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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 updated probability that the company will experience a decline is:

Given a set of prior probabilities for an event of interest, Bayes’ formula is used to update the probability of the event, in this case that the company we have already selected will experience a decline in earnings next year. Bayes’ formula says to divide the Probability of New Information given Event by the Unconditional Probability of New Information and multiply that result by the Prior Probability of the Event. In this case, P(company having a decline in earnings next year) = 0.20 is divided by 0.26 (which is the Unconditional Probability that a company having an earnings decline will have a negative ratio (90% have negative ratios of the 20% which have earnings declines) plus (10% have negative ratios of the 80% which do not have earnings declines) or ((0.90) × (0.20)) + ((0.10) × (0.80)) = 0.26.) This result is then multiplied by the Prior Probability of the ratio being negative, 0.90. The result is (0.20 / 0.26) × (0.90) = 0.69 or 69%.

John purchased 60% of the stocks in a portfolio, while Andrew purchased the other 40%. Half of John’s stock-picks are considered good, while a fourth of Andrew’s are considered to be good. If a randomly chosen stock is a good one, what is the probability John selected it?

Using the information of the stock being good, the probability is updated to a conditional probability:

P(John | good) = P(good and John) / P(good).

P(good and John) = P(good | John) × P(John) = 0.5 × 0.6 = 0.3.

P(good and Andrew) = 0.25 × 0.40 = 0.10.

P(good) = P(good and John) + P (good and Andrew) = 0.40.

P(John | good) = P(good and John) / P(good) = 0.3 / 0.4 = 0.75.

Bonds rated B have a 25% chance of default in five years. Bonds rated CCC have a 40% chance of default in five years. A portfolio consists of 30% B and 70% CCC-rated bonds. If a randomly selected bond defaults in a five-year period, what is the probability that it was a B-rated bond?

According to Bayes' formula: P(B / default) = P(default and B) / P(default).

P(default and B )= P(default / B) × P(B) = 0.250 × 0.300 = 0.075

P(default and CCC) = P(default / CCC) × P(CCC) = 0.400 × 0.700 = 0.280

P(default) = P(default and B) + P(default and CCC) = 0.355

P(B / default) = P(default and B) / P(default) = 0.075 / 0.355 = 0.211

The probability of A is 0.4. The probability of A^C is 0.6. The probability of (B | A) is 0.5, and the probability of (B | A^C) is 0.2. Using Bayes’ formula, what is the probability of (A | B)?

Using the total probability rule, we can compute the P(B):
P(B) = [P(B | A) × P(A)] + [P(B | A^C) × P(A^C)]
P(B) = [0.5 × 0.4] + [0.2 × 0.6] = 0.32

Using Bayes’ formula, we can solve for P(A | B):
P(A | B) = [ P(B | A) ÷ P(B) ] × P(A) = [0.5 ÷ 0.32] × 0.4 = 0.625