Open Access
May 2020 On the Probability That Two Random Integers Are Coprime
Jing Lei, Joseph B. Kadane
Statist. Sci. 35(2): 272-279 (May 2020). DOI: 10.1214/19-STS737


We show that there is a nonempty class of finitely additive probabilities on $\mathbb{N}^{2}$ such that for each member of the class, each set with limiting relative frequency $p$ has probability $p$. Hence, in that context the probability that two random integers are coprime is $6/\pi ^{2}$. We also show that two other interpretations of “random integer,” namely residue classes and shift invariance, support any number in $[0,6/\pi ^{2}]$ for that probability. Finally, we specify a countably additive probability space that also supports $6/\pi ^{2}$.


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Jing Lei. Joseph B. Kadane. "On the Probability That Two Random Integers Are Coprime." Statist. Sci. 35 (2) 272 - 279, May 2020.


Published: May 2020
First available in Project Euclid: 3 June 2020

MathSciNet: MR4106605
Digital Object Identifier: 10.1214/19-STS737

Keywords: coprime , Finitely additive probability , number theory , uniform distribution

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.35 • No. 2 • May 2020
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