An analyst expects that 20% 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

Using Bayes' theorem, we calculate the probability of a company experiencing a decline given that its ratio is negative. Let D be the event that a company will experience a decline, and N be the event that the ratio is negative. We are given:- P(D) = 0.20 (probability of decline)- P(N|D) = 0.90 (probability ratio is negative given decline)- P(N|D') = 0.10 (probability ratio is negative given no decline, where D' is the complement of D)We want to find P(D|N), the probability of a decline given a negative ratio.P(D|N) = [P(N|D) * P(D)] / [P(N|D) * P(D) + P(N|D') * P(D')]First, calculate P(D'):P(D') = 1 - P(D) = 1 - 0.20 = 0.80Now, substitute the values into the formula:P(D|N) = (0.90 * 0.20) / [(0.90 * 0.20) + (0.10 * 0.80)]       = 0.18 / (0.18 + 0.08)       = 0.18 / 0.26       = 0.6923...Rounded to the nearest whole number, this gives a 69% chance, which corresponds to choice D.

An Analyst Expects That 20% of All Publicly Traded Companies