According to Wikipedia, yes.
In probability and statistics, a realization, observation, or observed
value, of a random variable is the value that is actually observed
(what actually happened).
https://en.wikipedia.org/wiki/Realization_(probability)
But if we take the classic example of modeling a single toss of a coin, then I can model the experiment with a random variable $X$ such that $X(HEAD)=1$ and $X(TAIL)=0$.
Hence, the realizations of such a random variable are $1$ or $0$, but what I actually observe is $HEAD$ or $TAIL$.
Where is the fallacy in my reasoning?
I would say that the way I defined my random variable is wrong. Perhaps I should define it as $X(HEAD)=HEAD$ and $X(TAIL)=TAIL$?
But the way I defined at the very beginning is extremely common.
I am confused.
Best Answer
The importance of the word "realized" is to distinguish what happened from the collection of all the things that might have happened.
You could flip a coin and decide beforehand that $1$ means "H" and $0$ means "T". When the coin comes up heads, then your realized value is $1$.
You could also flip a coin and decide beforehand that "H" should map to $27$ (instead of $1$) and that "T" should map to $\pi$ (instead of $0$). Here, when the coin comes up heads, your realized value would be $27$.
To make things worse, you could flip a coin and decide beforehand that you'll just read the label of the side you see. In this case, when the coin comes up heads, your realized value would simply be "HEADS". (* -- see footnote)
The point is not that "HEADS" stands in opposition to $1$, or to $27$, since those are just labels; the point is that "HEADS" stands in opposition to "TAILS". That is what's meant by "realization." It's the distinction between talking about the entire collection of what a standard die could do if rolled, and what the die I just rolled on my desk actually did, which was to land on a 4.
(*) Some textbooks disallow this case; you will sometimes find a definition of random variable to require that the outcome map into $\mathbb R$ and not just into an arbitrary set like $\{\text{HEADS}, \text{TAILS}\}$. The point of this requirement is that if your object is real-valued, then you can talk about important concepts like density functions, expected values, etc.