Bayes' theorem for two events states: P(A|B) equals what?

• 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(A) P(B) / P(B).
• D. P(A ∩ B) = P(A|B) P(B).

Answer: (A)

Bayes' theorem updates what we believe about A after seeing B. It says 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 B: P(A|B) = P(B|A) P(A) / P(B). The numerator P(B|A)P(A) combines how likely B is if A is true with how likely A was to begin with, and dividing by P(B) normalizes this to a real probability by accounting for how common B is overall. In other words, we adjust our initial belief about A by how B supports A, then scale by how likely B is in the population.

Note that P(A ∩ B) = P(A|B) P(B) is a true identity but it doesn’t give the Bayes update in terms of P(B|A) and P(A); it’s a different relationship. The other forms either misplace terms or imply independence when that isn’t guaranteed.