[openasocket]

> Separately, what does "Since P probably does not equal NP" mean as a probabilistic statement?

It actually does have a formal meaning! This is one of the most interesting results (in my opinion) on the P vs NP problem. Given a random oracle R, Pr(P^R != NP^R) = 1. So given a random universe, it is almost certainly true that your version of P != your version of NP. At least that's how my theory prof liked to explain it.

Here's a link with the proof: http://theory.stanford.edu/~trevisan/cs254-14/lecture04.pdf