Question

Does P(A∩B|C)=P(A|C)P(B|C) imply that A and B are independent? Assume P(C)>0, so that the conditional probabilities are defined.

- yes

- no

Please explain the answer

Answer 1

The correct answer is no. [ANSWER]

Explanation:

P(A∩B|C)=P(A|C)P(B|C) does not imply that A and B are independent, it only implies that A and B are independent conditional on C.

The events A and B can be said to be independent if and only if P(A∩B)=P(A)P(B).

This is because the fact that if P(A∩B|C)=P(A|C)P(B|C) holds then it does not imply or guarantee that P(A∩B)=P(A)P(B) will hold.

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