Two dice roll with {1,2,3,4,5,6} and {10,20,30,40,50,60} and importance of RV mapping We're all too familiar with a two-dice-roll experiment where we start with a uniform sample space of $S_{die}=\{1,2,3,4,5,6\}$ and end up in a non-uniform pmf for the sum of the numbers on the two faces with $S_{sum}=\{2,3,4,5,6,7,8,9,10,11,12\}$
What I'm confused is that why the mapping of the faces to RV values not discussed in introductory discussion?
What if we assume for the second die (call it an "incdie" or increment die): $S_{incdie}=\{2,3,4,5,6,7\}$ ?
Well you'd say the results of the experiment don't change except for a shift in the sample space of the sum: $S_{sum}=\{3,4,5,6,7,8,9,10,11,12,13\}$. Fair enough.
But now what if the second die is a "decadie": $S_{decadie}=\{10,20,30,40,50,60\}$
This completely wrecks the experiment! e.g., first-die-outcome-1 + second-die-outcome-3 ($1 + 30 = 31$) is no longer equal to first-die-outcome-3 + second-die-outcome-1 ($3 + 10 = 13$). As a result, the pmf of the sum is not longer a pyramid shape, but a uniform pmf just like that of its constituents.
edit: I guess the question would be: what is the point of this "incomplete example" that is so prevalent in 90% of the texts? Because without the comprehensive discussion of its variations and caveats, I don't really understand the message being conveyed. I'm only more confused than before.
 A: The mapping that I know is from a n x m space to a one-dimensional space. 
$$\begin{array}{c|ccccc}
& d_{11} & d_{12} & \cdots &d_{1n} \\ \hline
d_{21} & d_{11} + d_{21} & d_{12} + d_{21} & \cdots & d_{1n} + d_{21}\\
d_{22} & d_{11} + d_{21} & d_{12} + d_{22} & \cdots & d_{1n} + d_{22}\\
\vdots & \vdots & \vdots & & \vdots \\
d_{2m} & d_{11} + d_{21} & d_{12} + d_{2m} & \cdots & d_{1n} + d_{2m}\\
\end{array} $$
The $\text{$P($sum $2$ rolls $= k)$}$ are not all the same.
Here the trick for the double dice roll is that some of the values $k$ may occur in multiple ways and you have to collect/count all the possible cases $d_{1i}+d_{2j}=k$.
This is probably one of the reasons why this example is so prevalent. It is a simple (and recognizable and intuitive) exercise of summing multiple pathways that lead to a single result (e.g. you get the sum equal to 4 when you roll 1+3, 2+2 and 3+1). 
I do not think that the two dice rolls exercise/example is used to particularly demonstrate the pyramid shape (which is indeed not general for all types of dice rolls), but more like showing a specific case of a convolution and practicing/showing how to compute it ( Why is the sum of two random variables a convolution? )
A similar problem (as in one that would require the student to make a table matching every case in the sample space with events) is Bertrand's box paradox
