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

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1. Jan 17, 2016

### Tomon

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. Jan 17, 2016

### Krylov

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
3. Jan 17, 2016

### Hornbein

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.

4. Jan 17, 2016

### Tomon

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.

5. Jan 17, 2016

### Krylov

You are welcome...