Such programmers almost never know they got it right, only believe so. And the difference may not be easy to perceive.
I remember how I used to program before I learned about unit testing. And I remember that feeling of confidence and trust on a wholly different level after I ran my first decent suite of unit tests. I hadn't known such level of confidence in my code existed before.
For someone who lacks this experience, it is impossible to explain the difference. So they may even go on developing in code-and-pray mode throughout their life, benevolently (and ignorantly) believing that they are doing their best considering the circumstances.
That said, there can indeed be great programmers and exceptional cases, when one really manages to hold the whole problem space in his/her mind, in a complete state of flow. I have experienced rare moments like this, when I perfectly knew what to write, the code just flew out of me effortlessly, I could foresee all special cases and boundary conditions, and the resulting code just worked. I have no doubt there are programming geniuses out there who can stay in such state of flow for extended periods or even most of their time, and what they produce is beautiful code, seemingly without effort. I guess such persons might feel no need to write puny unit tests to verify what they already know. And if you really are such a genius, it may be OK (although even then, you won't be around that project forever, and you should think about your successors...). But if not...
And let's face it, chances are you aren't. I, for myself, know I am not. I had some rare moments of flow - and countless hours of grief and sorrow, usually caused by my own mistakes. It's better be honest and realistic. In fact, I believe the greatest programmers are fully aware of their own fallibility and past mistakes, so they have consciously developed the habit of double checking their assumptions and writing those little unit tests, to keep themselves on the safe side. ("I am not a great programmer - just a good programmer with great habits." - Kent Beck.)