[humodz]

> 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?

[jncfhnb]

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)