It depends which kind of probability theory you're interested in. An introductory course on probability theory can either dwell on discrete probability or continuous probability.
Discrete probability, which deals with discrete events (e.g. the probability that if you throw a dice it comes up $6$ ten times in a row), only really needs elementary combinatorics. From set theory you need to know the definitions of basic concepts, and from combinatorics you need to know the likes of the binomial coefficient and its properties.
A little more is needed to understand Poisson random variables, namely Stirling's approximation, which is a topic you don't really learn anywhere; this is why these courses often just give the definition, which requires you to know the Taylor expansion of $e^x$. But this topic in its entirety is not necessarily covered.
Continuous probability deals with things like the normal distribution and the central limit theorem - distributions which may take "continuous" values (e.g. every real value rather than only integral values). Sometimes it is given as an addendum to a discrete probability course. To understand continuous probability you will need to know basic calculus (the kind you get from a first course, and then some).
Introductory courses don't usually cover multivariate Gaussians, but these require some linear algebra.
Summarizing, you will need to be confident about some fairly basic topics. Besides some familiarity with basic concepts, it's also best to have some "mathematical maturity", although not too much of it is actually needed in an introductory course.