why is probability not considered part of pure mathematics? wikipedia considers probability a part of applied mathematics, and it doesn't seem to fall under one of the four areas of mathematics (algebra, number theory, topology/geometry,analysis).
Nevertheless it seems to me to be a very fundamental mathematical concept, so why is it not considered a part of pure mathematics?
 A: The premise of your question is fundamentally flawed. Firstly, there is no clear distinction between pure and applied mathematics --- for example, is the study of PDEs like the Navier--Stokes equation pure or applied mathematics? One proves theorems and uses general theory just like in pure mathematics, but PDEs obviously have applications in physics. The classification is very imprecise.
Secondly, you say it doesn't fall under the "four areas of mathematics", algebra, number theory, topology/geometry, analysis. I have no idea where you got this "classification" from, but it's definitely far from comprehensive. For example, where is set theory? Dynamical systems? Combinatorics? Graph theory? Are you really suggesting these are not considered mathematics?
Probability is a part of pure mathematics, though of course it also has extremely significant applications in applied math, or even outside of mathematics (e.g. in economics, finance, etc). Both can be true at the same time.
A: I believe that probability and randomness are concepts of pure mathematical reasoning as fundamental as those of space, number and structure. In fact, it can be considered as a generalization of what is commonly referred to "pure math", like those four "branches" you mention.
Such "pure math" can be described as an object the part of probability theory concerned with events ocurring with probability 1 or 0. These events are then called theorems, things which always or never happen. Theorems with some additional hypothesis are events with conditional probability equal to 0 or 1, i. e., things that always or never happen, provided that some other thing has already happened (meaning, provided the hypotheses are true).
In truth, those kinds of divisions are artificial and useless. There's no real meaning to "pure" math. Branches are used by academics as a way to organize knowledge and an excuse to specialize and never learn anything outside "their" field. The only thing that actually exists are mathematical ideas, and all of them embrace each other. There's no meaningful separation between them as objects of the mind.
A: The way I interpret the applied and pure dichotomy in math is:
Applied math is taking established mathematical results and utilizing them in describing, understanding and solving real-world problems.
Pure math is a collection of mathematical results that serve the purpose describing, understanding and solving abstract problems. That is, problems which aren't a real-world problems.
They are not absolutely disjoint but rather are important parts of the whole of mathematics. Moreover, despite my above "definitions", I would say there is no strongly distinct line between the two. A research paper could have both pure math results and applied math results within it.
Probability falls under measure theory which is part of analysis, but also has the ability to be applied to real-world problems. I would say it is both.
