Question

Customers are used to evaluate preliminary product designs. In the past, 93% of highly successful products received good reviews, 51% of moderately successful products received good reviews, and 14% 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. Round your answers to four decimal places (e.g. 98.7654).(a) What is the probability that a product attains a good review?(b) If a new design attains a good review, what is the probability that it will be a highly successful product?(c) If a product does not attain a good review, what is the probability that it will be a highly successful product?

Answer 1

Answer:

a. 0.5855

b. 0.6354

c. 0.0676

Step-by-step explanation:

Be the events:

E: The product is highly successful

ME: The product is moderately successful

P: The product is poorly successful

B: The product received good reviews

MB: The product received bad reviews

You have then:

$P(E) = 0.4000$

$P(ME) = 0.3500$

$P(P) = 0.2500$

$P(B|E) = 0.9300$ and $P(MB|E) = 1 - P(B|E) = 1 - 0.9300 = 0.0700$

$P(B|ME) = 0.5100$

$P(P|E) = 0.1400$

a. invoking the total probability theorem, you have:

$P(B) = P(B|E)P(E) + P(B|ME)P(ME) + P(B|P)P(P) = (0.9300)(0.4000) + (0.5100)(0.3500) + (0.1400)(0.2500) = 0.5855$

b. invoking the Baye's theorem, you have:

$P(E|B) = \frac{P(B|E)P(E)}{P(B)} = \frac{(0.9300)(0.4000)}{0.5855} = 0.6354$

c. Using the result obtained in a. $P(MB) = 1 - P(B) = 1 - 0.5855 = 0.4145$, then:

$P(E|MB) = \frac{P(MB|E)P(E)}{P(MB)} = \frac{(0.0700)(0.4000)}{0.4145} = 0.0676$