How does Bayes' theorem work? I need a step-by-step CFA-style example.

Bayes' theorem is about updating your beliefs when you receive new information. It converts a prior probability into a posterior probability.

The Formula:

P(A | B) = [P(B | A) x P(A)] / P(B)

Investment Scenario:

An analyst at Crescent Capital classifies stocks as either "Growth" or "Value." Historical data shows:

• 40% of stocks are Growth: P(Growth) = 0.40
• 60% of stocks are Value: P(Value) = 0.60
• Among Growth stocks, 75% beat earnings: P(Beat | Growth) = 0.75
• Among Value stocks, 30% beat earnings: P(Beat | Value) = 0.30

Question: A stock just beat earnings. What's the probability it's a Growth stock?

Step 1: Find P(Beat) using the total probability rule:

P(Beat) = P(Beat | Growth) x P(Growth) + P(Beat | Value) x P(Value)

P(Beat) = 0.75 x 0.40 + 0.30 x 0.60 = 0.30 + 0.18 = 0.48

Step 2: Apply Bayes' theorem:

P(Growth | Beat) = [P(Beat | Growth) x P(Growth)] / P(Beat)

P(Growth | Beat) = (0.75 x 0.40) / 0.48 = 0.30 / 0.48 = 0.625

Interpretation:

Before the earnings announcement, there was a 40% chance the stock was Growth. After observing it beat earnings, the probability jumps to 62.5%. The earnings beat is evidence that updates our belief toward Growth.

The tree diagram approach (often easier on the exam):

P(Growth | Beat) = 0.300 / (0.300 + 0.180) = 0.300 / 0.480 = 62.5%

Exam tip: When you see phrases like "given new information," "after observing," or "updated probability" — that's Bayes. Draw the probability tree first, then use the formula.

Master Bayes' theorem with our CFA Level I Quantitative Methods course.