In this case study I attempt to untangle this pedagogical knot and illuminate the basic concepts and manipulations of probability theory.
Our ultimate goal is to demystify what we can calculate in probability theory and how we can perform those calculations in practice.
For example, making a seemingly innocent assumption known as the Axiom of Choice one can show that there exist sets in the real numbers that cannot be explicitly constructed.
In particular, one can show that these non-constructible sets feature some very odd properties.
Although superficially similar, a topology is distinct from a -algebra.
For example, the topologically open sets are not closed under the complement operation.
No finite number of disjoint rectangles will exactly cover the circle and hence yield the desired probability, but the limit of a countably infinite number of them will.
Consequently in general we need our probability distributions to be In other words, there is no way that we can engineer a self-consistent distribution of probability across non-constructible sets.
We begin with an introduction to abstract set theory, continue to probability theory, and then move onto practical implementations without any interpretational distraction.
We will spend time more thoroughly reviewing sampling-based calculation methods before finally considering the classic applications of probability theory and the interpretations of the theory that then arise.