There's a specific kind of probability problems in which in a certain sense, the sampling order doesn't matter to calculate probabilities. It is examplified in "sampling without replacement" problems like this one:
I have a deck of 52 cards. I shuffle the deck, and drop 20 cards randomly into a shredder. You then draw two cards from what remains. What is the probability that they are both aces?
Or this one :
A bowl contains twenty cherries, exactly fifteen of which have had their stones removed. A greedy pig eats five whole cherries, picked at random, without remarking on the presence or absence of stones. Subsequently, a cherry is picked randomly from the remaining fifteen. What is the probability that this cherry contains a stone?
In both of those problems, there are two sampling events from a set of objects of which we know the composition, let's call them $A$ and $B$. We don't know anything about the former, and we're trying to calculate the probability of the latter. The trick is to notice that the sampling order doesn't matter and that if we were to reverse the order of $A$ and $B$ it wouldn't change the result.
Now, this particular kind of order symmetry might seem obvious to you but I've found myself to be particularly resistant to building the kind of "primal intuition" that seems to be adequate for those kinds of problems. I can easily find a justification for the emergence of this kind of symmetry, indeed we can prove that any "total sampling sequence" (the full sequence of objects that we get from drawing every object from our original set) is equiprobable by commutativity of the multiplication. There are even more "intuitive" explanations that can be enough for other people, like "If you were playing cards and the dealer distributed your hand after dealing other people's, you wouldn't complain!".
However, what I'm trying to understand is what are the properties of this specific kind of problems that make this order symmetry happen and to what other classes of problems can we apply this symmetry reasoning apart from sampling without replacement? I think this is the missing piece in order for my intuition to become really primal.
(Sampling with replacement seems to be an obvious one for example...)