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

Question

Quizplus · Accepted Answer

Let's use Bayes' theorem to calculate the probability of a successful market research given a successful product.
P(Successful Market Research and Successful Product) = P(Successful Product) * P(Successful Market Research given Successful Product)
P(Successful Product) = 0.70
P(Successful Market Research given Successful Product) = 0.90
P(Failed Market Research given Failed Product) = 0.80
P(Failed Product) = 0.30

To calculate the probability of a successful market research overall, we need to consider the two ways in which a product can be successful: either the market research is successful and the product is successful, or the market research is failed and the product is successful.
P(Successful Market Research) = P(Successful Market Research and Successful Product) + P(Failed Market Research and Successful Product)
P(Failed Market Research and Successful Product) = P(Failed Market Research given Successful Product) * P(Successful Product) = 0.20 * 0.30 = 0.06
P(Successful Market Research) = 0.70 * 0.90 + 0.06 = 0.63

Therefore, the probability that the results indicate a successful market for the product and the product actually is successful is 0.63, which corresponds to answer choice D.

A Company Is Considering Producing Some New Gameboy Electronic Games