Which statement expresses independence of A and B?

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

Answer: (A)

Independence means the occurrence of one event doesn’t change the likelihood of the other. For events A and B, this is expressed by the joint probability equaling the product of their individual probabilities: P(A∩B) = P(A)P(B). This captures the idea that knowing B happened leaves A with its original probability P(A), so the joint chance of both happening is just the product of the separate chances.

The other statements don’t express independence in the necessary way. P(A∩B) = P(A|B)P(B) is always true by the definition of conditional probability, so it isn’t a characterization of independence by itself. P(A|B) = P(B|A) is not a standard criterion for independence and can occur even when the events are not independent. Finally, P(A∩B) = P(A) + P(B) contradicts the basic probability bound that a intersection cannot exceed either probability and is generally false.