I would respond that the question is moot: it is too difficult (at present) to talk in a meaningful way about the probability of a mathematical statement.
This is because one of the standard interpretations (or classes of interpretations) of probability is as quantifying over "possible states" - so that when we say that flipping a coin results in heads with probability 50%, what we mean is something like "50% of the possible outcomes of flipping a coin result in heads." Exactly what constitutes a "possible state" is a source of intense discussion, but the general theme is certainly a common one.
The problem with this type of interpretation is that it's not clear what "possible state" should mean in the context of mathematics, at least if we assume a "naive Platonist" assumption. A statement like "There is a contradiction in ZFC of length <10^10" is either true, full stop, or false, full stop. And it's exactly this type of sentence for which we need to develop a notion of probability in order to make sense of your question.
The notion of degrees of confidence rears its head now, but ultimately doesn't seem to do much better: this interpretation is tied to the notion of an ideal rational agent, and it's hard to pin down how such an agent can have limited mathematical knowledge.
I think this all becomes clearer when we replace Con(ZFC) with something not philosophically loaded. For example, how does verifying yet another case of Goldbach's conjecture affect (our assessment of) the probability of its truth? I think beyond a certain point it doesn't. Checking "small" cases of a given conjecture may verify that a wide range of ideas for producing counterexamples are not (trivially) successful, but beyond that it doesn't do much, and since ruling out larger counterexamples doesn't rule out additional ideas as far as we can tell, it doesn't yield a meaningful increase in confidence.
So how comfortable should we be about the consistency of ZFC?
Well, I personally would argue: not very! There are many arguments for its consistency, but they're all rather fuzzy (necessarily so). But I think this draws attention exactly to one of the strengths of the ZFC-style framework, namely that it provides not just a single system but a well-understood hierarchy of systems which give us strategies for handling discovered inconsistency. Fundamentally, very little would change in non-set-theoretic mathematics if an inconsistency were discovered in ZFC, because (i) ZFC is overkill for almost everything and (ii) we understand how ZFC is overkill for almost everything.
- Incidentally, the systems studied in reverse mathematics let us continue that hierarchy further down, that is further in the direction of confident consistency - roughly speaking, I'd say that the weakest of the natural ZFC-style systems is KP, and this is between the reverse mathematical systems ACAo and ATRo in a couple precise senses.
It's worth noting that this hierarchy extends upwards as well, via large cardinal axioms. Basically, ZFC/large-cardinal-style set theories provide both a framework for handling discovered inconsistencies and a framework for understanding results which, whether we realize it or not, are independent of ZFC. While of course this is not perfect (per Godel), it is incredibly rich and successful, and to my mind constitutes the real foundational advantage of ZFC and the large cardinal framework (at the moment ... :P).