Market Researchers, Inc

Market Researchers, Inc. has been hired to perform a study to determine if the market for a new product will be good or poor. In similar studies performed in the past, whenever the market actually was good, the market research study indicated that it would be good for 85% of the time. On the other hand, whenever the market actually was poor, the market study incorrectly predicted it would be good 20% of the time. Before the study is performed, it is believed there is a 70% chance the market will be good. When Market Researchers, Inc. performs the study for this product, the results predict the market will be good. Given the results of the study, what is the probability that the market will be good?

Solution. Denote by A the event that the market will be good. Denote by B the event that the market will be poor. Then by the given information, P(A)=70%=0.70, P(B)=1-P(A)=1-70%=0.30.
Denote by C the event that the market is actually good. So, we know that P(C|A)=85% and P(C|B)=20%.

Now we need to calculate P(C). By the whole probability formula, we have

P(C)=P(C|A)P(A)+P(C|B)P(B)
=85%*70%+20%*30%=0.655

So, given the results of the study, the probability that the market will be good is 0.655.