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Math Help - Probability Axioms

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    Probability Axioms

    How would you go about doing this? is it really obvious?
    Attached Thumbnails Attached Thumbnails Probability Axioms-picture-2.png  
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    Quote Originally Posted by n7615r View Post
    How would you go about doing this? is it really obvious?
    So where are you having trouble? Is it in:

    The first part : Reading set notations(about the infinitely often)

    The second part : The union bound axiom? the one that says probability of union of countable number of events is upperbounded by the sum of probability of those events

    In case of basic application problems like these, its beneficial to both (us and you), if you tell us where you are having trouble.
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    i'm having trouble understanding the set notation and how 'infintely often can be written in that way.

    and also showing how the P(A)≤P(Bn). is it because A is a subset of Bn?

    and how does this relate to the third part?
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  4. #4
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    Hello,

    If B_n happens, it means that there is at least one A_m in its union that happens.
    This can be translated as " \exists m\geq n, such that A_m happens"

    Then, for A=\bigcap_{n\geq 1} B_n to happen, B_n has to happen for any n\geq 1

    This can finally be translated as "for any n, there exists m>n such that A_m happens".
    In other words, there are infinitely many A_m events that happen.

    You can read this for more things : http://www.mathhelpforum.com/math-he...up-liminf.html in particular post #4, by Laurent


    and also showing how the P(A)≤P(Bn). is it because A is a subset of Bn?
    Probability axioms - Wikipedia, the free encyclopedia
    Since A is, as you noticed, a subset of B_n, then there exists a set E, disjoint from A, such that A\cup E=B_n
    By the third axiom, we have (the red equality) :
    \mathbb{P}(B_n)=\mathbb{P}(A\cup E){\color{red}=}\mathbb{P}(A)+\mathbb{P}(E)

    Now, by the first axiom, we know that \mathbb{P}(E)\geq 0

    And it follows that \mathbb{P}(B_n)\geq \mathbb{P}(A)


    and how does this relate to the third part?
    Note that P( A_n happens infinitely often)=0 \Leftrightarrow P(A)=0

    And note that if \sum_{m\geq 1} a_m<\infty, this implies that the partial reminder (I don't know the exact name) \sum_{m\geq n} a_m \xrightarrow[]{n\to\infty} 0
    It should be okay from now


    The second part : The union bound axiom? the one that says probability of union of countable number of events is upperbounded by the sum of probability of those events
    Hmm this axiom isn't stated in the probability axioms. Nor it is stated in the axioms defining a measure. Though it's a direct consequence. I know the French name, but not sure in English... I think it is called sigma subadditivity or something like that...
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    Thanks so much! this was a great help
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    Quote Originally Posted by Moo View Post
    Hmm this axiom isn't stated in the probability axioms. Nor it is stated in the axioms defining a measure. Though it's a direct consequence. I know the French name, but not sure in English... I think it is called sigma subadditivity or something like that...
    I know that, but since its equivalent to the axiom, I generally find it convenient to remember in that form
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