The probability of picking a rational number from [0, 1] under a uniform distribution is exactly zero. That’s because in a uniform distribution, the probability of picking a number from a given subset of [0, 1] is just the Lebesgue measure of that set.

The Lebesgue measure, like any other measure, has two characteristics:[ol][li]The Lebesgue measure of a set is non-negative, but it may be infinite (the infinity here is a formal symbol, not a transfinite number).The Lebesgue measure of a countable union of disjoint sets is the sum of the measures of the sets.[/ol]For an interval, the Lebesgue measure is just the length of the interval (whether it’s open or closed). Since the interval consisting of a single point has length zero, the Lebesgue measure of a set consisting of a single point is zero. [/li]

Take a countable set, such as the rationals. This can be expressed as the union of its points. Obviously, {a} = {b} only if a = b, so this is the countable union of disjoint sets. By 2) above, the Lebesgue measure of **Q** is zero–it’s the sum from 1 to infinity of zero, and that adds up to be nothing (in the formal limit sense, just like any other infinite sum).

Since the probability of picking from **Q** is just the Lebesgue measure of **Q**, and **Q** has Lebesgue measure zero, the probability of picking a rational number is zero. As I said before, this was discussed in the other thread referenced in the OP. Please read that thread.