Look, my question about "choosing" a random rational is acceptable, because It assumes that in principal it can be done. The "probability" of a natural number having some property is merely the asymptotic density of the numbers having that property. For example, since there are [sqrt(N)] = sqrt(N) + O(1) squares less than N, the formal probability of randomly selecting a square is lim (N->inf) 1/sqrt(N) = 0.

From any FINITE set of naturals, the probability of selecting a square is about 1/sqrt(N) if N is the largest number in the set.

So all my question asks is what the density of even-denom rationals is in the rationals, which is 1/3.

As to generating random reals, note that the good idea of taking a random real from [0,1] and then taking its reciprocal doesn't really help the "taking a random element from an infinite set" problem, since [0,1] and R have the same cardinality. But I think you know this.

I don't think it is meaningful to talk of "selecting at random from a set" in terms of actual methods, because I challenge one of you to "randomly" select an integer from {1,2,3}.

The real measures of randomness occur in measuring the degree to which a set's statistical properties correspond to those of some distribution. For example, a very simple test of how "random" a finite set of integers is is to merely count the number of evens in it and see if they duplicate the theoretical frequency of 1/2.

I have had some idea recently about measuring the "randomness" (or, since it is equivalent), amount of information provided by, finite sets of natural numbers (NOTE TO ACOLYTE: The goal is to prove a hardcore strong version of van der Waerden's theorem). Interestingly, the measure I have come up with corresponds closely to the conept of thermodynamic entropy that my physics teacher was so fond of haarping on. If anyone's interested, I might post them in all their beautiful incompleteness.