## About the RISK Battle Odds Calculator

NOTE: This site is currently being migrated to a different web hosting company; some links may be temporarily inactive. Please check back later if you have any difficulty accessing the calculator.

RISK Battle Odds Calculator ("RiskOdds") calculates the probability of an attacker winning a given battle sequence in the classic RISK board game (US/UK rules). The initial number of attacking and defending armies is required. RiskOdds also allows the user to set a predetermined attacker army threshold at which the attacker will voluntarily end the battle. The expected numbers of armies lost are also available for both the attacker and defender, as are the probabilities for all possible battle outcomes.

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?