Probabilistic vs. nondeterministic unary automata

We investigate and compare the descriptional power of unary probabilistic and nondeterministic automata (pfa's and nfa's, respectively). We show the existence of a family of languages hard for pfa's in the following sense: For any positive integer $d$, there exists a unary $d$-cyclic language such that any pfa for it requires $d$ states, as the smallest deterministic automaton. On the other hand, we prove that there exist infinitely many languages having pfa's smaller than nfa's which, in turn, are smaller than deterministic automata.