Stupid probability question

I got conned into doing this Eat Well, Live Well Challenge where I work, and one of the things you do is try to walk 10,000 steps a day. So I've been walking to a place around the corner for lunch, which is about 5,000 steps round trip.

On my walk today, I was thinking about probabilities. Although it blew my mind at the time, I've come to accept that a probability of zero only means "almost impossible", not actually "impossible". The classic example is that if you pick a random point inside a unit square (0<=x<=1, 0<=y<=1), the probability of any given point being chosen is, necessarily, zero -- but that doesn't mean it's impossible.

Okay, that's great. Of course, you can take any continuous segment of the unit square and give the probability of that, e.g. the odds of landing in (0<=x<=0.5, 0<=y<=0.5) would be 1/4... right? Interestingly, the odds of landing in (0<=x<0.5, 0<=y<0.5) are also 1/4, I think, but that's okay, because the odds of landing exactly on the lines x=0.5 or y=0.5 are that whole "almost impossible" zero again.

So now let's extend it to, pick any random point on an infinite 2-dimensional grid. Again, the probability of choosing any given point is zero. No problem. But now, the odds of choosing any finite continuous segment are also zero... which I guess I can handle.

Okay, but now what are the odds that the point chosen will be in the infinite continuous segment x>0? I don't know how to show the math (I know it involves integration, but I can't seem to quite figure it out), but it's got to be 50%, right?

Okay... so what are the odds that the point chosen will be in the infinite continuous segment x>1? It's also gotta be 50%, right? I think? Which I guess is okay, because the interval 0<=x<=1 has our good old friend "almost impossible" probability of zero.

Can anybody who knows probability theory tell me if that's correct? And if so, how do you sleep at night? This will piss me off almost as much as the Banach-Tarski Paradox...

Edit: I'm guessing this is how I do the math, but I have to get off the computer now so many I can figure this one out tomorrow...