State Bayes' theorem in simplest form.

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

Answer: (A)

Bayes' theorem shows how to update the probability of a hypothesis after seeing new evidence. In its simplest form, it states that the probability of A given B equals the probability of B given A times the prior probability of A, all divided by the probability of observing B: P(A|B) = [P(B|A) P(A)] / P(B). Here, P(A) is your initial belief about A, P(B|A) is how likely you would see B if A is true, and P(B) is the overall likelihood of observing B across all possibilities.

This form is best because it directly links how likely the evidence is under A to how plausible A is after seeing that evidence, normalized by how probable the evidence is in general. The other expressions either mix up the conditioning (P(B|A) vs P(A|B)), state a different rule for joint probability (P(A and B) = P(A) P(B|A)), or produce an impossible self-referential equation (P(A|B) = P(A) P(B) / P(A|B)).