Which formula represents Bayes' theorem for P(A|B) in terms of P(B|A), P(A), and P(B)?

Which formula represents Bayes' theorem for P(A|B) in terms of P(B|A), P(A), and P(B)?

• A. P(A|B) = [P(B|A)P(A)] / P(B)
• B. P(A|B) = P(A)P(B) / P(A|B)
• C. P(A|B) = P(B|A)P(B)/P(A)
• D. P(A|B) = P(A|B)P(B)/P(A)

Answer: (A)

Bayes' theorem lets you update the probability of A after seeing B. It follows from the definition of conditional probability: P(A|B) = P(A ∩ B)/P(B). The joint probability P(A ∩ B) can be written as P(B|A) P(A). Plugging in gives P(A|B) = [P(B|A) P(A)] / P(B). This is the form for expressing P(A|B) in terms of P(B|A), P(A), and P(B). The other expressions mix up the terms or introduce circulars, so they don’t represent the correct relationship.

Bayes' theorem lets you update the probability of A after seeing B. It follows from the definition of conditional probability: P(A|B) = P(A ∩ B)/P(B). The joint probability P(A ∩ B) can be written as P(B|A) P(A). Plugging in gives P(A|B) = [P(B|A) P(A)] / P(B). This is the form for expressing P(A|B) in terms of P(B|A), P(A), and P(B). The other expressions mix up the terms or introduce circulars, so they don’t represent the correct relationship.