Probabilities?

Question

Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products received good reviews, 60% of moderately succesful 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.

a) What is the probability that a product attains a good review?
b) If a new design attains a good review, what is the probabitliy that it will be 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?
d) Is "a product is poor" independent of "a product receives a good review"? Justify your answer with a mathematical test as well as with a statement of logic.

If any of them can be answered, I am very thankful. Please show your work for me to see. Thank you.

Answer 1

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)

c) P(Highly Successful | Bad Review)

= P(Highly successful)P(Bad Review | Highly successful) /
{P(Highly successful)P(Bad Review | Highly successful) +
P(Moderately successful)P(Bad Review | Moderately successful) +
P(Poor product)P(Bad Review | Poor product)}
again using Bayes' rule

= 0.40*0.05/(0.40*0.05 + 0.35*0.40 + 0.25*0.90)

d) If they are independent, then P(Poor product) = P(Poor product | Good Review). If they are not equal, then they are not independent.

P(Poor product) = 0.25

P(Poor product | Good Review)

= P(Poor Product successful)P(Good Review | 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)}
using Bayes' Rule

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

I think it's pretty likely that these two are different, so they are not independent.

Answer 2

a) is just the sum for each level of success times the chance that each level of success gets a good review

.95(.40)+.60(.35)+.10(.25)

b) is the probability that a design is highly successful given that it has a good review, a conditional problem

Pr (high success| good review) =
Pr (high succcess AND good review) / Pr (good review) =
.95(.40) / [whatever the answer to (a) turns out to be]

c) is similar to (b) in that it's a conditional problem

Pr (high success | bad review) =
Pr (high succcess AND bad review) / Pr (bad review) =
(1-.95)*.40 / [1 - your answer from answer (a)]

Answer 3

Lay out a 3 by 4 grid, columns labelled HS, MS, P, and total, rows labelled GR, NG, and total. In the last row, under HS put 40, under MS put 35, under P put 25. In the HS column, GR has to be 95% of 40 = 38, and NG = 2. In the MS column, GR has to be 60% of 35 = 21, NG = 14. In the P column, GR is 10% of 25 = 2.5, so NG is 22.5.

Total the rows and verify the total column add to 100, as does the last row.

So now, for a) prob(GR) = 61.5%
b) Given GR (look only at GR row, with total), prob(HS) = 38/61.5 = 0.6179
c) Given NG (look only at NG row with total), prob(HS) = 2/38.5 = 0.0519
d) for P to be independent of GR, we need prob(P) = prob(P given GR), that is, need 25/100 = 2.5/61.5. But 25/100 = 0.25 and 2.5/61.5 = 0.0407, so no, they're not independent.

Answer 4

how did you find the probability that it get a a bad review given its highly successful? please help..