Identical Dice and “Identical” Sum (or Boxes??)

Say you have two 6 sided dice numbered from 1 to 6. How would you calculate the probability of rolling 2, 3, 4, ..., 12?

My professor told me the generating function $$(x+x^2+x^3+x^4+x^5+x^6)^2$$ is incorrect for starting the problem because the dice are treated as distinct. That I should be looking for the answer using identical dice and "sum". How would this be solved?

• If the dice are treated as distinct, why does your title describe them as identical? – N. F. Taussig Dec 21 '18 at 23:23
• @N.F.Taussig My professor asked for identical dice while I provided distinct dice. – Violet Jung Dec 22 '18 at 1:58
• The coefficients of that generating function (when divided by $6^2$ for normalization), are the probabilities in question. – Math1000 Dec 22 '18 at 2:41

1 Answer

Well if your dice are identical then you can simply say that for finding the probability of rolling a certain sum $$s$$ we need to find the number of ways $$x_1 + x_2$$ can be equal to $$s$$ or rather we need to find the number of integral solutions to the equation $$x_1 + x_2 = s$$ where $$x_1,x_2 \equiv 1,2,3,4,5,6$$ We can solve this rather easily and then we can use the argument for probability.