Let's suppose that a language $L \in$ NSPACE($f(n)$) where $f(n)$ is $O(\log n)$. And now let's suppose that I have a probabilistic Turing machine. Can this machine run in $O(f(n))$ space and answer yes when $x \in L$ with $\Pr(yes) > 1/2$ and answer no when $x \not\in L$ with $\Pr(no) = 1$? Let's suppose I dont care about time as long as the machine halts.
My comment was incorrect, and in fact the answer to your question is YES. From Wikipedia:
Suppose C is the complexity class of problems solvable in logarithmithic space with probabilistic Turing machines that never accept incorrectly but are allowed to reject incorrectly less than 1/3 of the time; this is called one-sided error. The constant 1/3 is arbitrary; any x with 0 ≤ x < 1/2 would suffice.
It turns out that C = NL. Notice that C, unlike its deterministic counterpart L, is not limited to polynomial time, because although it has a polynomial number of configurations it can use randomness to escape an infinite loop. If we do limit it to polynomial time, we get the class RL, which is contained in but not known or believed to equal NL.