On the Probability That Two Random Integers Are Coprime

We show that there is a nonempty class of finitely additive probabilities on N2 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/π2. We also show that two other inte...

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Bibliographic Details
Published in:Statistical science 2020-05, Vol.35 (2), p.272-279
Main Authors: Lei, Jing, Kadane, Joseph B.
Format: Article
Language:English
Subjects:
Integers
Probability
Statistics
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Summary:We show that there is a nonempty class of finitely additive probabilities on N2 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/π2. We also show that two other interpretations of "random integer," namely residue classes and shift invariance, support any number in [0, 6/π2] for that probability. Finally, we specify a countably additive probability space that also supports 6/π2.
ISSN:0883-4237
2168-8745
DOI:10.1214/19-STS737