[stiff]

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)

[ouid]

What's an anagram for Banach Tarski?

Banach Tarski Banach Tarski.

Seriously though, you can do math without invoking the axiom of choice. The formulation of probability doesn't strictly depend on it.