Naturally, Eliezer has already thought of this, and wrote about it in Occam's Razor:
It seems to completely answer your question. That is, one can think about probabilities and formulate and test probabilistic hypotheses, without needing to generate any random numbers.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
Of course, for "every Monday", the last one should have been dated July 22-28. *cough*