In classical probability theory, the probability of all excluded middle instances (A v ~A) is 1 and the probability of all contradictions (A & ~A) is 0. This makes sense, given (some other, even more basic assumptions and) the fact that one of the basic organizing principles of classical probability theory is that for every A, P(A) and P(~A) have to add up to 1.
Of course, given some non-classical assumptions about logic, all that falls apart...as Hartry Field points out somewhere (in one of his essays in "Deflationism and Paradox," I think?), the dialetheist should accept that for some A's, the sum of P(A) and P(~A) can be high as 2, and the truth-value gap theorist should accept that it can be as low as 0.* But what I'm concerned with, at the moment, is *classical* logic and *classical* probability theory. This is my worry:
Even if you make thoroughly classical assumptions about logic--no truth-value gaps and no true contradictions--and take those assumptions to be extremely well-justified given current evidence, and the current arguments for and against them, it still seems to make sense to be a *fallibilist* about these matters. To admit that we can't be entirely sure, that considerations sometimes brought to bear against these assumptions (from the paradoxes of self-reference to Aristotle's sea battle problem to issues related to vagueness and "partially-defined predicates" or "over-defined predicates" or whatever) have at least some weight, even if we ultimately find them unconvincing, that we should be open to new arguments and new evidence that might be brought to bear by future dialetheists or by future gap theorists. To, in other words, grant at least *some* rational credence to belief in truth-value gaps and/or true contradictions, even if we firmly reject these beliefs, much the same way we grant at least *some* rational credence, however slight, to Lorentzian absolutism about space and time, or to extreme skeptical possibilities in epistemology, or to belief in ghosts.
Now, if you are a frequency theorist about probability, there's no problem about fallibilistically believing orthodox things about logic on the one hand, and accepting classical probability theory on the other hand, since, if there are no truth-value gaps and true contradictions, the number of each that will show up over the course of any sort of hypothetical limiting frequency will be 0, and thus it will always be correct to say that P(A v ~A)=1 and P(A & ~A)=0. On the other hand, I see no way for the Bayesian epistemologist, who takes probability to be intimately linked to *justification*, not a calculus of frequencies but of rational credences or some such, to have their cake and eat it too on this issue. Unless I'm missing something, it looks to me like the Bayesian has to give up on classical logic, classical probability theory or logical fallibilism, that they can't hold onto all three, given with their view of probability.
*Note that the sort of "truth-value gap" under discussion here--and this applies to the rest of the post--are exceptions not just to Bivalence but to the Law of the Excluded Middle.