Answer to Question #183515 in Statistics and Probability for nanthini

Question

The editor of a textbook publishing company is trying to decide whether to publish a proposed business statistics textbook. Information on previous textbooks publishes indicates that 10% are huge successes, 20% are modest successes, 40% break even and 30% are losers. However before a publishing decision is made, the book will be reviewed. In the past, 99% of the huge successes received favourable reviews, 70% of the moderate successes received favourable reviews, 40% of the break even books received favourable reviews and 20% of the losers received favourable reviews.

a) If the proposed textbooks receives a favourable review, how should the editor revise the probabilities of the various outcomes to take this information into account?

b) What proportion of textbooks receives favourable reviews?

Answer 1

We have given that,

10% are huge successes

20% are modest successes

40% break even

and 30% are losers.

99% of the huge successes received favourable reviews

70% of the moderate successes received favourable reviews

40% of the break even books received favourable reviews and

20% of the losers received favourable reviews.

P(Huge successes) = 0.1

P(Moderate successes) = 0.2

P(Break even successes) = 0.4

P(Losers) = 0.3

P(huge successes received favourable reviews) = 0.99

P( moderate successes received favourable reviews) = 0.7

P(break even books received favourable reviews) = 0.4

P(losers received favourable reviews) = 0.2

i.) Using the Baye's Theorem,

P (huge success/favorable review ) $= \dfrac{0.1\times 0.9}{0.1\times 0.9+0.2 \times 0.7+ 0.4 \times 0.4+ 0.3 \times 0.2}$

$= \dfrac{0.09}{0.459} = 0.2157$

P(moderate success/favorable review)  = $\dfrac{0.2\times 0.7}{0.1\times 0.9+0.2 \times 0.7+ 0.4 \times 0.4+ 0.3 \times 0.2}$

= 0.3050

P(break even/favorable review) = $\dfrac{0.4\times 0.4}{0.1\times 0.9+0.2 \times 0.7+ 0.4 \times 0.4+ 0.3 \times 0.2}$

= 0.3486

P(loser/favorable review) = $\dfrac{0.3\times 0.2}{0.1\times 0.9+0.2 \times 0.7+ 0.4 \times 0.4+ 0.3 \times 0.2}$

= 0.1307

ii.)Proportion of textbooks receives favourable reviews = $0.1\times 0.9+0.2 \times 0.7+ 0.4 \times 0.4+ 0.3 \times 0.2$ = 0.459