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Math Help - proving one probability is greater than another

  1. #1
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    Exclamation proving one probability is greater than another

    The claim is : Let S be a well defined sample space where E and F are events such that
    Pr(E) >1/2 implies Pr(F)<1/2

    I have no background on proving probability theory. help please!!
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  2. #2
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    Quote Originally Posted by nikie1o2 View Post
    The claim is : Let S be a well defined sample space where E and F are events such that
    Pr(E) >1/2 implies Pr(F)<1/2
    Of course that statement is false as written.
    Consider tossing a single die.
    Let E be the die shows at least three.
    Let F be the die shows at most five.
    P(E)=\frac{4}{6}~\&~P(F)=\frac{5}{6}.
    Both are more than \frac{1}{2}.

    Perhaps you misread the question or you left some bit of information out.
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  3. #3
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    Yes, you are correct - a great detail...prove or DISPROVE the following claim. Thank you very much for the counter example
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    What would be the outcome if i had the same claim as above but in addition E and F are mutually exclusive ? i.e. prove or disprove- Let S be a well defined sample space where E and F are events such that E and F are mutually exclusive. Pr(E)>1/2 implies Pr(F)< 1/2
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  5. #5
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    Quote Originally Posted by nikie1o2 View Post
    What would be the outcome if i had the same claim as above but in addition E and F are mutually exclusive ? i.e. prove or disprove- Let S be a well defined sample space where E and F are events such that E and F are mutually exclusive. Pr(E)>1/2 implies Pr(F)< 1/2
    Indeed, if E and F are mutually exclusive then the statement is correct.
    1\ge P(E\cup F)=P(E)+P(F)>\frac{1}{2}+P(F).
    So what about P(F)?
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  6. #6
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    my counter example

    Quote Originally Posted by Plato View Post
    Of course that statement is false as written.
    Consider tossing a single die.
    Let E be the die shows at least three.
    Let F be the die shows at most five.
    P(E)=\frac{4}{6}~\&~P(F)=\frac{5}{6}.
    Both are more than \frac{1}{2}.

    Perhaps you misread the question or you left some bit of information out.
    Pertaining to my first claim to prove or disprove: Let S be a well defined sample space where E and F are events such that
    Pr(E) >1/2 implies Pr(F)<1/2


    I don't know if this is right so maybe so can critique... As a counter example i said: Let S={1,2,3,4,5,6}. Let E={1,2,3,4} and Let F={1,2,3,4}. Then, the Pr(E)= 2/3 and the Pr(F)=2/3 which is >1/2 thus we have an implication with a true hypothesis and false conclusion which makes the implication false so it follows that the claim is also false. E.E.F
    Last edited by nikie1o2; September 9th 2010 at 07:40 PM. Reason: misleading
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