Risk Battle Odds Calculator
Why "Attack With Three Dice" Isn't Always Right
Risk's combat system compares the attacker's and defender's highest dice, one pair at a time, and the defender wins every tie — a rule that quietly favors the defender more than casual players expect. This calculator simulates every possible outcome of a given dice matchup exhaustively (not by sampling) to report the exact expected unit losses on both sides.
The Method
Sort each side's dice highest to lowest
Compare pairs: highest vs highest, second-highest vs second-highest
Defender wins ties; higher die wins otherwise
Result = average units lost per side, across all possible dice outcomes
Exact Odds by Matchup
| Attacker dice | Defender dice | Attacker avg. loss | Defender avg. loss |
|---|---|---|---|
| 1 | 1 | 0.5833 | 0.4167 |
| 1 | 2 | 0.7454 | 0.2546 |
| 2 | 1 | 0.4213 | 0.5787 |
| 2 | 2 | 1.2207 | 0.7793 |
| 3 | 1 | 0.3403 | 0.6597 |
| 3 | 2 | 0.9209 | 1.0791 |
Even the best case for the attacker — 3 dice against a lone defending die — still loses an average of 0.34 units per exchange, since the defender only needs one high roll to win a comparison.
Why the Defender Has an Edge
With 2 dice on each side, the defender loses 0.7793 units on average versus the attacker's 1.2207 — a meaningfully better ratio for the defender despite both sides rolling the same number of dice. That asymmetry comes entirely from the tie-goes-to-the-defender rule, which effectively gives the defender's dice a small edge on every single comparison.
Where This Gets Used
- Deciding whether to attack — comparing expected losses before committing armies to an assault, especially against a well-defended territory.
- Defensive troop allocation — understanding how much a second defending die actually improves survival odds when deciding how to garrison a border territory.
- Teaching probability — Risk combat is a commonly used real-world example of order statistics and combined probability in introductory probability discussions.
How to Use This Calculator
- Select the number of attacking dice (1–3).
- Select the number of defending dice (1–2).
- Select Calculate to see the average units each side loses per battle, computed across every possible outcome.
Related Calculations
Check probability on other dice-based mechanics with the Dice Probability Calculator, or see resource-tile odds with the Catan Resource Probability Calculator.