[jncfhnb]

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

[thaumasiotes]

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?