To make the question clearer, I will write the context in which I formulated it myself.
I was studying the twelve coin puzzle, in which you are given 12 coins and by using a balance certain number of times you need to decide something about the coins. The details actually don't matter. The problem I wanted to solve was not the puzzle itself but giving a bound on the maximum number of coins $n$ you can have if you want to solve the puzzle using the balance $k$ times.
This is a classical problem, which can be approached using techniques from combinatorics, for example. For my approach, however, I used Information Theory to get the bound. My argument started like this:
Let $X$ be the random variable representing the possible states of the coins. What we want is to be able to determine $X$ out from $k$ uses of a balance, for any distribution of $X$....
What remains of the proof, just like the details of this problem itself, are not relevant. What matters is the question I made to myself at this point: Is it OK to assume some kind of distribution from the states of the coins? This looks weird, because even though the proof gives the same bound you can get with combinatorial methods, these do not assume anything about the underlying distribution of the coins. On the other hand, I'm assuming that the coins must have some kind of distribution! Even if we don't know it, it must be there.
I am aware of other contexts where you assume data distributions must exist. In Machine Learning for example, you make predictions based on the assumption that the underlying data has some distribution, which although unknown, is there! (So statistical arguments can be applied).
The question
I will do my best to abstract my question in a concise way out from my examples above.
Is it natural to regard any outcome from a process as a random variable with an underlying distribution? Are there some studies in this direction? Is this beyond the scope of Mathematics/Statistics itself (and more related to how we interpret results in these fields in "real life")?
It is the nature of statistics itself. Maybe when dealing with deterministic processes one could argue that the random variable framework is too much (you would be dealing with degenerate distributions...)
It doesn't make sense to deal with a "random variable without a underlying distribution" because Random variables are defined as functions from sample spaces (i.e. given an experiment, the possible outcomes and the distribution associated with them).