An analyst expects that 10 percent 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' theorem,the posterior probability that the company will experience a decline is ____.
A)3%
B)7%
C)18%
D)28%

Question

Quizplus · Accepted Answer

Let A be the event that the company will experience a decline in earnings next year and B be the event that the ratio is negative. We want to find P(A|B).

We know that P(A) = 0.1 (given in the problem statement). 
We also know that P(B|A) = 0.7 (if the company is headed for a decline, there is a 70% chance that the ratio will be negative) and P(B|A') = 0.2 (if the company is not headed for a decline, there is only a 20% chance that the ratio will be negative).

Using Bayes' theorem,

P(A|B) = P(B|A) * P(A) / [P(B|A) * P(A) + P(B|A') * P(A')] 
           = 0.7 * 0.1 / [0.7 * 0.1 + 0.2 * 0.9] 
           = 0.28 or 28%

Therefore, the answer is D) 28%.

An Analyst Expects That 10 Percent of All Publicly Traded