Is there a noncomputable set which can be described by a probabilistic Turing machine with bounded error?

Every such decision problem is computable, even in the harder version of the problem, assuming that the transition probabilities are, say, fixed rational numbers. A deterministic algorithm can calculate the probability distribution on the set of states of this stochastic TM after each $t$ time steps, and then step through $t$ until the probability of halt at either yes or no exceeds $1/2$.