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Math Help - Probability proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    Probability proof

    Q: If A and B are two events, prove that P(A\cap\\B)\geq\\1-P(A^{c})-P(B^{c}).

    I am having a rough start with this one. Any direction would be much appreciated.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Replace on the right, P(A')=1-P(A) and the similar equation with B.

    Hence you need to show that

    P(A)+P(B)-1\le P(AB)

    On the left use that P(A)+P(B)=P(A\cup B)+P(AB)

    So, you need to show that

    P(A\cup B)+ P(AB)-1\le P(AB)

    and that reduces to

    P(A\cup B)\le 1 WHICH is where you should start this proof.
    -----------------------------------------------------------------------------

    P(A\cup B)\le 1

    P(A)+P(B)-P(AB)\le 1

    \bigl(1-P(A')\bigr)+\bigl(1-P(B')\bigr)-P(AB)\le 1

    1-P(A')+1-P(B')-P(AB)\le 1

    1-P(A')-P(B')\le P(AB)
    Last edited by matheagle; September 8th 2009 at 11:57 PM.
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