What is the density of a regular language $L$ over an alphabet $\Sigma$ in $\Sigma^n$?
If I understand the question, your problem is the following:


  Given a regular language $L$ over alphabet $\Sigma$ and a positive integer $n$, compute the probability that a word chosen uniformly at random from $\Sigma^n$ will be in $L$.


That's equivalent to computing $|L \cap \Sigma^n|$, i.e., the cardinality of the language $L \cap \Sigma^n$.  Note that $L \cap \Sigma^n$ is regular, so your problem is equivalent to the problem of counting the number of words in a finite regular language.  This is a standard problem that is well-studied; see https://cstheory.stackexchange.com/q/8200/5038, https://cstheory.stackexchange.com/q/32473/5038, Why isn&#39;t it simple to count the number of words in a regular language?, Counting the number of words accepted by an acyclic NFA.  Here are some results:


If the language $L$ is specified as a DFA or as an unambiguous regexp, then the problem can be solved in polynomial time.
If the language $L$ is specified as a NFA and $n$ is specified in unary, the problem is $\#P$-complete.  Thus, there is an exponential-time algorithm but you should not expect a polynomial-time algorithm.
If the language $L$ is specified as a NFA and $n$ is specified in binary, the problem is $PSPACE$-complete.  Thus, you should not expect a polynomial-time algorithm.


There are approximation algorithms that you might be able to use in practice to estimate the probability you're seeking (using a SAT solver as a subroutine).