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Math Help - Deriving independence from conditional probability

  1. #1
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    Deriving independence from conditional probability

    If [A|B] = [A|B complement], show that A and B are independent.

    Here's what I've got so far:

    Using the definition of conditional probability and the multiplicative law, I arrived at this:

    P[A|B] = (P[B complement|A]*P[A])/P[B complement]

    I'm not sure how I'm supposed to arrive at the definition of independence (P[A|B] = P[A]) from this.
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  2. #2
    Member courteous's Avatar
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    We use the known equality: P(A|B)=\frac{P(AB)}{P(B)}=\frac{P(AB')}{P(B')}.
    We also know that P(B')=1-P(B) and use that on last fraction: P(AB')=P(A|B)-P(A|B)P(B)=P(A|B)-P(AB) \Rightarrow P(A|B)=P(AB')+P(AB)=P(A)
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