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Math Help - law of total probability

  1. #1
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    law of total probability

    use the law of total probabilty to prove that
    a. if P (A l B) = P (A l B,) then A and B are independent
    b. if P (A l C ) > P (B l C) and P (A l C' ) > P( B l C' ), then P (A) > P (B)
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  2. #2
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    Hello,

    The law of total probability says that P(A)=P(A|B)P(B)+P(A|B')P(B')

    So here, P(A)=P(A|B)[P(B)+P(B')]=P(A|B) since B and B' are complementary events.

    Provided that P(B)\neq 0, it follows that P(A|B)=\frac{P(A\cap B)}{P(B)}=P(A) \Rightarrow P(A\cap B)=P(A)P(B)



    For the second one... :

    P(A)=P(A|C)P(C)+P(A|C')P(C')
    But from the two inequalities, we can say that P(A)>P(B|C)P(C)+P(B|C')P(C'), and the RHS is exactly P(B)
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