We roll two dice. Let the random variable X be the difference between larger and smaller number of dots

Question

We roll two dice. Let the random variable $X$ be the difference between larger and smaller number of dots. What's the probability scheme for the random variable $X.$

Answer 1

Maybe don't look for fancy combinatorial methods here. Instead, just write the sample space as:

11 12 13 14 15 16 
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
51 62 63 64 65 66

Then write 36 (absolute) differences in a similar array, and count.

0  1  2  3  4  5
1  0  1  2  3  4
2  1  0  1  2  3
...

A simulation of many rolls of a pair of dice produced the histogram below showing a good approximation to the distribution of absolute differences $Y.$ [Each 'brick' in the histogram has area $1/36.]$