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Math Help - Help Proving that not all points of an infinite set can be equally likely

  1. #1
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    Help Proving that not all points of an infinite set can be equally likely

    "Consider an experiment whose sample space consists of a countably infinite number of points. Show that not all points can be equally likely. Can all points have positive probability of occurring?"

    The only way I can think of proving this is by contradiction, but I don't know if I believe my own proof:

    If each point is equally likely then there is a 1/n chance of it occurring. However:
    lim n->infinity of 1/n = 0
    All the probabilities of the sample space added together should equal 1, but if the limit is 0 then it follows that 0+0+0+... = 0 is not equal to 1.

    I think the problem with this proof is that 1/n does not really equal 0...it equals epsilon. And epsilon*n should be 1. Is this proof completely off and is there a better way?
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  2. #2
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    I think you want to avoid a "1/n" argument because otherwise you will have to define n, which seems difficult.

    Ask yourself the following questions:

    1. Can the points all have probability p where p > 0 ? Will that work?

    2. If not, how about p = 0?
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  3. #3
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    I meant to define n before - it is simply the number of points.

    For your questions:
    2) Can't be true, because if the probability of each point was 0 then nothing would have any chance of occurring, right?

    1) I'm not sure about, I would like to think it's possible but I'm not sure I grasp the concept of "countably infinite."

    The problem makes sense when I apply it, for example, the instruction "Pick a random positive integer" does not mean anything because there are an unlimited number of positive integers so it's not really possible to randomly pick one. There are some integers trillions of digits long, and I guess I could say the chances of one of those being picked is 0. But if the question doesn't make sense in the first place, I don't see why I have to analyze it.
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    Quote Originally Posted by paulrb View Post
    I meant to define n before - it is simply the number of points.

    For your questions:
    2) Can't be true, because if the probability of each point was 0 then nothing would have any chance of occurring, right?

    1) I'm not sure about, I would like to think it's possible but I'm not sure I grasp the concept of "countably infinite."

    The problem makes sense when I apply it, for example, the instruction "Pick a random positive integer" does not mean anything because there are an unlimited number of positive integers so it's not really possible to randomly pick one. There are some integers trillions of digits long, and I guess I could say the chances of one of those being picked is 0. But if the question doesn't make sense in the first place, I don't see why I have to analyze it.
    If you have an infinite number of points, then the number of points can't be an integer; so you can't "let n be the number of points".

    For (2), what would be the sum of the probabilities of all the events in the space?

    As for the problem not making sense, that's what you are trying to prove-- that no such space can exist.
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    For 2) the sum of the probabilities of all the events in the space would be 0. It should be 1, which means there's a contradiction.

    What I don't understand is the problem is asking me to prove that the events in an infinite set cannot be equally likely. I think it would make more sense if the problem asked me to prove it's not possible to evaluate the probability of events in an infinite set, since doing so receives nonsensical results.
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  6. #6
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    What is

    \sum_{n=1}^\infty p

    if p > 0?
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  7. #7
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    This requires a well-known result from real analysis.
    \left( {\forall \varepsilon  > 0} \right)\left( {\exists N} \right)\left[ {\frac{1}{N} < \varepsilon } \right] \Rightarrow \quad \left[ {1 < N\varepsilon } \right].
    But that means that if we let p=\varepsilon  > 0 then \sum\limits_{k = 1}^N p  = Np > 1.
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