A market research firm conducts studies regarding the success of new products.The company is not always perfect in predicting the success.Suppose that there is a 50% chance that any new product would be successful (and a 50% chance that it would fail).In the past, for all new products that ultimately were successful, 80% were predicted to be successful (and the other 20% were inaccurately predicted to be failures).Also, for all new products that were ultimately failures, 70% were predicted to be failures (and the other 30% were inaccurately predicted to be successes).If the market research predicted that the product would be a success, what is the probability that it would actually be a success?
A)0.27
B)0.73
C)0.80
D)0.24
E)1.00

Question

Quizplus · Accepted Answer

To find the probability that a product is actually successful given that it was predicted to be a success, we use Bayes' theorem. Let's denote S as the event that a product is successful and F as the event that a product fails. Let PS be the event that a product is predicted to be successful and PF as the event that a product is predicted to fail.We are given:- P(S) = 0.5 (probability of success)- P(F) = 0.5 (probability of failure)- P(PS|S) = 0.8 (probability of being predicted successful given it's successful)- P(PF|F) = 0.7 (probability of being predicted a failure given it's a failure)- P(PS|F) = 0.3 (probability of being predicted successful given it's a failure)We want to find P(S|PS), the probability that a product is actually successful given that it was predicted to be successful.Using Bayes' theorem:$$P(S|PS) = \frac{P(PS|S) \cdot P(S)}{P(PS|S) \cdot P(S) + P(PS|F) \cdot P(F)}$$Substituting the given values:$$P(S|PS) = \frac{0.8 \cdot 0.5}{0.8 \cdot 0.5 + 0.3 \cdot 0.5} = \frac{0.4}{0.4 + 0.15} = \frac{0.4}{0.55} = \frac{4}{5.5} = \frac{8}{11}$$$$P(S|PS) = \frac{8}{11} \approx 0.727$$Therefore, the probability that a product is actually successful given that it was predicted to be a success is approximately 0.73.

A Market Research Firm Conducts Studies Regarding the Success of New