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's theorem,the posterior probability that the company will experience a decline is

Question

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Let A be the event that a company will experience a decline in earnings next year, and let B be the event that the company has a negative ratio. We want to find P(A|B).
By Bayes's theorem, we have P(A|B) = P(B|A) * P(A) / P(B). 
We know that P(A) = 0.20, and P(B|A) = 0.90 (the probability that the ratio is negative if the company is headed for a decline). 
To find P(B), we use the law of total probability: 
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A) 
where P(not A) = 1 - P(A) = 0.80, and P(B|not A) = 0.10 (the probability that the ratio is negative if the company is not headed for a decline). 
Therefore, P(B) = (0.90 * 0.20) + (0.10 * 0.80) = 0.26. 
Plugging in the values, we get P(A|B) = (0.90 * 0.20) / 0.26 = 0.6923, or approximately 69%. Therefore, the answer is D.

An Analyst Expects That 20% of All Publicly Traded Companies