I am just diving into some set theory again and need some helping starting this problem. I am given this hint:
Recall that two events A and B are conditionally independent given an event C if:
$$P(A \cap B \mid C) = P(A \mid C)P(B \mid C)$$
and asked to prove:
$$P(A \mid B \cap C) = P(A \mid C)P(B \mid C) \Leftrightarrow P(A \mid B \cap C) = P(A \mid C)$$
I am completely unsure of where to start. Thanks!
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Hint: When dealing with these kind of problems, get rid of the "conditioning" part. In this case $$P(A|B \cap C)= P(A\cap B \cap C)/P(B\cap C)=\cdots$$
Can you now cleverly break the above and below terms using conditional probability so that a certain term cancels? Then again break it so that you are just staring at the answer?
Also i think there is a typo in the original post. Please check your question. I just gave you a general way to tackle these problems.
I assume there's a typo and you want to show $$ P(A\cap B\mid C) = P(A\mid C)P(B\mid C)\iff P(A\mid B\cap C)=P(A\mid C)$$
By using the definition of conditional probability three times, $$ P(A\mid B\cap C) = \frac{P(A\cap B\cap C)}{P(B\cap C)} = \frac{P(A\cap B\mid C)P(C)}{P(B\cap C)}= \frac{P(A\cap B\mid C)}{P(B\mid C)}.$$