A different level explanation, which may or may not be helpful:
Read up on affine space, convex combinations, and maybe this article about torsors.
If you are frustrated with hand waving in calculus, read a Real Analysis textbook. The magic words which explain how the heck you can have a probability distributions over real numbers is measure theory).
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.