Yvain said the finiteness well, but I think the “infinitely many possible arrangements” needs a little elaboration.
In any continuous probability distributions we have infinitely many (actually uncountably infinitely many) possibilities, and this makes the probability of any single outcome 0. Which is the reason why, in the case of continuous distributions, we talk about probability of the outcome being on a certain interval (a collection of infinitely many arrangements).
So instead of counting the individual arrangements we calculate integrals over some set of arrangements. Infinitely many arrangements is no hindrance to applying probability theory. Actually if we can assume continuous distribution it makes some things much easier.
Very useful considering that many variables can be approximated as a continous with a good precision.
Small nitpicking about “or any actual measurement of a continuous quantity”. All actual measurements give rational numbers, therefore they are discrete.