Why is it assumed that the results seen in the double slit experiment are probabilistic and not just a statistical result of some unknown variable or set of variables within the system.
Ever since the origination of quantum mechanics, some theorists have searched for ways to incorporate additional determinants or "hidden variables" that, were they to become known, would account for the location of each individual impact with the target.
In my opinion, the "were they to become known" is the tricky bit (to put it mildly). And, as things stand, for prediction purposes one might as well assume an inherently probabilistic nature.
(I'll add to this later.)
You are asking can the probabilistic results of quantum mechanics be reproduced by a deterministic classical mechanics with added random noise of some sort to reproduce the statistical results. This question has been considered in leonard susskinds lectures on quantum entanglement1 (see videos on Stanford continuing education website). The answer he gives is "no certainly not!". He points out that the logic of classical physics is the logic of set theory, states of systems being represented as points in a set; while the logic of quantum mechanics is the logic of vector spaces which is not deducible from the logic of sets. We get states (entangled states) in the latter which are impossible in the former (hence bell's inequalities which classical states must obey but which some quantum states violate) Nature has been shown to violate Bell's inequalities (Aspect's experiments etc) so the quantum description is closer to observation than classical stochastic type explanations.