If P(A|B) = P(A) and P(B|A) = P(B), what can we conclude?

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Multiple Choice

If P(A|B) = P(A) and P(B|A) = P(B), what can we conclude?

Explanation:
Independence of two events means that knowing one happened doesn’t change the chance the other happened. If P(A|B) equals P(A), then B provides no information about A. Similarly, if P(B|A) equals P(B), then A provides no information about B. When both equalities hold, the joint chance of both events occurring is the product of their individual chances: P(A∩B) = P(A|B)P(B) = P(A)P(B). This is the hallmark of independence. So A and B do not influence each other; the occurrence of one tells you nothing about the likelihood of the other. It’s not about one happening before the other, and mutual exclusivity would typically force P(A∩B) = 0 (which would only align with independence if at least one event has probability zero).

Independence of two events means that knowing one happened doesn’t change the chance the other happened. If P(A|B) equals P(A), then B provides no information about A. Similarly, if P(B|A) equals P(B), then A provides no information about B. When both equalities hold, the joint chance of both events occurring is the product of their individual chances: P(A∩B) = P(A|B)P(B) = P(A)P(B). This is the hallmark of independence.

So A and B do not influence each other; the occurrence of one tells you nothing about the likelihood of the other. It’s not about one happening before the other, and mutual exclusivity would typically force P(A∩B) = 0 (which would only align with independence if at least one event has probability zero).

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