What is the mathematical foundation for the probabilistic nature of chemistry? Is this best answered through quantum theory, Boltzmann statistics, or what? 

For example, in a reaction energy diagram, only some of the reactants go over to the side of the products, and then get stuck there. So the progression of the reaction can be thought of in probabilities - what are the chances that something will go to products, and what are the chances the products will revert back. So you'd use the Arrhenius equation here. Could you dig deeper and find a quantum mechanical reason for this?
Actually, a huge amount of the probablistic nature of chemistry is simply the Law of Large Numbers (LLN). A mole of any substance is approximately $6\times10^{23}$ molecules, which is 600 sextillion molecules! The LLN kicks in at quantities much smaller than the sextillion level. Put simply, there are so many interactions occurring in any macro-level reaction that these are inevitably going to converge to the theoretical mean by the LLN.

Now your question adds in the issue of activation energy, but I still think it fits this model, because activation energy tends to produce a sharp difference: too low, and little reaction happens; but high enough, and we mostly go to completion. Once activation energy is high enough, the LLN applies.