Risk Battle Odds Calculator
Enter the armies on each side to get the exact chance that the attacker conquers the territory, how many armies each side should expect to keep, and the probability of every possible ending.
10 attacking armies (9 can fight, 1 stays behind) against 10 defending armies. When the attack succeeds the attacker has on average 5.25 armies left in the attacking territory (before moving in). When it fails the defender keeps on average 4.37. Expected losses: attacker 6.96, defender 7.73.
| Final outcome | Probability | This or better for attacker |
|---|---|---|
| Attacker wins, 10 left | 0.7% | 0.7% |
| Attacker wins, 9 left | 2.1% | 2.8% |
| Attacker wins, 8 left | 3.9% | 6.7% |
| Attacker wins, 7 left | 5.9% | 12.6% |
| Attacker wins, 6 left | 7.7% | 20.3% |
| Attacker wins, 5 left | 8.7% | 29.0% |
| Attacker wins, 4 left | 9.6% | 38.6% |
| Attacker wins, 3 left | 6.3% | 44.9% |
| Attacker wins, 2 left | 3.1% | 48.0% |
| Defender holds, 1 def. left | 4.3% | 52.3% |
| Defender holds, 2 def. left | 8.0% | 60.3% |
| Defender holds, 3 def. left | 8.7% | 69.0% |
| Defender holds, 4 def. left | 7.7% | 76.7% |
| Defender holds, 5 def. left | 7.5% | 84.2% |
| Defender holds, 6 def. left | 5.9% | 90.1% |
| Defender holds, 7 def. left | 4.8% | 94.9% |
| Defender holds, 8 def. left | 2.9% | 97.8% |
| Defender holds, 9 def. left | 1.6% | 99.5% |
| Defender holds, 10 def. left | 0.5% | >99.9% |
Risk win probability table
Chance that the attacker wipes out the defender when fighting to the end with the maximum number of dice every roll. Rows are the armies in the attacking territory (one of them must stay behind), columns are the defending armies.
| Att \ Def | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 41.7% | 10.6% | 2.7% | 0.7% | 0.2% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% |
| 3 | 75.4% | 36.3% | 20.6% | 9.1% | 4.9% | 2.1% | 1.1% | 0.5% | 0.3% | 0.1% | 0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% | <0.1% |
| 4 | 91.6% | 65.6% | 47.0% | 31.5% | 20.6% | 13.4% | 8.4% | 5.4% | 3.3% | 2.1% | 1.3% | 0.8% | 0.5% | 0.3% | 0.2% | 0.1% | 0.1% | <0.1% | <0.1% | <0.1% |
| 5 | 97.2% | 78.5% | 64.2% | 47.7% | 35.9% | 25.3% | 18.1% | 12.3% | 8.6% | 5.7% | 3.9% | 2.6% | 1.7% | 1.1% | 0.7% | 0.5% | 0.3% | 0.2% | 0.1% | 0.1% |
| 6 | 99.0% | 89.0% | 76.9% | 63.8% | 50.6% | 39.7% | 29.7% | 22.4% | 16.2% | 11.8% | 8.3% | 5.9% | 4.1% | 2.9% | 1.9% | 1.4% | 0.9% | 0.6% | 0.4% | 0.3% |
| 7 | 99.7% | 93.4% | 85.7% | 74.5% | 63.8% | 52.1% | 42.3% | 32.9% | 25.8% | 19.3% | 14.7% | 10.7% | 8.0% | 5.7% | 4.1% | 2.9% | 2.1% | 1.4% | 1.0% | 0.7% |
| 8 | 99.9% | 96.7% | 91.0% | 83.4% | 73.6% | 64.0% | 53.6% | 44.6% | 35.7% | 28.7% | 22.2% | 17.3% | 13.0% | 10.0% | 7.3% | 5.5% | 4.0% | 2.9% | 2.1% | 1.5% |
| 9 | >99.9% | 98.0% | 94.7% | 88.8% | 81.8% | 73.0% | 64.3% | 54.7% | 46.4% | 38.0% | 31.2% | 24.7% | 19.7% | 15.2% | 11.9% | 9.0% | 6.9% | 5.1% | 3.8% | 2.8% |
| 10 | >99.9% | 99.0% | 96.7% | 93.0% | 87.3% | 80.8% | 72.6% | 64.6% | 55.8% | 48.0% | 40.0% | 33.4% | 27.0% | 21.9% | 17.3% | 13.8% | 10.6% | 8.3% | 6.2% | 4.8% |
| 11 | >99.9% | 99.4% | 98.1% | 95.4% | 91.6% | 86.1% | 80.0% | 72.4% | 65.0% | 56.8% | 49.4% | 41.7% | 35.3% | 29.0% | 24.0% | 19.2% | 15.5% | 12.2% | 9.7% | 7.4% |
| 12 | >99.9% | 99.7% | 98.8% | 97.2% | 94.3% | 90.5% | 85.2% | 79.4% | 72.3% | 65.4% | 57.6% | 50.7% | 43.3% | 37.1% | 30.9% | 25.9% | 21.0% | 17.3% | 13.7% | 11.1% |
