Think about it like this - what are the all the possible errors due to mutation that could happen? And what is the likelihood that one of them will be beneficial? Has this been ever calculated mathematically?

There are certainly a good deal of "errors" that could happen. Some small (ie one base affected) and some large (thousands of bases inserted, deleted and altered, all in one go). In order to calculate the likelyhood of one being beneficial one would probably first have to determine empirically the fraction of mutations that are positive and those that are negative. I've only seen something similar to this being done for simple point-mutations, which does not reflect what really happens in real life. As far as I'm aware, a realistic model for this has not been done so far. But hey, there are thousands of articles published daily(?), so who knows? If you find one that does go into detail about this, I'd certainly be keen to know.

That's a bit of a non-answer, I suppose. So the absense of any kind of research into that area would mean that any kind of discussion concerning it would be speculative.