@WalterPlinge wrote:
Keep in mind that my explanation is very informal, since it basically amounts to saying "even though they're both infinite, there are infinitely many more real numbers than rational numbers".
Part of the problem (essentially, what makes it counterintuitive) is that there's a lot of machinery needed to even correctly state the question. Standard probability computations work fine when you're dealing with finite situations, but they run into trouble when you're looking at something "sufficiently infinite".
That is, assuming everything is random, the odds of getting a coin to come up heads twice in a row is 1/4. There are 4 possible outcomes (HH, HT, TH, TT) and 1 favorable outcome (HH). So, you can just divide 1 by 4 to get the answer.
In the real numbers case, however, you can't just divide by infinity, so you need to correctly define what it means to determine the probability of hitting a number before you can even ask what is the probability of hitting a real number.
The correct approach is to define a specific way of measuring probability for continuous spaces (i.e. ones that don't have any gaps; consider real numbers vs. rational numbers or finite sets). This is again somewhat complicated, and it's where you might encounter probability distributions (e.g. the normal distribution) or even probability measures (special functions that mimic the standard probability computation outside of finite/discrete cases).
Using these notions it is possible to rigorously state what we mean by "the probability of an arbitrary number on the real number line being rational is 0" (the actual statement would look more like: "the measure of the set of rational numbers on the real number line is 0").
