Probability of dice double - why not count duplicates?

Question

What is the probability of getting double dice? (one throw)

All possible dice rolls are 36 (counting even rolls like 1-4 and 4-1).

But when counting the doubles, why not count 1-1 twice, for example?

My book says the probability is 6/36.

Answer 1

You don't count (1,1) twice, because the situation "both dice show a 1" can only occur in one way - the first dice is 1, and the second dice is 1.

By contrast, the situation "one dice shows a 4, the other shows a 1" can occur in two ways - either the first dice is 4 and the second is 1, or the first dice is 1 and the second is 4.

The easiest way to convince yourself of this is to tabulate all 36 possibilities:

1 1    1 2    1 3    1 4    1 5    1 6
2 1    2 2    2 3    2 4    2 5    2 6
3 1    3 2    3 3    3 4    3 5    3 6
4 1    4 2    4 3    4 4    4 5    4 6
5 1    5 2    5 3    5 4    5 5    5 6
6 1    6 2    6 3    6 4    6 5    6 6

Answer 2

Before you roll your dice, colur one of them red and the other blue. Then red $1$ and blue $1$ is the same thing as blue $1$ and red $1$.

But red $1$ and blue $5$ is not the same as blue $1$ and red $5$.