Something that I find really interesting about probability in continuous systems is that you can have events that are technically possible despite having a probability of zero.
For example, if I'm trying to pick a number between 0 and 1, and the space is continuous, there are an infinite number of numbers available. The odds of it being any particular number in the set one out of infinity, which is to say, zero, yet at the end of the day its still one of those numbers.
A second thing that is interesting about probability in continuous spaces, that is faaaaaar weirder than even the above, is that there can be two events that both have a probability of zero, while one is infinitely more likely than the other.
Imagine a right triangle.
Imagine you are throwing a dart at this right triangle.
The odds of it landing in any particular region is just the area of the region divided by the area of the triangle. One dimensional points don't have an area, so the odds of it hitting any point is, obviously, zero, but lines also don't have an area, so the odds of hitting one of those is, again, zero. But! The odds of hitting anywhere on the line x=0.5 is infinitely more likely than hitting the exact point (0.5,0.5). And then being weirder still, the odds of hitting somewhere on the line x=0.5 is exactly half of what they'd be hitting somewhere on the line x=1, what with the vertical line at x=0.5 being half the length of the vertical line at x=1. So you can have a zero probability even that is infinitely more probably than another zero probability event, and also, exactly twice as likely as another zero probability event. But they're all still zero probability. Which doesn't mean that they can't happen, just that they won't.
(Not a mathematician, this was just sort of casually explained in my Baby Quantum for Dummy Electrical Engineers and we all left the room like this:)
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