The notion of a *limiting recursive set* (Gold 1965, *J. Symb. Log.* **30**: 28–48) or *trial and error predicate* (Putnam 1965, *J. Symb. Log.* **30**: 49–57) is defined as follows. A *guessing function* is a total recursive $g: \mathbb{N} \times \mathbb{N} \rightarrow \{ 0,1 \}$. 


A set $S \subseteq \mathbb{N}$ is limiting recursive just in case $$n \in S \Leftrightarrow \exists i (\forall j>i) \ g(n,j)=1$$ and $$n \notin S \Leftrightarrow \exists i (\forall j>i) g(n,j)=0.$$


Gold and Putnam showed (independently) that the limiting recursive sets are exactly the $\varDelta\_2$ sets in the arithemetical hierarchy. 


Suppose that we weaken the notion as follows. Let $\delta$ range over positive reals. Let us say that a set $S \subseteq \mathbb{N}$ is probabilistically limiting recursive just in case $$n \in S \Leftrightarrow (\forall \delta < 1)\exists i (\forall j>i) \ \frac{ \{ k \leq j: g(n,k) = 1\}}{j}>1-\delta$$ and $$n \notin S \Leftrightarrow (\forall \delta < 1)\exists i (\forall j>i) \ \frac{ \{ k \leq j: g(n,k) = 0\}}{j}>1-\delta.$$


Are all probabilistically limiting recursive sets of natural numbers limiting recursive? 



Probabilistic generalization of trial-and-error predicates
I am not sure one should really call your family of sets "probabilistically limiting recursive" since there is no real randomness involved here, but in any case the answer to your question is yes. Suppose you have a function $g$ witnessing that $S$ is probabilistically limiting recursive. Consider the function $h$ defined by $h(n,j)=1$ if $|\{k \leq j : g(n,k)=1\}|>j/2$ and $h(n,j)=0$ otherwise. It is easy to see that $h$ is total recursive, and it witnesses that $S$ is limiting recursive. 


