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.
from Hot Weekly Questions - Mathematics Stack Exchange
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