  We study the emptiness and $\lambda$-reachability problems for unary and
binary Probabilistic Finite Automata (PFA) and characterise the complexity of
these problems in terms of the degree of ambiguity of the automaton and the
size of its alphabet. Our main result is that emptiness and
$\lambda$-reachability are solvable in EXPTIME for polynomially ambiguous unary
PFA and if, in addition, the transition matrix is binary, we show they are in
NP. In contrast to the Skolem-hardness of the $\lambda$-reachability and
emptiness problems for exponentially ambiguous unary PFA, we show that these
problems are NP-hard even for finitely ambiguous unary PFA. For binary
polynomially ambiguous PFA with fixed and commuting transition matrices, we
prove NP-hardness of the $\lambda$-reachability (dimension 9), nonstrict
emptiness (dimension 37) and strict emptiness (dimension 40) problems.
