic Games