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What lies left of a random number on a line of integers

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  1. Jan 17, 2016 #1
    When I pick a random number on a number line made out of integers, starting from zero and expanding infinite to the right, what can I say about the position of this random number ?
    To the right the amount of numbers is infinite.
    To the left is an amount, a number, so that is finite, but it has an ´almost´ infinite big chance of being the biggest number ever. (and of course if could also be, for example, 103 or a googolplexian, but there is an almost infinite small chance it will be that small...).

    Does that make sense ? Please comment on this.
     
  2. jcsd
  3. Jan 17, 2016 #2

    Krylov

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    You have to start by setting up a correct probability model. The sample space ##\Omega := \mathbb{Z}_+##, the set of non-negative integers. The collection ##\mathcal{A}## of events ("measurable subsets of ##\Omega##") is simply the power set of ##\Omega##, so ##\mathcal{A} := 2^{\Omega}##. Now, concerning the probability measure, there is no such measure that assigns equal probability to all non-negative integers, because that measure would assign infinite mass to ##\Omega## itself. Indeed, any valid probability measure ##P## on ##\mathcal{A}## assigns to a singleton ##k \in \Omega## the probability ##p_k \in [0,1]## in such a way that
    $$
    \text{probability of occurrence of event } E = P(E) = \sum_{k \in E}{p_k}, \qquad P(\Omega) = 1
    $$
    An example is the often encountered Poisson distribution, for which ##p_k := e^{-\lambda}\frac{\lambda^k}{k!}## for ##k \in \Omega##, where ##\lambda > 0## is a fixed parameter. So, given any admissible ##P## ,
    • the probability of finding exactly the number ##m## is ##p_m \in [0,1]##,
    • the probability of finding a number ##\le m## is simply ##\sum_{k=0}^m{p_k} < \infty##,
    • the probability of finding a number ##> m## is ##\sum_{k=m+1}^{\infty}{p_k} < \infty##.
    All these probabilities are finite.
     
    Last edited: Jan 17, 2016
  4. Jan 17, 2016 #3
    When people say "pick a random number" it is usually implied of picking a number from a set, with equal probability of choosing any number. It is impossible to do this with an infinite set. Assuming that it is possible leads to a contradiction, which is what you seem to be wrestling with.
     
  5. Jan 17, 2016 #4
    Thanks for your reply. The problem came up for me because I was thinking about the possibility the universe is in an infinite loop of expanding and contracting
    (the closed model) since the first time ´once´ in the ´past´ . Looking at this theory, I was thinking about the number of times the universe would have expanded and contracted before our existence in this cycle. That is, as I concluded, the same as choosing a random number on a line of integers, so a came up with the idea the chance is very high it is a very large number.
     
  6. Jan 17, 2016 #5

    Krylov

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    You are welcome...
     
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