Which expression correctly defines P(B|A)?

Which expression correctly defines P(B|A)?

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

Answer: (A)

Conditional probability asks what the chance of B is when we already know A has occurred. It is defined as P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0. Intuitively, you’re focusing only on the outcomes where A happened and asking what fraction of those also include B. The numerator counts the outcomes where both A and B occur, while the denominator counts all outcomes where A occurs. This is different from the probability of A given B, which uses P(B) in the denominator, and from the idea of independence, which states P(A ∩ B) = P(A)P(B). It also differs from the union rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The given expression is precisely the definition of P(B|A).

Conditional probability asks what the chance of B is when we already know A has occurred. It is defined as P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0. Intuitively, you’re focusing only on the outcomes where A happened and asking what fraction of those also include B. The numerator counts the outcomes where both A and B occur, while the denominator counts all outcomes where A occurs. This is different from the probability of A given B, which uses P(B) in the denominator, and from the idea of independence, which states P(A ∩ B) = P(A)P(B). It also differs from the union rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The given expression is precisely the definition of P(B|A).