Customer are used to evaluate preliminary product design. In the past, 95% of hi

Question

Customer are used to evaluate preliminary product design. 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. I n addition, 40% of products have been highly successful, 35% have been moderately successful, and 25% have been poor products.

a) What is the probability that a product attains a goog review?

b) If a new design attains a good review, what is the probability that it will be a highly successful product?

Explanation / Answer

It's good to start out by stating the probabilities that you already know.

P(Good Review | Highly successful) = 0.95
P(Good Review | Moderately successful) = 0.60
P(Good Review | Poor product) = 0.10
P(Highly successful) = 0.40
P(Moderately successful) = 0.35
P(Poor product) = 0.25

a) P(Good Review)

= P(Good Review and Highly successful) + P(Good Review and Moderately successful) + P(Good Review and Poor product)

= P(Highly successful)P(Good Review | Highly successful) +
P(Moderately successful)P(Good Review | Moderately successful) +
P(Poor product)P(Good Review | Poor product)
(These are using the general multiplication rule.)

= 0.40*0.95 + 0.35*0.60 + 0.25*0.10

b) P(Highly successful | Good Review)

= P(Highly successful)P(Good Review | Highly successful) /
{P(Highly successful)P(Good Review | Highly successful) +
P(Moderately successful)P(Good Review | Moderately successful) +
P(Poor product)P(Good Review | Poor product)}
using Bayes' Rule

= 0.40*0.95/(0.40*0.95 + 0.35*0.60 + 0.25*0.10)

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