How To Approach Dice Rolls When asked about rolling $2$ dice, do we count the result that both dice have the same number just one time and not two?
If it is because we can distinguish between the two, would it change the result if the dice were two different colors? 
 A: If you have two (six-sided) dice of different colors, then there are $36$ distinct outcomes, because you can tell the dice apart.  A red $1$ and a green $3$ is a different outcome than a green $1$ and a red $3$.
If they're the same color, then there are only $36 - 15 = 21$ distinct outcomes, because the $15$ combinations for which the dice have different values can't be distinguished from each other.
Hope this helps!
A: Your informal explanation is a good one: the probability that both dice show a $3$ should be unaffected by colouring, as should the probability of a $3$-$4$ split. 
It is possible to imagine a world in which a double $3$ and a $3$-$4$ split are equally likely. And indeed in the mysterious world of quantum interactions (please see Bose-Einstein Statistics) something like that  can happen, and assuming the equivalent of equiprobability provides a good model of reality.
Instead of colouring, we can imagine rolling one of the dice, and a few seconds later rolling the other one. Again, the intuition is that the long term frequency of a double $3$, and of a $3$-$4$ split, should be unaffected by the small time separation between the rolls. 
So part of the explanation comes from the assumption that the outcomes of rolling the two dice are independent. The experimental evidence is that they do behave as if they were independent. 
