  We consider probabilistic automata on infinite words with acceptance defined
by parity conditions. We consider three qualitative decision problems: (i) the
positive decision problem asks whether there is a word that is accepted with
positive probability; (ii) the almost decision problem asks whether there is a
word that is accepted with probability 1; and (iii) the limit decision problem
asks whether for every epsilon > 0 there is a word that is accepted with
probability at least 1 - epsilon. We unify and generalize several decidability
results for probabilistic automata over infinite words, and identify a robust
(closed under union and intersection) subclass of probabilistic automata for
which all the qualitative decision problems are decidable for parity
conditions. We also show that if the input words are restricted to lasso shape
(regular) words, then the positive and almost problems are decidable for all
probabilistic automata with parity conditions. For most decidable problems we
show an optimal PSPACE-complete complexity bound.
