Calculations of a priori probabilities of this sort are completely specious? And that even if they weren't, they're the sort of odds we find against everyday occurences?
Take Boltzmann's equation S = k ln W. Now invert it: W = exp(S/k). Take a process whose entropy is -100 J/K. With Boltzmann's constant k = 1.3 X 10^-23 J/K, that gives you a probability of the process of exp(-10^25) or so; in other words, about 1 in 1 followed by 5 X 10^24 zeroes, give or take a factor of 10.
That's the a priori probability of a couple of ice cubes forming from water in your freezer after you've filled the trays up with water. You could either learn to love luke-warm drinks, like the English, or be skeptical about a priori probability calculations.
Please "stupid proof" your calcs here for me if you would. I confess I did not understand the mechanics of your argument, though I did get the picture of what you are saying. My problem comes in trying to use Boltzmann to define probabilities is something as dense as water. Be gentle with me please. It has been almost 30 years since PChem. My (very limited) knowledge here translates to a rough approximation of "Bolzmann is great when it comes to ideal gasses but his equation is less and less accurate as higher density materials are in view" (don't bother googling for that..., no one else is so thickheaded to put it in such crass terms). Again, this may just be that I dont' understand Boltzmann. If so, my ego is not on the line here, and I can take correction.
So tell me, Professor, does Boltzmann's equation have a material cause? How about natural logarithms -- are they the product of matter in its modifications/motions? Or the British taste for luke-warm beer for that matter? Or the fact that I like mine icy cold?
Personally, I don't think Penrose's probability calculation is at all specious.