Question: Let $\Sigma:=\{0,1\}.$ A finite string $,s,$ over $\Sigma$ is any finite sequence composed of $0s$ and $1s.$ The length of $s$ denoted $|s|$ counts the number of symbols comprising $s.$ The Hamming weight of $s;$ which I will denote by $\mathcal{H}(s),$ counts the number of symbols in $s$ that differ from $0.$ What are some mathematical and more importantly physical interpretations of $$ \frac{1}{|s|}\mathcal{H}(s) \label{a}\tag{1} $$ ? Note I am not asking for an account of histories or common usages of Hamming weights. I am asking what averaging over Hamming weights informs us.
Edit: See Phicar's answer. It would appear $\ref{a}$ gives a density, not a probability, of the number of $1s$ in $s.$
I would think $\ref{a}$ gives the probability of drawing a $1$ among the symbols ( -bits) of $s.$
Notice that $\frac{\mathcal{H}(s)}{|s|}$ gives you a value in between $0$ and $1$ that defines the density of $1's$ in the string. This is not quite a probability(you are evaluating non random points), but lets you compare strings of different sizes. If you do not divide by the length, the word $00100$ would have the same value as $1$ but the later seems more populated of $1's$ than the former.