TITLE: Must every event have a probability?
QUESTION [8 upvotes]: Is it possible for an event to simply happen for which it is impossible to define any probability?
(Note: By "impossible" I don't mean just "impractical" -- I really mean that the event should not follow any probability distribution.)
Somewhat similarly: can a "random" number generator exist which does not follow any probability distribution?

REPLY [3 votes]: Consider the following situation. Fix a group $G$ and a subset $S$ of $G$. Pick an element of $G$ and ask yourself: What is the probability of getting an element of $S$?
Example 1: If $G$ is a finite group, then the unique natural intrinsic notion of probability is given by the cardinality of $S$ divided by the cardinality of $G$.
Example 2: If $G$ is compact, the unique natural thing to do is to look at the Haar measure of $S$.
Example 3: If $G$ is amenable, the unique natural thing to do is to look at the value that an invariant mean takes over $S$. Since there are might be many invariant means, this already leads to some ambiguity. 
In general, it seems that what we need is just a measure on $G$ which leaves invariant the characteristic function of $S$.
Now, have a look here https://mathoverflow.net/questions/60247/when-is-non-amenablity-witnessed-by-a-single-non-measurable-set It is proved that there is a group $G$ and a subset $S$ such that there are NO measures on $G$ which leave invariant only the characteristic function on $S$. In my opinion, this is a case where it is not possible to define the probability of getting an element of $S$ when picking casually an element of $G$.