It depends what you mean by "probabilistic". There are at least two interpretations. First, the algorithm has some probability of success for every input. Second, the algorithm succeeds on a certain fraction of inputs.
For the first interpretation, it is easy to rule out such an algorithm: probabilistic computation can be simulated (inefficiently but effectively) using deterministic computation, by trying all possible coin tosses.
For the second interpretation, we will have to work a bit harder. Suppose that your algorithm is guaranteed to work with an asymptotic success probability of $2/3$. That means that the fraction of inputs in $[1,N]$ for which it gives the correct answer is some $p_N \to 2/3$. Now suppose you're interested in a certain program $P$. It seems that under a reasonable encoding of programs, you would be able to come up with a long stretch $[M,10M]$ (say) of programs equivalent to $P$. By taking $M$ large enough, it should be the case that by taking the majority vote on the answer of a "probabilistic" algorithm on all of these equivalent programs, you will be able to ascertain whether $P$ halts or not. This rules out even this interpretation of a "probabilistic" algorithm.