> 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_.

This is all in your head. Who are you responding to? Where did your three claims ("sampling uniformly by distance from 0" / "sampling uniformly by element count" / "randomly sampling the line") come from? What does "sampling uniformly by distance" mean? Uniform sampling is done by count for discrete sets and by area for continua. You have yet to mention a discrete set.

It is the difference between picking a random point along a line and picking a random number from a set. A dart throw will not land in the range of [9,10) more often than [1,9) simply because we are considering irrationals in the former.

These are both uniform. But the outcome is different

You're never going to get anywhere without defining your terms.

As you originally pointed out, a physical dart can't hit a single point on a number line. It will hit an infinite number of them simultaneously. This is true whether you're worrying about rationals or reals.

But if you have a dart so sharp that its tip is zero-dimensional, one that can hit a single point on a real line, and you throw it at a composite of the rationals from [0,9] and the reals from [9,10], it will have a 10% chance of hitting an irrational number (within [9,10]), and it will have a 90% chance of missing the line entirely, striking one of the holes in the rational interval [0,9]. The chance of hitting a rational number will not improve from 0.

Do you have a model of uniform selection in mind, or do you find that it's easier to say the words without assigning them any particular meaning?

> Sampling uniformly such that each distance is equally likely across the line gives at least a 90% chance of choosing a rational.

Let's say the numbers are targets on the line. Your distribution implies the range 1-9 is less dense with targets than the range 9-10. Doesn't that mean you're less than 90% likely to hit something between 1-9?

> 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.

If we turn this around, by forbidding a bunch of values in the 1-9 range from being hit, then won't the probabilities get skewed towards the 9-10 range?

No, because a dart throw is not a uniform draw from set elements. Is a uniform draw of length which the inclusion of irrational numbers does not affect. You are 90% likely to throw something in the first 90% of the line. It doesn’t matter if we say we will round anything in [1,2) to 1. There’s a ten percent chance of falling in that range.

Not a 0% chance because there happens to be an uncountable infinity number of options in [9,10)