Customers are used to evaluate preliminary product designs, In the past, 95% of

Question

Customers are used to evaluate preliminary product designs, In the past, 95% of highly successful products received good reviews, 60% of moderately successful products received good reviews, and 10% of poor products received good reviews. In addition, 40% of products have been highly successful, 35% have been moderately successful, and 25% have been poor products. What is the probability that a product attains a good review? If a new design attains a good review, what is the probability that it will be a highly successful product? If a product does not attain a good review, what is the probability that it will be a highly successful product?

Explanation / Answer

Let H shows the event that product is highly successful, M shows the event that product is moderately successful and P shows the event that product is poor. So we have

P(H) = 0.40, P(M) = 0.35, P(P) = 0.25

And let G shows the event that product receive good review. So we have

P(G|H) = 0.95, P(G|M) = 0.60, P(G|P)= 0.10

(a)

By the law of total probability, the probability that a product attains a good review is

P(G) = P(G|H)P(H)+ P(G|M)P(M)+ P(G|P)P(P) = 0.95 * 0.40 + 0.60* 0.35 + 0.10 * 0.25 = 0.615

(b)

By the Baye's theorem, the probability that it will be a highly successful product, given that new design attain a good review so

P(H|G) = [P(G|H)P(H)] / P(G) = [0.95 *0.40 ] / 0.615 = 0.6179

(c)

Let N shows the event that product do not receive a good review. By the complement rule we have ,

P(N|H)=1-P(G|H) = 0.05, P(N|M)=1-P(G|M) = 0.40, P(N|P)=1-P(G|P)= 0.90

By the complement rule we have

P(N)=1-P(G) =1- 0.615 = 0.385

So the requried probability is

P(H|N) = [P(N|H)P(H)] / P(N) = [0.05*0.40] / 0.385 = 0.0519

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