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' theorem, the posterior probability that the company will experience a decline is ________.
A) 3%.
B) 7%.
C) 18%.
D) 28%.

Question

Quizplus · Accepted Answer

Using Bayes' theorem:
P(D | R-) = P(R- | D) * P(D) / P(R-)
Where:
P(D) = Probability of a decline = 0.10
P(R- | D) = Probability of a negative ratio given a decline = 0.70
P(R- | ~D) = Probability of a negative ratio given no decline = 0.20
P(~D) = Probability of no decline = 0.90

P(R-) = P(R- | D) * P(D) + P(R- | ~D) * P(~D)
P(R-) = 0.70 * 0.10 + 0.20 * 0.90
P(R-) = 0.23

Now we can solve for P(D | R-):
P(D | R-) = 0.70 * 0.10 / 0.23 = 0.304
So the posterior probability that the company will experience a decline is 30.4%, which rounds to 28% (answer D).

An Analyst Expects That 10% of All Publicly Traded Companies