Question

The editor of a textbook publishing company is deciding whether to publish a proposed textbook. Information on previous textbooks published show that 10 % are huge​ successes, 20 % are modest​ successes, 50 % break​ even, and 20 % are losers. Before a decision is​ made, the book will be reviewed. In the​ past, 99 % of the huge successes received favorable​ reviews, 60 % of the moderate successes received favorable​ reviews, 40 % of the​ break-even books received favorable​ reviews, and 20 % of the losers received favorable reviews. If the textbook receives a favorable​ review, what is the probability that it will be huge​ success?

Answer 1

Answer:

If the textbook receives a favorable review, there is a 21.57% probability that it will be a huge success.

Step-by-step explanation:

We have the following probabilities:

-A 10% probability that the textbook is a huge success.

-A 20% probability that the textbook is a modest success.

-A 50% probability that the textbook breaks even

-A 20% probability that the textbook is a loser

-If the book is a huge success, there is a 99% probability that it receives favorable reviews.

-If the book is a moderate success, there is a 60% probability that it receives favorable reviews.

-If the book breaks even, there is a 40% probability that it receives favorable reviews.

-If the book is a loser, there is a 20% probability that it receives favorable reviews.

If the textbook receives a favorable​ review, what is the probability that it will be huge​ success?

This can be formulated as the following problem:

What is the probability of B happening, knowing that A has happened.

It can be calculated by the following formula

$P = \frac{P(B).P(A/B)}{P(A)}$

Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.

So, for this problem.

What is the probability that the book is a huge success, given that it received favorable reviews?

P(B) is the probability that the book is a huge success. So:

$P(B) = 0.1$

P(A/B) is the probability that the book receives favorable reviews when it is a huge success. So:

$P(A/B) = 0.99$

P(A) is the probability that the book receives favorable reviews:

$P(A) = P_{1} + P_{2} + P_{3} + P_{4}$

$P_{1}$ is the probability that a book that is a huge success is chosen and receives favorable reviews. So:

$P_{1} = 0.1*0.99 = 0.099$

$P_{2}$ is the probability that a book that is a moderate success is chosen and receives favorable reviews. So:

$P_{2} = 0.2*0.6 = 0.12$

$P_{3}$ is the probability that a book that breaks even is chosen and receives favorable reviews. So:

$P_{3} = 0.5*0.4 = 0.20$

$P_{4}$ is the probability that a book that is a loser is chosen and receives favorable reviews. So:

$P_{4} = 0.20*0.20 = 0.04$

So

$P(A) = P_{1} + P_{2} + P_{3} + P_{4} = 0.099 + 0.12 + 0.20 + 0.04 = 0.459$

If the textbook receives a favorable​ review, what is the probability that it will be huge​ success?

$P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.1*0.99}{0.459} = 0.2157$

If the textbook receives a favorable review, there is a 21.57% probability that it will be a huge success.