Fellow nerds, I have a risky question

Good afternoon, fellow nerds! I'm working on a game and I need your assistance!
The game is basically a spin on "Risk." In the quest for world domination, two players roll opposing dice, and the one with the higher roll wins (defender wins in case of a tie). Now, to balance the units, I want to figure out the winning probabilities for each player with each type of unit, and it's giving me a hard time...
Here's the scenario:
Infantry = 1d6
Tanks = 1d12
Planes = 1d20

So, the question is: What are the winning probabilities for a player using a 1d20 both in defense and offense against the other dice? What are the winning probabilities for a player using a 1d12 both in defense and offense against the other dice? And what are the winning probabilities for a player using a 1d6 both in defense and offense against the other dice?
Something like:

I've been digging through my old stats class notes and online pages to find the answer, and it's probably simpler than I think, but I'm stuck...
Any help is appreciated!