Bayes' theorem expresses the relationship between P(A|B), P(B|A), P(A) and P(B). Which of the following formulas correctly represents Bayes' theorem?

Bayes' theorem expresses the relationship between P(A|B), P(B|A), P(A) and P(B). Which of the following formulas correctly represents Bayes' theorem?

• A. P(A|B) = P(B|A) * P(A) / P(B)
• B. P(A|B) = P(A|B) * P(B) / P(A)
• C. P(A|B) = P(B|A) * P(B) / P(A)
• D. P(A|B) = P(B) / (P(A) + P(B))

Answer: (C)

Bayes' theorem shows how to update the probability of A after observing B by combining how likely B is if A is true (the likelihood) with how likely A was to begin with (the prior), then normalizing by how common B is (the marginal likelihood). The correct relation is P(A|B) = [P(B|A) P(A)] / P(B). This follows from the joint probability equality P(A ∩ B) = P(B|A) P(A) = P(A|B) P(B); solving for P(A|B) gives the stated formula. The normalization by P(B) ensures the result is a proper probability and reflects the new evidence B. Other forms mix up the order of the terms or place the conditional incorrectly, which would violate the equality of the joint probability.

Bayes' theorem shows how to update the probability of A after observing B by combining how likely B is if A is true (the likelihood) with how likely A was to begin with (the prior), then normalizing by how common B is (the marginal likelihood). The correct relation is P(A|B) = [P(B|A) P(A)] / P(B). This follows from the joint probability equality P(A ∩ B) = P(B|A) P(A) = P(A|B) P(B); solving for P(A|B) gives the stated formula. The normalization by P(B) ensures the result is a proper probability and reflects the new evidence B. Other forms mix up the order of the terms or place the conditional incorrectly, which would violate the equality of the joint probability.