Risk Battle Odds: Calculating Attack Probability Exactly
In Risk, the attacker rolls up to three dice and the defender up to two, the highest dice are compared in pairs, and the defender wins ties. Chaining those single-round probabilities across every possible army state gives the exact chance an attack conquers a territory.
Single-round probabilities (3 vs 2 dice)
- Attacker loses two armies: 2275/7776, about 29.3%.
- Defender loses two armies: 2890/7776, about 37.2%.
- Each side loses one: 2611/7776, about 33.6%.
How the defender's edge works
Ties going to the defender is worth a substantial advantage, but it is partly offset by the attacker rolling a third die and keeping the best two. At equal army counts the defender is the favourite; the attacker becomes the favourite somewhere around a 1.2 to 1.5 times numbers advantage, and by two to one the attack wins the large majority of the time.
Practical play guidance
- Always attack with three dice when you can — attacking with fewer wastes your only structural advantage.
- One army must stay behind, so ten armies on a territory commit only nine to the attack.
- Continent bonuses compound each turn, so a marginal attack that secures a continent is often worth worse odds.
- Count expected survivors, not just win probability: winning with one army left usually loses the territory next turn.
Key takeaways — Risk Global Domination Battle Calculator
- Exact odds come from recursing the per-round dice distribution, not from simulation.
- The defender wins ties, so parity favours the defence.
- Aim for roughly a 1.5x army advantage before committing to an attack.