Does there exist any noncomputable set $A$ and probabilistic Turing machine $M$ such that $\forall n\in A$ $M(n)$ halts and outputs $1$ with probability at least $2/3$, and $\forall n\in\mathbb{N}\setminus A$ $M(n)$ halts and outputs $0$ with probability at least $2/3$? What if you only require that $M(n)$ is correct with probability greater than $1/2$?



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 $1/2$.


