Belief in the axioms of probability theory is justified by the fact that someone with inconsistent beliefs can be Dutch-booked.
If you're willing to put money on your beliefs (i.e. bet on them), then you ought to believe in the axioms in the first place, otherwise your opponent will always be able to come up with a combination of bets that will cause you to lose money.
This fact was proved by Bruno de Finetti in 1930-ties. See e.g. AI: A Modern Approach for an easily approachable technical discussion.
I think De Finetti's justification is fine as far as it goes, but it doesn't go quite as far as people think it does. Here's a couple dialogues to illustrate my point.
Dialogue 1
A: I have secretly flipped a fair coin and looked at the result. What's your probability that the coin came up heads?
B: I guess it's 50%.
A: Great! Will you accept a bet against me that the coin came up heads, at 1:1 odds?
B: Hmm, no, that doesn't seem fair because you already know the outcome of the coinflip and chose the bet accordingly.
A: So rational agents shouldn't necessarily ac...
I've raised arguments for philosophical scepticism before, which have mostly been argued against in a Popper-esque manner of arguing that even if we don't know anything with certainty, we can have legitimate knowledge on probabilities.
The problem with this, however, is how you answer a sceptic about the notion of probability having a correlation with reality. Probability depends upon axioms of probability- how are said axioms to be justified? It can't be by definition, or it has no correlation to reality.