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# Probabilities in Risk watch

1. [size-14]Probabilities in Risk[/size-14]

I thought I would start a new thread for those interested in board-games - Risk in particular - and who might (for some weird reason ) want to think about the probabilities of events on the board in Risk!

Let me lay down the rules for those of you who haven't played the glorious game of Risk:

a) An attacker attacks, and a defender defends. The attacker can attack with any number of pieces less than or equal to how many he has, and a defender defends with all the pieces he has.

b) When the attacker attacks, he rolls with 1 die (if he is attacking with 1 piece), 2 dice (if he is attacking with 2 pieces) or 3 dice (if he is attacking with 3 pieces or more). When the defender defends, he rolls with 1 die (if he has one piece) or 2 dice (if he has 2 or more pieces). The roll of the attacker, and the roll of the defender, are independent of one another, i.e. they essentially occur simultaneously.

c) Of the rolls of the attacker's die/dice, align them in order from highest roll to lowest roll. e.g. {2,5,5} should be aligned {5,5,2}. Of the rolls of the defender's die/dice, align them in order from highest roll to lowest roll. e.g. {5,1}. Now compare the attacker's first, highest score to the defender's first, highest score: if the attacker's is higher, the defender loses a piece; if the defender's is higher or the same as the attacker's, the attacker loses a piece. Then compare the attacker's second score to the defender's scond score: if the attacker's is higher, the defender loses a piece; if the defender's is higher or the same as the attacker's, the attacker loses a piece. In my example, 5v5 so the defender wins the first, but 2v1 so the attacker wins the second, and the net result is that each side loses 1 piece. Note that, the defender having 2 dice total maximum, the attacker can only lose a maximum of 2 pieces per turn, and same for the defender.

d) The next turn, the defender proceeds with his new number of pieces, and the attacker with his new number of pieces, and the same thing happens again. This continues until one side runs out of pieces.

And now you're ready to play Risk! solve probability questions to do with Risk.

I'll post links here to any of the cool proofs people do.
2. The ultimate question, I suppose, would be "the attacker has dice, the defender has dice; what is the probability of the attacker winning in the end?" An easier start might be "the attacker has dice, the defender has dice; what are the probabilities of each outcome for the first roll?" (the outcomes being a) defender wins both, b) attacker wins both, c) defender wins 1, attacker wins 1).

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Updated: May 15, 2013
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