How to Win Risk โ the odds
Risk is decided by dice, and the dice are exactly solvable. We enumerated every possible roll โ all 7,776 outcomes of the 3-attacker-vs-2-defender clash โ to get the true combat odds. Headline: a 1-vs-1 attack wins just 41.67% of the time, because the defender wins ties. You need a numbers edge โ here's exactly how much.
The headline roll: 3 attackers vs 2 defenders
This is the matchup you'll roll most. With the maximum dice on both sides, here's how a single round breaks down (7,776 equally-likely dice outcomes):
Always roll the maximum dice. The attacker wins both armies 37.2% of the time and comes out even-or-ahead (wins both or splits) about 71% of rounds. More dice is always strictly better for the attacker โ there's never a reason to hold dice back.
Every single-roll matchup
Exact win probabilities for every legal dice combination. "Attacker wins all" means the defender loses every contested army that round; ties always go to the defender:
| Atk vs Def dice | Atk wins all | Split | Def wins all | Exp. loss A / D |
|---|---|---|---|---|
| 1 vs 1 | 41.67% | โ | 58.33% | 0.58 / 0.42 |
| 1 vs 2 | 25.46% | โ | 74.54% | 0.74 / 0.26 |
| 2 vs 1 | 57.87% | โ | 42.13% | 0.42 / 0.58 |
| 2 vs 2 | 22.76% | 32.41% | 44.83% | 1.22 / 0.78 |
| 3 vs 1 | 65.97% | โ | 34.03% | 0.34 / 0.66 |
| 3 vs 2 | 37.17% | 33.58% | 29.26% | 0.92 / 1.08 |
"Exp. loss A / D" = expected armies lost by attacker / defender that round. Note 1-vs-1 (41.7% attacker) and 1-vs-2 (25.5%) are bad attacker odds โ never attack without a dice and army advantage.
Conquer-probability matrix (attack to the death)
The big one: the exact probability the attacker conquers the territory, fighting until one side is wiped out. Attacking armies down the side, defending armies across the top. Darker = more likely the attacker wins:
| A ๏ผผ D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 42 | 11 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 75 | 36 | 21 | 9 | 5 | 2 | 1 | 0 | 0 | 0 |
| 3 | 92 | 66 | 47 | 32 | 21 | 13 | 8 | 5 | 3 | 2 |
| 4 | 97 | 78 | 64 | 48 | 36 | 25 | 18 | 12 | 9 | 6 |
| 5 | 99 | 89 | 77 | 64 | 51 | 40 | 30 | 22 | 16 | 12 |
| 6 | 100 | 93 | 86 | 74 | 64 | 52 | 42 | 33 | 26 | 19 |
| 7 | 100 | 97 | 91 | 83 | 74 | 64 | 54 | 45 | 36 | 29 |
| 8 | 100 | 98 | 95 | 89 | 82 | 73 | 64 | 55 | 46 | 38 |
| 9 | 100 | 99 | 97 | 93 | 87 | 81 | 73 | 65 | 56 | 48 |
| 10 | 100 | 99 | 98 | 95 | 92 | 86 | 80 | 72 | 65 | 57 |
Values are conquer probability (%). E.g. 3 attackers vs 2 defenders โ 65.6%; even armies favour the defender until the stacks get large.
Equal armies are not a fair fight โ they favour the defender. Because ties go to the defender, you need to outnumber them to be a favourite. The matrix shows exactly how the odds climb with each extra attacking army.
How big an edge do you need?
Reading off the matrix, here's the smallest attacking force that gives you a coin-flip (โฅ50%) and a confident (โฅ70%) chance to conquer each defender count:
| Defenders | Attackers for โฅ50% | Attackers for โฅ70% |
|---|---|---|
| 1 defenders | 2 | 2 |
| 2 defenders | 3 | 4 |
| 3 defenders | 4 | 5 |
| 4 defenders | 5 | 6 |
| 5 defenders | 5 | 7 |
| 6 defenders | 6 | 8 |
| 7 defenders | 7 | 9 |
| 8 defenders | 8 | 10 |
| 9 defenders | 9 | 11 |
| 10 defenders | 10 | 12 |
Bring more than you think. A bare +1 army edge is only a coin flip on small stacks; for a battle you actually expect to win, aim for roughly +2 attackers over the defenders, and remember you also lose armies conquering โ keep enough to hold the new territory.
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How we got these numbers
Exact enumeration of dice, not simulation. For a single battle round we enumerate every possible roll of the attacker's dice (up to 3) against the defender's dice (up to 2) โ all 7,776 outcomes for the 3-vs-2 case โ sort each side high-to-low, compare highest-vs-highest and 2nd-vs-2nd, and give ties to the defender (the attacker must roll strictly higher). That yields the exact per-round loss probabilities, which validate against the published Risk tables (1v1 attacker wins 15/36 = 41.67%; 3-vs-2 wins both 2890/7776 = 37.17%). We then solve the full 'attack to the death' battle exactly by recursion over army counts to get the probability the attacker conquers and the expected armies each side loses. Standard Risk rules; the attacker rolls the maximum dice allowed each round. A guide to the odds โ individual battles still swing on the dice.
Validation: our enumeration reproduces the published Risk tables exactly โ 1-vs-1 attacker wins
15/36 (41.67%); 3-vs-2 wins both 2890/7776
(37.17%), splits 2611/7776, loses both 2275/7776. Source code
lives in our sims/ folder (risk_sim.py). See also the full Risk entry.
Common questions
What are the odds of winning a Risk battle?
In a single round with the maximum dice (3 attacker vs 2 defender), the attacker wins both armies 37.2% of the time (2890/7776), splits 1-1 33.6%, and loses both 29.3%. A lone 1-vs-1 attack is a losing proposition โ the attacker wins only 41.7% because the defender wins ties. Over a full attack-to-the-death battle, the conquer-probability table on this page gives the exact odds for any army count.
Should you always roll 3 dice in Risk?
Yes, whenever you're allowed to. Rolling the maximum number of dice strictly improves the attacker's expected outcome every round โ 3 attacker dice beat 2 defender dice 37.2% of the time outright and break even or better about 71% of rounds. There is no situation where rolling fewer attacker dice helps you.
How many armies do you need to conquer a territory in Risk?
You generally need a numbers edge, because ties go to the defender. For a coin-flip-or-better chance you usually want at least one more army than the defender on small stacks; for a confident (~70%) conquest you want roughly +2 on average. The conquer matrix here gives the exact probability for every attacker-vs-defender count.