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# Risk Battles

## Question

In the popular board game, Risk, players try to occupy as much territory as possible by moving around their armies and attacking territory owned by other players. These battles are settled through a series of dice rolls.

The attacker rolls up to three dice, while the defender only rolls two. First, the highest values rolled by each player are compared. If the attacker rolled a higher value, the defender loses a unit. Otherwise, the attacker loses a unit. Then, the second highest values are compared in the same manner and one of the players loses a unit. This goes on until one of the armies becomes depleted.

If the attacker has fewer than three units, he may only roll the same number of dice as the number of units. Likewise, if the defender has fewer the two units, he may only roll one die.

1) What is the probability that a dice roll results in a draw?

Okay, so now pretend there is a battle where the attacker has n units, and the defender has m units.

2) What is the probability that the attacker wins the exchange?

3) What is the average/best/worst case running time of the entire exchange?

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Okay, so now pretend there is a battle where the attacker has n units, and the defender has m units.

Does "battle" imply multiple attacks until either the defender looses all his armies or the attacker gets down to one army and can no longer attack?.

2) What is the probability that the attacker wins the exchange?

3) What is the average/best/worst case running time of the entire exchange?

By "running time" do you mean number of dice rolls?

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What are the initial numbers of opposing armies?

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If you're strictly asking about dice equalities (because draws technically go to the defender in Risk) and if all dice comparisons need to equal in a given roll to count as a draw, then:

3v2 = 4.8997% (381 / 7776)

3v1 = 16.6667% (216 / 1296)

2v2 = 5.0926% (66 / 1296)

2v1 = 16.6667% (36 / 216)

1v1 = 16.6667% (6 / 36)

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