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I was wondering how you can calculate the outcome of dice rolls in the board game Risk, but with modifiers, assuming 3 attackers and 2 defenders.

In Risk the attacker rolls 3 dice, and the defender rolls 2. The highest two of each sets of dice (the attacker's and defender's) are compared, the highest to the highest, and the second highest to the second highest. If a die is higher than the one it is compared to, the opposing player loses a unit. The defender wins on a tie. Each player can lose up to two units this way, or each can lose one.

I want to know how to calculate the probability if you add a value to the 1st, 2nd, or 3rd highest dice of each set. For example, how do the probabilities change if you add 2 to the attacker's 2nd highest die and add 1 to the defender's highest die, after the dice have already been paired for comparison. I would like to know how to calculate the odds when adding any value to any of the attacker's or defender's dice.

Thanks in advance for anyone who can help me with this problem!

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The following link provides a robust analysis of Risk probability:

http://arxiv.org/pdf/1204.4082v1.pdf

The following equation can be adjusted for added values to the dice rolls:

$P(X_{(k)}{\le}m) = {\sum \limits_{j = 0}^{n-k} {\binom{n}{j} (\frac{7-m} 6)^j (\frac {m-1} 6)^{n-j}}}$

It is numbered as equation 6 in the article.

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