Well, the probability is computed by an algorithm that is itself imperfect. "I'm 50% confident that the probability is 1/1000" means something like "My computation gives a probably of 1/1000, but I'm only 50% confident that I did it right". For example, if given a complex maths problem about probabilities of getting some card patterns from a deck with twisted rules of drawing and shuffling, you can do that maths, ends up with a probability of 1/5 that you'll get the pattern, but not be confident you didn't make a mistake in applying the laws of probability, so you'll only give a 50% confidence to that answer.

And there is also the difference between belief and belief in belief. I can something "I believe the probability to be of 1/1000, but I'm only 50% confident that this is my real belief, and not just a belief in belief".