In probabilistic questions with "real life" context, why can we ignore defining the sample space?

Question

Consider, for the sake of argument, the following question: we are coloring every side and every diagonal of a regular hexagon with one of three colours (say white $W$, red $R$ and black $B$). Everytime each colour is equally probably and colorings are independent. Let $X$ denote number of triangles in one colour. Compute $E(X)$.

I'd be satisfied with a solution going like: enumerate all 15 considerated edges with numbers from $1$ to $15$ and all 20 triangles with numbers from $1$ to $20$.
Step 1: define a "compelling" sample space, like a product space of 15 spaces $\Omega=\{W, R, B\}$ with $\frac{1}{3}$ probability for every singleton.
Step 2: define random variables $X_{1}, X_{2}, \dots, X_{20}$ setting values $1$ if corresponding triangle is in one colour, $0$ otherwise. Then $X=X_{1}+\dots X_{20}$ and we proceed from here.

Usually we ignore step $1$ and go directly to step $2$. I'd kindly ask you to tell me why is it considered as as good of a solution. When dealing with similar questions, do we start by assuming that there is some "compelling" (what would that even mean?) sample space? How else would one be able to define random variables? Does one usually postulate anything when begins to model using probability theory?

Also, in the usual way we continue by kind of guessing (without explicit sample space isn't that so? Or maybe it is arbitrary and equivalent to defining sample space?) distributions of defined random variables and say things like: $P(X_{i}=1)=\frac{1}{3} \cdot \frac{1}{3}$ because u fix one colour of one edge and then other two edges have $\frac{1}{3}$ probability of having this colour and those two colorings are independent, so we multiply. Why are those kind of solutions correct, formally? With sample space defined it seems so much more elegant and precise to me. Is it just my lack of practice? Why should i solve problems that way?

Any help would be much appreciated. I'd like the answers to be as technical as it is necessary. My knowledge exclude things like Markov chains, martingales, stochastic processes (i.e. Kolmogorov's existence theorem) and statistics, but please use it if it's helpful, i'll then just get back to it again in the future.

Answer 1

Terence Tao has some interesting related commentary in his blog post "A review of probability theory":

Elements of the sample space $\Omega$ will be denoted $\omega$. However, for reasons that will be explained shortly, we will try to avoid actually referring to such elements unless absolutely required to.

...

In order to have the freedom to perform extensions every time we need to introduce a new source of randomness, we will try to adhere to the following important dogma: probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space. (This is analogous to how differential geometry is only “allowed” to study concepts and perform operations that are preserved with respect to coordinate change, or how graph theory is only “allowed” to study concepts and perform operations that are preserved with respect to relabeling of the vertices, etc..) As long as one is adhering strictly to this dogma, one can insert as many new sources of randomness (or reorganise existing sources of randomness) as one pleases; but if one deviates from this dogma and uses specific properties of a single sample space, then one has left the category of probability theory and must now take care when doing any subsequent operation that could alter that sample space. This dogma is an important aspect of the probabilistic way of thinking, much as the insistence on studying concepts and performing operations that are invariant with respect to coordinate changes or other symmetries is an important aspect of the modern geometric way of thinking. With this probabilistic viewpoint, we shall soon see the sample space essentially disappear from view altogether, after a few foundational issues are dispensed with.

I get the impression that for people who deeply understand probability theory, it is "bad form" to focus too much on the underlying sample space, because the sample space might be changed (extended) anyway. Personally, I always need to define the sample space explicitly in my mind (as in your step 1) in order to feel like I really understanding what I'm doing.

Answer 2

The point is that often-times, your question only depends on the distributions of the involved random variables, not about their behaviour as a map.

For instance, if $X:\Omega\to \mathbb{R}$ has the property that $X$ only ever assumes a finite number of values, $1,2,3,...,n$, then any question relating to $X$ really reduces to questions about the $n$ sets $A_j=\{\omega|X(\omega)=j\},$ which is, naturally a partitition of $\Omega$. In fact, the only thing I really need to know about $A_j$ is its probability.

So say $\mathbb{P}(A_j)=p_j$ for some collection of parameters $p_j$ such that $\sum_{j=1}^{n} p_j=1$. And let $\Omega_2$ be any other set with a partition $(B_j)_{1\leq j\leq n}$ and a probability measure $\mathbb{P}_2$ such that $\mathbb{P}(B_j)=p_j$. Then, we could define a random variable $Y:\Omega_2\to \mathbb{R}$ by $Y(\omega)=j$ if $\omega\in B_j$. Then, what really is the difference between $X$and $Y$?

Surely, the background spaces are potentially different in all sorts of relevant ways, but $X$and $Y$ can't tell us the difference. They give rise to the same probabilities, the same moments (expectation, variance, so on and so forth), so unless we have some reason to care about them as maps, their different background spaces won't matter - although, it does happen in probability theory that we do care about things on the level of maps - see the concept of almost sure convergence, for instance.

This generalises to non-discrete variables - whenever you are asking about probabilistic traits of a single variable, the only thing that's gonna matter is its distribution, and there are potentially many different background spaces that have a variable with a specific distribution - and it doesn't matter which one you use. For this reason, we usually don't care too much about the background space in question.