> As you roll the dice an infinite number of times, what is the probability that your die will ever deviate from the necessary strict repetition of the repeating segment? It is literally 100%, because no matter how many times it happens to conform to the repeating sequence, there remain an infinite number of rolls remaining in which to deviate.
While I think that accumulating as much intuition as possible is a good idea, I'm afraid that this intuition may be misleading. The problem is that there's no roll at which we can give up and say "oh well, we deviated from the pattern and aren't rolling a rational number", because we have no idea in advance when we've entered the repeating portion of the decimal expansion. That is, if we roll 0.000123456789, and then roll a 6 rather than a 1, we can't conclude that we're rolling an irrational number; all we can conclude is that, if we roll a rational number, then the repeating portion of its decimal expansion either isn't 123456789, or else hasn't begun yet.
For any given rational number, my argument holds. I meant to type "suppose we're going to roll a particular rational number", but I see that I neglected to edit that in.
For any given roll up to a certain point, there are a finite number of rational numbers that you could be rolling. For any given one of those numbers, my argument holds. There is never a point at which any rational number you could be rolling has a probability greater than zero.
While I think that accumulating as much intuition as possible is a good idea, I'm afraid that this intuition may be misleading. The problem is that there's no roll at which we can give up and say "oh well, we deviated from the pattern and aren't rolling a rational number", because we have no idea in advance when we've entered the repeating portion of the decimal expansion. That is, if we roll 0.000123456789, and then roll a 6 rather than a 1, we can't conclude that we're rolling an irrational number; all we can conclude is that, if we roll a rational number, then the repeating portion of its decimal expansion either isn't 123456789, or else hasn't begun yet.