If P(A|B) = P(A), what does this imply about A and B?

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Multiple Choice

If P(A|B) = P(A), what does this imply about A and B?

Explanation:
If knowing that B occurred doesn’t change the chance of A, then A and B are independent. That’s exactly what P(A|B) = P(A) says: B provides no information about A’s likelihood. When two events are independent, observing one doesn’t alter the probability of the other. Thinking about alternatives helps see why this fits. If A and B were mutually exclusive, then seeing B would typically make A impossible (P(A|B) would be 0 whenever P(B) > 0), which usually isn’t equal to P(A). Independence is the relationship that aligns with P(A|B) staying equal to P(A). Additionally, independence implies P(B|A) = P(B) (when those probabilities are defined and positive), but the key point is the idea that the occurrence of B does not affect the probability of A.

If knowing that B occurred doesn’t change the chance of A, then A and B are independent. That’s exactly what P(A|B) = P(A) says: B provides no information about A’s likelihood. When two events are independent, observing one doesn’t alter the probability of the other.

Thinking about alternatives helps see why this fits. If A and B were mutually exclusive, then seeing B would typically make A impossible (P(A|B) would be 0 whenever P(B) > 0), which usually isn’t equal to P(A). Independence is the relationship that aligns with P(A|B) staying equal to P(A). Additionally, independence implies P(B|A) = P(B) (when those probabilities are defined and positive), but the key point is the idea that the occurrence of B does not affect the probability of A.

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