This is how I thought this through, outloud (but in my head):

In the first example, let's suppose there are some balls in the bag. But which balls could they be? For ball 1 cannot be supposed in the bag, we can be sure that it is taken out in the first step. Likewise, ball 2 cannot be in the bag, since it is taken out in the second step. And so on for ball n, which is taken out in the nth step. Thus, by contradiction, there cannot be any balls in the bag.

The second example seems similar. Can we not use the same reasoning? Perhaps not. For if we ask when ball 1 is taken out, we are immediately ignorant. We don't know when ball 1 is taken out! Indeed, we don't know that it is ever taken out.

So it seems like the mere randomness of the ball-addition-algorithm (so to speak) led to this pseudo-paradox. Weird!

I can't wrap my brain around that, but at least I understand why Georg Cantor went insane.

I've always felt (intuitively, I suck at math) that infinity is a self-refuting concept because anything that is infinitely big is also infinitely small. Any portion of an infinitely large object (like, hypothetically, the universe) is infinitely small in comparison to the whole. Which means it is nothing. Which means that no part of the universe exists. Which is rather obviously wrong.

SPOILER ALERT: My solution (without looking at the answer).

For each step, n, we've added 10n balls, but taken out n balls. So for each n, the number of balls is 10n-n = 9n. Taking this as a sequence, it diverges to infinity.

The problem here is if you take something that's growing infinitely large, and subtract from that something also gowning infinitely large, you don't always get 0. And, this is one of those cases.

Well, infinity is a hard thing to wrap your head around. They don't get into really studying infinity until upper division mathematics classes. If you stopped at calculus or DE, then I'd understand the confusion.

Anyway, your example is wrong. For instance, the real number line, and the segment between 0 and 1, denoted (0, 1), both have infinitely many numbers in them. But since (0, 1) is only a "portion" of the number line, you'd think it's infinity smaller than the number line as a whole.

But the function "f(x) = 2/Pi * arctan(x)" maps every point on the real number line to one, and only one point in (0, 1). Moreover, every point in (0, 1) gets mapped to, i.e. for every y, a number in (0,1) theres a point on the the real number line, x, such that f(x) = y.

Anyway, this function pairs, in a one-to-one fashion, every point on the real number line with every point in (0, 1). So, they both have the same "number" of points in.

The key in dealing with infinities is using function like f. We compare sizes of infinities (yes, there are different sizes of infinity), by trying to find a function that matches points.

But the numbers between 0 and 1 are abstracts. It is unsurprising that we can conceive of abstract infinities, for instance, I can imagine a perfect square even though such a thing could never exist in the real world.  I (sort of) understand that the math works, but just because we can describe a concept mathmatically, does that make it real? My example of the square would seem to refute the idea.

I'm saying that the actual universe can't be infinite. Do your numbers show that it can? 

I'm familiar with mathematical induction, but I don't know if it would be useful in showing the bag would be empty. Go ahead and give it a try.