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In a RISK battle, an attacker and a defender each have a certain number of armies at the start of a battle sequence.
In each round of battle, the attacker may roll a number of dice equal or less than the number of attacker armies minus one,
up to a maximum of three dice. Under US/UK rules, the defender may roll a number dice equal or less than the number of defender
armies, up to a maximum of two dice. The rolled dice are paired, one from the highest remaining attacker die and one from the
highest remaining defender die, until no additional pairs can be made. For each pair, the person with the lower die roll loses
an army, with ties in favor of the defender. After each round the attacker decides whether to continue the battle for another
round, but the battle *must* end when the defender has 0 armies remaining or the attacker has 1 army remaining.

**RiskOdds** assumes that the attacker and defender always choose to roll the maximum number of dice allowed and that the dice are
independent and fair. For an excellent discussion of RISK rules and strategies see the
RiskFAQ. The FAQ also includes an interesting section on
probabilities in RISK.

I find the battle probabilities most useful when viewed in tables such as this one that cover a range of values for attacker and defender armies. By checking the table entries against the desired level of confidence, the attacker can determine how many armies constitute a reasonable margin of safety in any given battle. In addition, after any round of battle the same table can be consulted to help the attacker decide whether or not to proceed.

*Technical notes:* The online version of **RiskOdds** is written in perl with an HTML interface. Calculations start with the
determination of the complete matrix of 'end state' probabilities. This is accomplished through combination formulas and, for low
numbers of armies, automated reversed branching. Once the matrix is determined, the probability of winning the battle and the expected
army losses are easily calculated. All probabilities are rounded for display (5 always rounds up); the degree of rounding may be selected
on the input form. The online demonstration version of **RiskOdds** does not accept input values that exceed 50 armies.

Designed by
Scott Bartell; last updated 9/9/01. | Comments or suggestions? |