$\newcommand{\Sym}{\operatorname{Sym}}$ I was studying $\Sym(\mathbb{N})$, the set consisting of all the bijections from $\mathbb{N}$ to itself. Since it is a group, the concept of "period of an element" has a sense, and it is the smallest positive integer $n$ such that $f^n = e$, where $f$ is one of those bijections and $e$ is the identity of the group (the identity function).
I was interested in the subset of all the elements of the group that have finite period. My question is: if I randomly choose an element of $\Sym(\mathbb{N})$, is there a way to know... if it's more likely to get an element of infinite period, or an element of finite period? The problem is that, according to the results I got, both $\Sym(\mathbb{N})$ and its subset I'm interested in are infinite sets that have the cardinality of the continuum.
Am I unawarely asking a stupid/impossible question, or are there any mathematical tools to know what that probability is?