Understanding Sample Space in Statistics

In statistics, we often work with independently and identically distributed (iid) samples. For instance, consider a set of observations: $ (X_1, X_2,\dots,X_{10}). $ However, what is the underlying sample space (a foundational concept from probability) associated with these samples?

Specifically, let’s take an element $ \omega\in\Omega, $ where this $ \omega $ represent an individual like Alice. How, then, could we use $ X_1 $ to denote Alice and, simultaneously, $ X_2 $ to denote another individual, say Bob?

This question once confused me for years. If you are having the same trouble, check out the video!