I got this one without realizing the complex basis for it... there's an infinite number of real numbers between 0 and 1, compared with a limited proportion of rational numbers.
A way to intuit why you have a zero probability of choosing a rational is to consider randomly picking digits one at a time. Suppose you start with a "3". What is necessary to get one third as your random choice? You must now uniformly randomly select an infinite number of 3s. You can think of that as just a contradiction without much loss here, in that selecting an infinite number of 3s is infinite proof that your process must not be random.
(Putting rigor on that might be a challenge but I think it's a fine intuition.)
This argument similarly holds for any rational, because no matter how large any given rational's repeat may be, it is always finite, and in order for that rational to be your choice you have still fully determined the result of a putatively uniformly random process for an infinite number of exact picks, and if it is truly random, the probability of that process producing the exact necessary repeat infinitely is always 0.
That argument works for any number. The probability of picking any particular number is zero, there's nothing special about rationals from that perspective. This has no bearing on the measure of the set as a whole.
Which is why I very, very explicitly presented it as a way to gain some intuition and not as a rigorous mathematical argument. That wasn't an accident.
The probability of picking a single number can't be used to determine the probability of picking a set. You can't build intuition this way.
This is almost exactly the same thing as saying integrating x^3 from minus infinity to plus infinity is zero because each zero width line has zero area, so adding them all up is zero too.
Gives the right answer by coincidence, but also works to show incorrect results.
I agree with you, except your claim that it proves too much. You will indeed never pick pi or any other specific number you can name. I don't know if you were trying to say my argument leads to those results you consider absurd, but they are in fact the correct results.
One way of interpreting this result is to say that randomly selecting a real number isn't physically meaningful and this does not fit into our human brains well. You can do math on the "random number", but you can't ever actually have one in hand (as a sibling comment to yours points out correctly, it is guaranteed to be uncomputable as well), not even mathematically, so you should expect counterintuitive results. Which is generally true of the real numbers when you get close enough to them anyhow. They are popular for a reason but they have rather more spiky bits and strange behaviors than one might expect from a number called "real".
I think both of you may be thinking of "random number" as something like what you get out of a RNG or something, but random reals are much stranger than random ints from a finite range.
No, I mean that if you start with "you won't land on any given rational" and sum that up to mean "you land on any rationals", that argument also seems to prove that you can't ever land on any irrationals either, so as an intuitive argument it doesn't really point toward why you'll always land on an irrational and not a rational.
("You'll never land on Pi because the odds of picking out the infinitely many digits is zero"- repeat that for every irrational, and naively sum that up, and it looks like you'll never land on an irrational either, which is not true.)
Suppose you construct a fat Cantor set of measure 0.5. You then select a random real from [0,1]. Similar to the rationals, any particular number in the fat Cantor set has zero chance of being selected. Does that mean that the chance of selecting a member of that set is 0?
