Customers are used to evaluate preliminary product design, 90% of highly success

Question

Customers are used to evaluate preliminary product design, 90% of highly successful products received good reviews, 50 % of moderately successful products received good reviews, 10 % of poor products received good reviews. In addition, 60% of products have been highly successful, 30% of products have been moderately successful, and 10 % of products have been successful. (a) What is the probability that a product receives good review and is highly successful? (b) If a product receives a good review, what is the probability that it will be highly successful? (c) If a product doesn't receive good review, what is the probability that it will be highly successful? (d) What is the probability that a product receives good review? (e) Is the event receiving good review is independent from the event becoming highly successful product?

Explanation / Answer

(A)
Probability that a product receives good reviews and is highly successful = P(G and S) = P(G) * P(S) = 0.9 * 0.6 = 0.54

(B)
Probability that a product receives good reviews = P(G) = 0.6*0.9 + 0.3*0.5 + 0.1*0.1 = 0.7
The required probability P(S|G) = P(S and G) / P(G) = (0.6*0.9)/0.7 = 0.7714

(C)
P(G') = 1 - P(G) = 1 - 0.7 = 0.3
P(S and G') = 0.1 * 0.6 = 0.06
Required probability, P(S|G') = P(S and G') / P(G') = 0.06/0.3 = 0.2

(D)
Probability that a product receives good reviews = P(G) = 0.6*0.9 + 0.3*0.5 + 0.1*0.1 = 0.7

(E)
Yes, the events are independent.

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