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 results indicate a successful market for the product and the product actually is successful?
A)0.90
B)0.54
C)0.60
D)0.63
E)None of the above

Question

Quizplus · Accepted Answer

We need to use Bayes' theorem here. Let A be the event that the market research indicates a successful market for the product, and B be the event that the product is successful. We want to find P(B|A).

Using Bayes' theorem:

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

We know that P(B) = 0.7 (given in the problem) and P(A|B) = 0.9 (from the problem statement "successful products were predicted to be successful based on market research 90% of the time"). We also need to find P(A).

To find P(A), we need to consider the cases where the product is successful and where it fails. Let's use a table to organize this:

|              | Market research says successful | Market research says failure |
|--------------|--------------------------------|------------------------------|
| Successful   | 0.9                            | 0.1                          |
| Failure      | 0.2                            | 0.8                          |

From this table, we can see that:

P(A) = P(A|B) * P(B) + P(A|B') * P(B') = 0.9 * 0.7 + 0.2 * 0.3 = 0.66

Now we can substitute into Bayes' theorem:

P(B|A) = P(A|B) * P(B) / P(A) = 0.9 * 0.7 / 0.66 = 0.95 (rounded to two decimal places)

Therefore, the probability that the market research indicates a successful market for the product and the product actually is successful is 0.63 (rounded to two decimal places).

A Company Is Considering Producing Some New Gameboy Electronic Games