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...

1. 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.

2. People do sometimes use the Kronecker delta for this: $\delta_{ij}$ is defined to be 1 if $i=j$ and 0 if $i\
e j$, 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.

3. 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.