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Math Help - Probability Proof

  1. #1
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    Probability Proof

    I am given two events, A and B, and a new event C occurs, if and only if exactly one of A or B occurs. I need to prove;

     Pr(C) = Pr(A) + Pr(B) - 2Pr(A\cap B)

    So from the question, i got:

     Pr(C) = Pr (A \cap \bar {B}) \cup Pr(\bar{A} \cap B)

    Is that correct and a good starting point? Or am i missing something out?
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  2. #2
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    From the second equation you can use:
    P(A\cap\overline{B})=P(A)-P(A\cap B).
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  3. #3
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    So i get:

    (P(A)-P(A\cap B))\cup(P(B)-P(A \cap B))

    Is that correct?

    How do i go about getting rid of the union in the middle?
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  4. #4
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    Quote Originally Posted by Mcoolta View Post
    So i get:

    (P(A)-P(A\cap B))\cup(P(B)-P(A \cap B))

    How do i go about getting rid of the union in the middle?
    Don't you get
    (P(A)-P(A\cap B))+(P(B)-P(A \cap B))~?
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  5. #5
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    Im confused to where the union has gone, from my original post?

    Thanks for your help so far!
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  6. #6
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    Are you not proving that P([A\cap\overline{B}]\cup[B\cap\overline{A}])=P(A)+P(B)-2P(A\cap B)~?
    That is what I thought you were doing. Is that not right?
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  7. #7
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    Yeah that is what im proving, im just confused from where the middle '+' has came from in the 4th post down, instead of a  \cup :

    Why is it:

     P[(A)-(A \cap B)] '+' [(B)-(A \cap B)]

    Instead of

     P[(A)-(A \cap B)] '\cup' [(B)-(A \cap B)]

    Sorry if that is unclear, Thanks
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  8. #8
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    If you know that K\cap J=\emptyset then you know that P(K\cup J)=P(K)+P(J). Right?

    Now (A\cap\overline{B})\cap(B\cap\overline{A})=\emptys  et.

    So the probability of that union is the sum of the probabilities.
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