| 13 | >99.9% | 99.8% | 99.3% | 98.2% | 96.4% | 93.4% | 89.6% | 84.5% | 79.0% | 72.3% | 65.8% | 58.4% | 51.8% | 44.8% | 38.7% | 32.6% | 27.6% | 22.8% | 18.9% | 15.2% |
| 14 | >99.9% | 99.9% | 99.6% | 98.9% | 97.6% | 95.6% | 92.5% | 88.9% | 83.9% | 78.7% | 72.3% | 66.1% | 59.2% | 52.8% | 46.1% | 40.2% | 34.2% | 29.3% | 24.4% | 20.5% |
| 15 | >99.9% | >99.9% | 99.8% | 99.3% | 98.5% | 97.0% | 94.9% | 91.8% | 88.2% | 83.5% | 78.4% | 72.4% | 66.5% | 59.9% | 53.8% | 47.3% | 41.6% | 35.7% | 30.8% | 25.9% |
| 16 | >99.9% | >99.9% | 99.9% | 99.6% | 99.0% | 98.1% | 96.4% | 94.3% | 91.2% | 87.7% | 83.1% | 78.3% | 72.5% | 66.9% | 60.5% | 54.7% | 48.4% | 42.8% | 37.1% | 32.3% |
| 17 | >99.9% | >99.9% | 99.9% | 99.8% | 99.4% | 98.7% | 97.6% | 95.9% | 93.8% | 90.7% | 87.2% | 82.8% | 78.2% | 72.6% | 67.2% | 61.1% | 55.5% | 49.4% | 44.0% | 38.4% |
| 18 | >99.9% | >99.9% | >99.9% | 99.9% | 99.6% | 99.2% | 98.4% | 97.2% | 95.5% | 93.3% | 90.2% | 86.9% | 82.6% | 78.1% | 72.8% | 67.6% | 61.7% | 56.3% | 50.4% | 45.1% |
| 19 | >99.9% | >99.9% | >99.9% | 99.9% | 99.8% | 99.5% | 98.9% | 98.1% | 96.9% | 95.0% | 92.8% | 89.8% | 86.5% | 82.4% | 78.1% | 72.9% | 68.0% | 62.3% | 57.0% | 51.3% |
| 20 | >99.9% | >99.9% | >99.9% | 99.9% | 99.9% | 99.7% | 99.3% | 98.7% | 97.8% | 96.5% | 94.6% | 92.5% | 89.5% | 86.3% | 82.2% | 78.1% | 73.1% | 68.3% | 62.8% | 57.7% |
75%+ 50–75% 25–50% under 25%
How the odds are calculated
The calculator does not simulate thousands of random battles. It works out the exact answer by treating the battle as a Markov chain: every position is a pair of numbers (attacking armies, defending armies), and each roll moves the battle to a new position with a known probability.
Those single-roll probabilities come from checking every possible combination of dice, up to 7,776 combinations for three attacking dice against two defending dice. The highest attacker die is compared with the highest defender die, then the second highest with the second highest, and the defender wins every tie. A three-against-two roll ends with the defender losing two armies 37.2% of the time, one army each 33.6% of the time and the attacker losing two 29.3% of the time.
Starting from your armies, the calculator pushes the probability forward roll by roll (dynamic programming) until every path has ended, either because the defender has no armies left or because the attacker is down to a single army and can no longer attack. Adding up the end positions gives the win probability, the expected survivors and the full outcome distribution shown above, with no rounding until the final display.
Army convention: the attacker number is the total in the attacking territory. One army must always stay behind, so 10 attacking armies means 9 can actually fight, and the attacker rolls three dice while it has 4 or more armies, two dice with 3 and one die with 2. The defender rolls two dice whenever it has at least 2 armies. Both sides are assumed to roll the maximum dice every time. Under this convention 10 against 10 is a 48.0% win for the attacker and 5 against 5 is 35.9%. If you count only the armies that can move, add one to the attacker number.
Want to see how it plays out? Use blitz mode on the dice roller to fight the whole battle with real random dice.