Difference between tossing $2$ dice that are identical as compared to tossing $2$ dice that are distinct

Question

I came across a basic math question. $2$ dice are tossed simultaneously, what is the probability that the sum of the top faces of the dice is equal to $x$, where $x$ can be {2,3,...12}.

Intuitively, there are $36$ possible outcomes of the tosses, and we can count the favourable outcomes for each $x$ and thus get the probability.

Will the answer differ if the dice are identical? I assume they won't, as the underlying probability distribution will not change.

If the answer will not differ in this case, in which cases will it matter if the dice are identical or not?

Answer 1

If the question is only about probability of a specific sum, it does not matter whether they are identical or distinct by appearance. If you have to get a sum of $3$, the probability will be $\dfrac{2}{36}$ as there are two possible outcomes of {2,1} and {1,2} with equal probability of $\dfrac{1}{6} \times \dfrac{1}{6}$ each.

In fact what matters is if both are fair dice or not, meaning if the probability of each face is $\dfrac{1}{6}$ for both dice or not.

Answer 2

With two "distinct" dice, say one green and the other red, the 36 ways they can come up, writing the number on the green die first, are (1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), etc. If the two dice are "identical" you cannot distinguish between "(1, 2)" and "(2, 1)".

Removing (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6) leaves 30 pairs where the first and second numbers are different so, with "identical" dice, cannot be distinguished from there dual. With "identical" dice there 30/2+ 6= 15+ 6= 21 possible outcomes rather than 36.