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Math Help - Probability Proof

  1. #1
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    Probability Proof

    Problem:

    Let A and B be two events, show that:

    P(A) + P(B) -1 $\leq$ P(A U B) $\leq$ P(A)+P(B)

    Questions/Attempts:

    Am I supposed to assume that A and B are the only events in the sample space, and that therefore P(A)+P(B)=1?

    If I assume that, and begin the proof with the first axiom of probability, I get this far:

    1. 0 $\leq$ P(A) $\leq$ 1

    2. P(A)+P(B)=1 (Since P(S)=1, where P(S) is the sample space)

    so

    3. 0 $\leq$ P(A) $\leq$ P(A) + P(B)


    There's a couple things I can do from here but it seems to make things more complicated than they need to be.

    Thanks for any help in advance!
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  2. #2
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    Use P(A)+P(B)-P(A\cap B)=P(A\cup B)\le 1 and 0\le P(A\cap B)\le 1
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  3. #3
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    Questions:

    1. If I reduce the problem statement to an identity, is that considered "proving" it?

    2. In this problem, can I assume P(A)+P(B)=1?

    Attempt:

    1.P(A) + P(B) -1 $\leq$ P(A U B) $\leq$ P(A)+P(B)

    2. P(A) + P(B) -1 $\leq$ P(A)+P(B)- P(A \cap B ) $\leq$ P(A)+P(B)

    but P(A)+P(B)=1 so,

    3. 0 $\leq$ 1 - P(A \cap B ) $\leq$ 1

    and P(A \cap B ) is between 0 and 1.


    It almost seems like the proof went backwards here, from a statement to an identity. Is this correct? Thanks for your help Plato I appreciate it.
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  4. #4
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    Quote Originally Posted by divinelogos View Post
    1. If I reduce the problem statement to an identity, is that considered "proving" it?

    2. In this problem, can I assume P(A)+P(B)=1?
    No to both of those.
    From the axioms we get:
    P(A)+P(B)=P(A\cup B)+P(A\cap B)\le P(A\cup B)+1

    OR P(A)+P(B)-1\le P(A\cup B).
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  5. #5
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    How do you start the proof? What axiom do you start with?
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