The battle
Odds
Single roll odds
Each roll compares the highest attacker die with the highest defender die, then the second highest with the second. Ties go to the defender. The battle odds follow every roll to the end, exactly, assuming the attacker always rolls as many dice as allowed.
A Risk dice odds calculator tells you whether that attack is a good idea before you roll. Ten armies attacking eight defenders take the territory about 72% of the time. Three against three is a coin flip at 47%. A lone army attacking a lone defender wins only 41.7% of the time, because the defender wins ties. The free calculator above works out the exact odds for any battle, with how many armies you’ll typically have left.
How the Risk dice odds calculator works
Each roll, the attacker throws up to three dice and the defender up to two. The highest dice are compared, then the second highest, and each comparison kills one army: the defender wins ties. The calculator follows every possible sequence of rolls from your starting armies until one side runs out (or until you stop at the number you choose), adding up the probability of each outcome exactly.
Single roll odds
| Dice | Attacker wins all | Split | Defender wins all |
|---|---|---|---|
| 3 v 2 | 37.2% | 33.6% | 29.3% |
| 3 v 1 | 66.0% | – | 34.0% |
| 2 v 2 | 22.8% | 32.4% | 44.8% |
| 2 v 1 | 57.9% | – | 42.1% |
| 1 v 2 | 25.5% | – | 74.5% |
| 1 v 1 | 41.7% | – | 58.3% |
Three dice against two is the only match-up where the attacker loses fewer armies on average, about 0.92 to the defender’s 1.08. Below that the defender has the edge. That’s why big stacks win and small attacks bleed: keep rolling three dice and the maths is on your side.
Attacking armies here means armies that can attack: the territory’s total minus the one that has to stay behind. Playing a house rule where the defender can roll three? Switch the defender dice and the odds update.
For other dice questions, the dice probability calculator covers any pool of dice.
Risk is a trademark of Hasbro. This is an independent odds tool, not affiliated with or endorsed by Hasbro.