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Math Help - Independent Prob Proof

  1. #1
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    Independent Prob Proof

    If A and B are independent events each with positive probability, prove that they cannot be mutually exclusive.

    Not to sure about this one but this is what I have:

    \displaystyle P(A\cap B)=P(A)P(B)\Rightarrow P(A|B)=P(A) \ \ \text{and} \ \ P(B|A)=P(B)

    This is all I have.
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by dwsmith View Post
    If A and B are independent events each with positive probability, prove that they cannot be mutually exclusive.

    Not to sure about this one but this is what I have:

    \displaystyle P(A\cap B)=P(A)P(B)

    This is all I have.
    Okay so,

    The events are mutually exclusive if P(A \cap B)=0 . Here you have P(A) \;and\; P(B) are both greater than 0, so P(A \cap B) \neq 0.
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  3. #3
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    Quote Originally Posted by harish21 View Post
    Okay so,

    The events are mutually exclusive if P(A \cap B)=0 . Here you have P(A) \;and\; P(B) are both greater than 0, so P(A \cap B) \neq 0.
    Proof by contradiction:

    P\land\sim Q

    P(A\cap B)=P(A)P(B)>0

    We have reached a contradiction since P(A\cap B)=P(A)P(B)\neq 0.

    Therefore, A and B cannot be mutually exclusive.
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