I considered two questions about the board game Risk.

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 who has 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.

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