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Math Help - Discrete Math Probability Proving.

  1. #1
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    Discrete Math Probability Proving.

    Suppose that

    a) P(A ∩ B ∩ C) = P(A) P(B) P(C)
    P(A*∩ B ∩ C) = P(A*) P(B) P(C)
    P(A ∩ B ̅ ∩ C) = P(A) P(B*) P(C)
    P(A ∩ B ∩ C) = P(A) P(B) P(C*)
    Prove that A B and C are independent.

    b) Suppose that A B and C are independent. Then so are A* B* and C.

    Note: A* means bar under A. Same for B & C.


    I've no idea how to begin with, so please help me out.

    Thanks.
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  2. #2
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    Re: Discrete Math Probability Proving.

    Quote Originally Posted by blackZ View Post
    a) P(A ∩ B ∩ C) = P(A) P(B) P(C)
    P(A*∩ B ∩ C) = P(A*) P(B) P(C)
    P(A ∩ B ̅ ∩ C) = P(A) P(B*) P(C)
    P(A ∩ B ∩ C) = P(A) P(B) P(C*)
    Prove that A B and C are independent.
    Note: A* means bar under A. Same for B & C.
    \begin{align*} P(A\cap B) &=P(A\cap B\cap C)+ P(A\cap B\cap \overline{C})\\ &= P(A)P(B)P(C)+P(A)P(B)P(\overline{C})\\ &=P(A)P(B)[P(C)+P(\overline{C})]\\ &=P(A)P(B)[1] \\ &=P(A)P(B)\end{align*}
    So A~\&~B are independent.

    Now you have to do the other two cases on your own.
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  3. #3
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    Re: Discrete Math Probability Proving.

    For a) In order to prove B&C as independent, do I need to replace A & B to B & C as you did. And later add all of them.
    Also what is [1]? Is it part of the proof?
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  4. #4
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    Re: Discrete Math Probability Proving.

    Quote Originally Posted by blackZ View Post
    Also what is [1]? Is it part of the proof?
    That is the number 1: P(X)+P(\overline{X})=1
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