  We introduce a new complexity measure for finite strings using probabilistic
finite-state automata (PFAs), in the same spirit as existing notions employing
DFAs and NFAs, and explore its properties. The PFA complexity $A_P(x)$ is the
least number of states of a PFA for which $x$ is the most likely string of its
length to be accepted. The variant $A_{P,\delta}(x)$ adds a real-valued
parameter $\delta$ specifying a required lower bound on the gap in acceptance
probabilities between $x$ and other strings. We prove $A_{P,\delta}$ is
$\delta$-computable for all $\delta$, relate $A_P$ to the DFA and NFA
complexities, and obtain a complete classification of binary strings with
$A_P=2$. Finally, we discuss several other variations on $A_P$ with a view to
obtaining additional desirable properties.
