Let $X = X_{1} + X_{2}$ be a random variable where $X_{1}$ is the result of rolling a first die and the $X_{2}$ of rolling a second die. The possible outcomes are
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
It is easy to see that there are $11$ different possible outcomes: $2, 3, 4, \dots, 12$.
I would like to calculate the probability mass function of $X$. From the table, I can see that $$n(2) = 1, n(3) = 2, n(4) = 3, n(5) = 4, n(6) = 5, \\ n(7) = 6, n(8) = 5, n(9) = 4, n(10) = 3, n(11) = 2, n(12) = 1$$
where $n(x)$ is the number of occurrences of a particular outcome.
Since there are $11$ different outcomes, I would intuitively think that the pmf is $p(x) = n(x) / 11$. For example, $p(2) = 1/11$ since there is $11$ possible outcomes and only one way of having $2$ occurring.
However, $$\frac{1+2+3+4+5+6+5+4+3+2+1}{11} = \frac{36}{11}$$
so my logic is wrong since it should come as $1$.
What is the correct way of thinking about this? (I can see what the answer is but not how to get there.)
