An analyst expects that 10% of all publicly traded companies will experience a d
ID: 3363826 • Letter: A
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 ________.
Explanation / Answer
Let A be the event that a publicly traded companies will experience a decline in earnings next year and B be the event that the ratio will be negative.
P(A) = 0.1 P(B|A) = 0.7 P(B|A') = 0.2
P(A') = 1 - 0.1 = 0.9
P(A and B) = P(B|A) P(A) = 0.7 * 0.1 = 0.07
P(A' and B) = P(B|A') P(A') = 0.2 * 0.9 = 0.18
=> P(B) = P(A and B) + P(A' and B) = 0.07 + 0.18 = 0.25
P(A|B) = P(A and B) / P(B) = 0.07 / 0.25 = 0.28