In probability theory, when dealing with continuous sample spaces / random variables, events with probability 0 still have a chance of occurring, and events with probability 1 stil l have a chance of NOT occurring, see:

https://en.wikipedia.org/wiki/Almost_surely

This strange property comes from strange properties of the real numbers (and uncountably infinite sets) that give rise to things like:

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Measure theory deals with resolving this:

https://en.wikipedia.org/wiki/Measure_(mathematics)