[hypersoar]

The probability of an event being 1 is _not_ the same thing as that event being completely certain. For example, if you pick a random real number between 0 and 1, the probability of getting something rational is zero. It's clearly not impossible, though.

[gizmo686]

Do you know of any proof for that? My math intuition is telling me that randomly picking a real number is guarenteed to be irrational, based on the fact that there is an uncountable infinity real numbers, but only a countable infinity of rational numbers. But, without assuming a probability of 0 means impossible, I do not know how to go about proving/disproving this.

[hypersoar]

We can't really pick "random real numbers" in any practical sense, so this is pretty much a theoretical distinction. It's essentially a matter of definition. The rational numbers have probability zero of being drawn, but they still lie in the sample space.

One way to see it is this: The probability of picking any particular point in the interval is 0. But that doesn't mean that picking that point is impossible. _Some_ point has to show up when you pick one at random.