Yes, and it includes the Halting Problem, but it's not very useful.
The solution to the Halting problem is known in the particular case of the program that ends immediately. Actually there are arbitrarily many programs that are known to halt; for example all programs with no looping instructions.
So suppose we write the following program: "If the input program obviously halts according to one or more quick tests, return True. If not, return a coin flip." The coin flip gets 50% of the non trivial cases (which are the vast majority), and the reasoned test gets 100% of of the trivial cases. By simple arithmetic, the routine as a whole will "solve" the halting problem strictly more successfully than 50%. Such a construction works for any problem for which there are any decidable special cases.
The reason that this is not very useful is that the randomness doesn't actually add any information, and so it doesn't stack informatively. When we talk about probabilistic algorithms we either expect that independent runs of the routine give independent answers which are each more reliable than not. The notion is that if you have a mere 51% successful independent random guesser run ten thousand times, the majority guess will be right with 97% probability. However, if you have a 51% successful random guesser that just sticks to its first guess, after 10,000 attempts you're still at 51%! Independence is critical. In particular, you need independence in your >50% chance of success. In this construction, only bit that gets better than mere guessing is deterministic and reproducible, while the random bit is completely disconnected from reality. That means that although we can get something to a 50%+ε for all problems with some decidable special cases, we can't leverage that fact to get arbitrarily high.