An analyst expects that 20 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 90% chance that this ratio will be negative.If the company is not headed for a decline,there is only a 10% chance that the ratio will be negative.The analyst randomly selects a company with a negative ratio.Based on Bayes' theorem,the posterior probability that the company will experience a decline is ____.
A)18%
B)26%
C)44%
D)69%

Question

Quizplus · Accepted Answer

Let A be the event that a company is randomly selected and B be the event that the selected company has a negative ratio. We want to find the probability of A given B, i.e., P(A | B).
Using Bayes' theorem,
P(A | B) = (P(B | A) * P(A)) / P(B)
P(B | A) = 0.9 (given)
P(B | A') = 0.1 (given)
P(A) = 0.2 (given)
P(B) = P(B | A) * P(A) + P(B | A') * P(A') = 0.9 * 0.2 + 0.1 * 0.8 = 0.26
Therefore,
P(A | B) = (0.9 * 0.2) / 0.26 = 0.6923 ≈ 69%
Hence, the answer is (D) 69%.

An Analyst Expects That 20 Percent of All Publicly Traded