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solipsist comments on Open thread, January 25- February 1 - Less Wrong Discussion
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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.