What is P(A|B) defined as?

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

Answer: (A)

Conditional probability measures how likely A is once we know B has occurred. It is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. This formula reflects narrowing the sample space to the outcomes in B and asking what fraction of those outcomes also belong to A. The expression for B given A is P(B|A) = P(A ∩ B) / P(A), which is a different conditioning. The alternative P(A|B) = P(A ∩ B) / P(A) would mix up the conditioning by dividing by P(A) instead of P(B). And P(A ∩ B) = P(A) P(B) is a statement about independence, not the definition of conditional probability—it only holds when A and B are independent.