How to write down formally number of occurrences? I want to write the following sentence formally:

The sequence $S$ contains elements of the set $A$. The probability value $P(a)$ for an element $a$ is defined as the number of its occurrences in the sequence $S$, divided by the count of all its elements.

I can write it the following manner:
$$ S = (s_1, s_2, ..., s_n) : s_i \in A.$$
$$ P(a) := {{ \left| \lbrace i \in \lbrace 1, 2, ..., n \rbrace : s_{i} = a \rbrace \right| } \over {n}}, \text{ given } n > 0\text{ and }a \in A. $$
It's, however, quite long and rather not elegant. Is there a simpler way to write this?

Edit:
There's always a solution, which involves breaking the formula to smaller parts:
$$ \text{Let } C(x) = \left| \lbrace i \in \lbrace 1, 2, ..., n \rbrace : s_i = x \rbrace \right|.$$
$$ P(a) := {C(a) \over n}. $$
It's more readable, but it's still not what I'm searching for...
 A: It's also not quite correct. How about $|\{i\in\{1,\ldots,n\}\colon s_i=a\}|$ in the numerator? If you view the sequence $S$ as a function $S\colon\{1,\ldots,n\}\to A$, $i\mapsto s_i$, then you might even write $|S^{-1}(a)|$ for the numerator. And the denominator should simply be $n$ (which you implicitly defined for $S$). As $S$ is not (primarily) a set, $|S|$ looks strange.
A: Using the number symbol "#",
$$ 
\DeclareMathOperator*{\countif}{\#}
P(a) := \dfrac{\countif\limits_{i=1}^{n} (s_i=a)}{n}, \text{ given } n > 0\text{ and }a \in A. 
$$
See John Fox, Applied Regression Analysis and Generalized Linear Models (3rd edition), Section 21.2.3
A: *

*If you are willing to use the "Iverson bracket notation", popularized by Knuth and others, you can say $$C(x) = \sum_{i=1}^n [s_i = x]$$
Here $[\ldots]$ are the Iverson brackets.  $[P]$ is defined to be 1 if $P$ is true, and 0 if it is false.

*People do sometimes use the Kronecker delta for this: $\delta_{ij}$ is defined to be 1 if $i=j$ and 0 if $i\ne j$, so you would have:
$$C(x) = \sum_{i=1}^n \delta_{xs_i}$$ or
$$C(x) = \sum_{i=1}^n \delta(x, s_i)$$
but I think the Iverson bracket is more straightforward.

*Most straightforward would be to write

Let $C(x)$ be the number of elements of $s_1,\ldots,s_n$ that are equal to $x$.  Then…

The idea that this is somehow less "formal" than something involving a bunch of funny symbols is a common misapprehension.
A: 2 ideas:
1) Using the hashtag or number symbol "#", e.g:
$$
\#(a \in S)
$$
It is fairly common within literature, see:
2008. Elements of Statistical Learning 2nd Ed, Chapter 9.2.2. Hastie, Tibshirani, Friedman
2)  Using the indicator function (returns 1 if condition true, else 0) in a sum, e.g:
$$
\sum\limits_{i=1}^{n}{\textbf{1}_{a_i \in S}}
$$
or
$$
\sum\limits_{i=1}^{n}{I({a_i \in S})}
$$
It is probably the most common and 'mathematical', see:
2008. Elements of Statistical Learning 2nd Ed, Chapter 9.2.3. Hastie, Tibshirani, Friedman

Usage example for original post:
$$
\mathbb{P}(a) = \frac{\sum\limits_{i=1}^{n}{\textbf{1}_{a_i \in S}}}{n} \forall \ a \in S \in A
$$
where
$n$ = number of elements in S
$\textbf{1}_a$ = indicator function that returns 1 if the element is a
