1. What are the odds of winning a particular dice roll? (like 3 dice vs. 2 dice, 3 dice vs. 1 die, etc.)
2. What are the odds of conquering a territory? (for example, 20 men invading 17 men)

Anyone who has played risk has an intuitive sense of the answers to the first question.  Odds of winning a 3 dice vs. 2 dice battle are about 50/50.  The invading army gets the advantage of an extra die, but ties go to the defender.  It turns out that these advantages roughly balance each other out.  (Full results.)

Question #2 is harder because battles between a large number of soldiers are complicated.  It all comes down to which player is forced to roll with a reduced number of dice.  For example, if a large army is cut down early with a string of bad luck, its odds of winning go down much faster.

This table gives the odds of winning a whole series of dice rolls and capturing a territory.  The number of invading soldiers is along the side, defending soldiers along the top. Their corresponding entry gives the invaders’ odds of wiping out the defending army.

Click to enlarge!

Only the truly dedicated would want to memorize some of this. But the top-left corner gives us a feel for how typical situations will likely play out.

Some footnotes:
• Of course there is never a 100% probability of success.  I round to 100% when the probability is greater than 99.5%.
• This table considers up to 20 soldiers, but it could easily be extended.
• It’s obvious that more soldiers = better odds.  But following along the diagonals reveals interesting features. Notice how 2 vs. 1 is better than 3 vs. 2 but not as good as 4 vs. 3.
• To generate this table I computed 3 800 000 simulated conquests (10 000 per entry) and tabulated their outcomes in Mathematica.

Previous Work. There is a web article by Daniel C. Taflin (2001) that considers Question #1 and explains the underlying mathematics of his approach in detail.

Other important life lessons. Alliances are made to be broken; Asia is weak; never leave Australia unattended.