Taking rational numbers at random

TL;DR

The paper proposes methods to define and analyze probability distributions on the rational numbers in [0,1], exploring their properties and the asymptotic behavior of probabilities assigned to individual numbers and intervals.

Contribution

It introduces simple prescriptions for distributions on rational numbers in [0,1] and investigates their properties and asymptotic behavior, including the challenge of defining a uniform distribution.

Findings

Probability of a single rational number vanishes asymptotically.

Probability of rationals in an interval [a,b] approaches b-a.

No proper uniform distribution exists on all rational numbers in [0,1].

Abstract

We outline some simple prescriptions to define a distribution on the set $\mathbb{Q}_0$ of all the rational numbers in $[0,1]$, and we then explore both a few properties of these distributions, and the possibility of making these rational numbers asymptotically equiprobable in a suitable sense. In particular it will be shown that in the said limit -- albeit no uniform distribution can be properly defined on $\mathbb{Q}_0$ -- the probability allotted to a single $q\in\mathbb{Q}_0$ asymptotically vanishes, while that of the subset of $\mathbb{Q}_0$ falling in an interval $[a,b]$ goes to $b-a$. We finally give some hints to completely sequencing without repetitions the numbers in $\mathbb{Q}_0$ as a prerequisite to the laying down of more distributions on it