> This seems to be just you being confused over the concept of a uniform distribution.
Try and follow the example again.
The distribution is all rationals 1-9 and all numbers 9-10.
Sampling uniformly such that each distance is equally likely across the line gives at least a 90% chance of choosing a rational.
Sampling uniform by elements of the set gives a 0% chance of choosing a rational.
The problem with the latter is that even though you’re claiming to be randomly sampling the _line_ you are never going to sample the first 90% of the line length because you are instead sampling the _distribution of set elements_.
You are NOT more likely to throw a dart that lands in 9+ just because you have magically introduced an infinitely tense series of irrationals in that range.
Try and follow the example again.
The distribution is all rationals 1-9 and all numbers 9-10.
Sampling uniformly such that each distance is equally likely across the line gives at least a 90% chance of choosing a rational.
Sampling uniform by elements of the set gives a 0% chance of choosing a rational.
The problem with the latter is that even though you’re claiming to be randomly sampling the _line_ you are never going to sample the first 90% of the line length because you are instead sampling the _distribution of set elements_.
You are NOT more likely to throw a dart that lands in 9+ just because you have magically introduced an infinitely tense series of irrationals in that range.