I don't entirely agree with this. In the dice analogy it would be more accurate to say that once a "1" is rolled the person who rolled the "1" will no longer roll another "1" because "1" has been proven to not be the solution to the problem at hand. That is, he would reduce the number of faces on the dice before rolling again. I agree that each attempt has equal chance of being the right solution in that the chance of rolling a "1" was just as good as rolling a "6" and that on the successive roll the chance of rolling a "5" will be just as good as rolling a "6" but chance of rolling another "1" has now been made zero and the available pool of possible answers has been reduced. Since the pool of possible answers is finite and thus every attempt proven incorrect will not be reattemtped the next attempt does run a statistically higher chance of being the correct answer than the previous attempt.
I think you are mistaken. There is no guarantee that a any solution will work for any given block. While the pool of possible answers is finite, it is so large that it can essentially be considered infinite for the purposes of considering an increase in statistical probability. If we consider the pool of potential hashes to be "finite" then we also have to consider that a resulting hash is not removed from the pool, since that same hash could be calculated again with a different set of data.
The next hash you calculate after incrementing the nonce is no more or less likely to result in a solution than then previous or the next hash.