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 ____.

Question

Quizplus · Accepted Answer

Let A be the event that the company is headed for a decline (10% chance) and B be the event that the ratio is negative. We need to find P(A|B), the probability that the company is headed for a decline given that the ratio is negative.
Using Bayes's theorem:

P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|not A)P(not A)]

P(B|A) = 0.7 (given)
P(A) = 0.1 (given)
P(B|not A) = 0.2 (given)
P(not A) = 0.9 (complement of A)

Substituting the values:

P(A|B) = 0.7 x 0.1 / [0.7 x 0.1 + 0.2 x 0.9]
= 0.07 / 0.25
= 0.28

Therefore, the posterior probability that the company will experience a decline is 28%, which is option D.

An Analyst Expects That 10% of All Publicly Traded Companies