  Automata with monitor counters, where the transitions do not depend on
counter values, and nested weighted automata are two expressive
automata-theoretic frameworks for quantitative properties. For a well-studied
and wide class of quantitative functions, we establish that automata with
monitor counters and nested weighted automata are equivalent. We study for the
first time such quantitative automata under probabilistic semantics. We show
that several problems that are undecidable for the classical questions of
emptiness and universality become decidable under the probabilistic semantics.
We present a complete picture of decidability for such automata, and even an
almost-complete picture of computational complexity, for the probabilistic
questions we consider.
