Hence why I said "near enough" 50/50.
Also, as you demonstrate, as we increase the resolution (by which I mean the amount of possible numbers we can generate between one integer and the next) towards infinity, our probability of success will invariably tend towards 50/50.
Not only that, but since the minimum number you generate is 0 and the maximum will always be even (being 2 times your attribute), you will always have an odd number of possibilities - so either you will never have a black check, or it has to class scenarios like this where you are required to role equal to your attribute or higher as the black check.
Assuming that the dice is in no way weighted - which they may well be.