Fun with probability and dice

My 3rd grader's class is doing money, fractions, and probability, and I've been thinking up a brief parent volunteer lesson plan to talk about probability with dice, around the idea of having two people each roll a die with a different number of sides, and talking about how likely it is that one or the other roll the higher number.

I'd start with d4 vs. d6; there are 24 possibilities and it's easy enough to work with the kids to enumerate all of them.  There are 14 wins for the d6, 6 for the d4, and 4 ties.

d6 vs. d8 is tougher to do in a short lesson; we'll work out that there are 48 possible outcomes, and there are 6 ties in those outcomes.  Maybe we'll have time to work out that the number of wins for the d6 is 1+2+3+4+5, or the sum of number from 1 to (number of sides-1), and therefore the number of wins for d8 is 48 - 6 - 15 = 27, but that's probably too much to get through in a single day. 

So we're all set, except... in d4 vs. d6, 14 - 6 = 8 = 2 * 4; in d6 vs. d8, 27 - 15 = 12 = 2 * 6.  Hunh.

Dice          Larger  Smaller  Ties
              Wins    Wins
d4  vs. d6    14      6        4
d6  vs. d8    27      15       6
d8  vs. d10   44      28       8
d10 vs. d12   65      45       10
d12 vs. d20   162     66       12

The difference in the number of wins is the number of sides on the smaller die times the difference in the size of the dice.  I'd never noticed this before.  Cool.

The visual explanation for this relationship is as follows (looking at the possible outcomes of d4 vs. d6):

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

The red area above the diagonal shows where the smaller die wins, the blue on the diagonal shows the ties, and the green below the diagonal shows where the larger die wins.  In the first four rows, the numbers of wins are balanced, but once past the possibility of a tie, the larger die wins every possible outcome in the row.  The number of outcomes in each row is the size of the smaller die, and the number of rows in which the the larger die wins all possible outcomes is the difference in the sizes of the dice.

Ah, of course.  It looks obvious from this angle, but wasn't immediately so coming at it from the other side.