My Friend Andrew wrote:

Initially, I thought it was dependent on your definition of “random rational number”, but then I saw your second example that removed duplicates (1/2, 2/4, etc..).

I tried a different method by sampling from a real distribution and then rationalizing it:

Mean@Mod[Denominator@Rationalize[#, 1*^-6] & /@ RandomReal[{0, 1}, 100000], 2]

Then, I tried changing the distribution, and still get the same result:

Mean@Mod[Denominator@Rationalize[#, 1*^-6] & /@ RandomReal[NormalDistribution[0, 1], 100000], 2]

— Andrew

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