Probabilistic automata are a generalization of DFAs and NFAs, and the set of languages that they recognize are the "stochastic languages".
Wikipedia has that these are a superset of the "regular languages" recognized by DFAs and NFAs; but how much of an improvement are we talking? Is it known how stochastic languages compare to the recursively enumerable languages recognized by Turning machines for instance?
You should read the wikipedia article more carefully. It is explained that the set of stochastic languages is uncountable, contrary to the set of recursively enumerable languages.
On the other hand, it is shown here that the language $$ \{a^{n_1}ba^{n_2} \dotsm ba^{n_k}ba^n \mid \text{there exists $i> 1$ such that $n_i = n_1$}\} $$ is context-free but not stochastic.