An analyst expects that 10% of all publicly traded companies will experience a d

Question

An analyst expects that 10% of all publicly traded companies will experience a decline in earnings next year. The analyst has developed a ratio to help forecast this decline. If the company is headed for a decline, there is a 70% chance that this ratio will be negative. If the company is not headed for a decline, there is only a 20% chance that the ratio will be negative. The analyst randomly selects a company and its ratio is negative. Based on Bayes's theorem, the posterior probability that the company will experience a decline is ______. 18% 28% 7% 3%

Explanation / Answer

Let N = negative , D = Decline , N' = non negative , D' = non decline

P(D) = 0.1 ,P(D') = 1-0.1 = 0.9

p( ratio negative if decline )= p(N/D) = 0.7

p( ratio non negative if decline )=p(N' /D) = 1- 0.7 = 0.3

p(ratio negative if not decline)= p(N /D') = 0.2

p(ratio not negative if not decline) = p(N'/D')= 1 - 0.2 = 0.8

p(D/N) = ?

P(D/N) = P(D) P(N/D) / {P(D) P(N/D) + P(D') P(N/D') }

= 0.1 * 0.7 / { 0.1*0.7 + 0.9 *0.2}

= 0.07/0.25

= 0.28

= 28%

2nd option that is 28% is correct answer.

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