RISK Dice Odds: Every Attack, Calculated

Three dice against two is the best attack RISK offers, and it still costs you both armies 29% of the time. That one number explains most of the bad turns you have ever had.

This page has the exact odds for every roll and for whole battles. Nothing here is simulated: the numbers come from listing every possible roll and then working the battle through to the end. They apply to the classic rules, which our RISK-style game CONQUEST calls Wild Dice. The last part covers Tactical Dice, where the math is different on purpose.

One roll has six possible matchups

The rule fits in a paragraph. The attacker rolls up to three dice, but always one fewer than the armies on the attacking territory, because one army has to stay home. The defender rolls up to two. The highest dice are compared, then the second highest, and the official RISK rules settle the rest in one line: "In case of a tie, the defender always wins."

That leaves six combinations of dice. When both sides roll at least two, two armies are at stake.

Attacker dice	Defender dice	Defender loses 2	One army each	Attacker loses 2
3	2	37.2%	33.6%	29.3%
2	2	22.8%	32.4%	44.8%

When either side rolls a single die, one army is at stake.

Attacker dice	Defender dice	Attacker wins the roll	Defender wins the roll
3	1	66.0%	34.0%
2	1	57.9%	42.1%
1	2	25.5%	74.5%
1	1	41.7%	58.3%

These are the same values Jason Osborne published in Markov Chains for the RISK Board Game Revisited, where the three against two case appears as 2,890, 2,611 and 2,275 out of 7,776 rolls. We recomputed all six by enumeration before putting them in a table, and they agree to the last digit.

Look at the second row of the first table. With two attacking dice against two defending dice you lose both armies almost twice as often as the defender does. Three dice is the only two-army roll where the attacker comes out ahead.

A whole battle depends on the stacks, not on one roll

Nobody stops after one roll. The question during a turn is whether your 8 can take their 5.

The table below gives the chance of capturing the territory if you keep attacking with as many dice as you are allowed until it falls or you are down to one army. "Your armies" is the number on the attacking territory, including the one that cannot leave.

Your armies	vs 1	vs 2	vs 3	vs 4	vs 5	vs 6	vs 8	vs 10
2	41.7%	10.6%	2.7%	0.7%	0.2%	0.0%	0.0%	0.0%
3	75.4%	36.3%	20.6%	9.1%	4.9%	2.1%	0.5%	0.1%
4	91.6%	65.6%	47.0%	31.5%	20.6%	13.4%	5.4%	2.1%
5	97.2%	78.5%	64.2%	47.7%	35.9%	25.3%	12.3%	5.7%
6	99.0%	89.0%	76.9%	63.8%	50.6%	39.7%	22.4%	11.8%
7	99.7%	93.4%	85.7%	74.5%	63.8%	52.1%	32.9%	19.3%
8	99.9%	96.7%	91.0%	83.4%	73.6%	64.0%	44.6%	28.7%
9	100.0%	98.0%	94.7%	88.8%	81.8%	73.0%	54.7%	38.0%
10	100.0%	99.0%	96.7%	93.0%	87.3%	80.8%	64.6%	48.0%
12	100.0%	99.7%	98.8%	97.2%	94.3%	90.5%	79.4%	65.4%
15	100.0%	100.0%	99.8%	99.3%	98.5%	97.0%	91.8%	83.5%
20	100.0%	100.0%	100.0%	99.9%	99.9%	99.7%	98.7%	96.5%

So 8 against 5 works about three times in four. 6 against 5 is a coin flip. 5 against 5 is a bad idea that feels like a fair fight.

The second defender matters more than any other. Taking a territory held by 1 army with a stack of 4 succeeds 91.6% of the time. Put a second army on that territory and the same attack drops to 65.6%. If you have armies to spare on a border, the jump from 1 to 2 is the cheapest defense in the game.

The break-even points are worth memorizing

You will not have a table open in the middle of a turn. These are the stack sizes where a battle tips in your favor.

Defenders	Stack for better than even	Stack for 80%
1	3	4
2	4	6
3	5	7
4	6	8
5	6	9
6	7	10
8	9	13
10	11	15
15	15	20
20	19	25

The short version: their count plus two is a coin flip, their count plus four or five is a good attack.

The pattern at the bottom of the table is the interesting part. Against 20 defenders, 19 armies already win more often than they lose. The defender's tie advantage matters most in small fights. In big ones the attacker's third die wins out, and Osborne's paper reaches the same conclusion about expected losses in large battles.

That is why turtling behind one huge stack is weaker than it looks. A neighbor who matches your stack is the favorite on the day they choose to attack.

Tactical Dice rewards the bigger stack

CONQUEST has a second dice mode for players who would rather not lose 10 armies to 3.

In Tactical Dice the pips do not decide a comparison. Each comparison is won by the attacker with a probability that depends on the two stacks: the attacking armies (minus the one that stays home) raised to the power 1.4, divided by that same value plus the defending armies raised to the power 1.4. The result is held between 18% and 82%, so nothing is ever certain. The dice you see on screen are then set to match the outcome.

Here is what that does to the same battles.

Battle	Wild Dice	Tactical Dice
3 against 2	36.3%	50.0%
4 against 3	47.0%	50.0%
6 against 5	50.6%	50.0%
8 against 4	83.4%	96.6%
10 against 3	96.7%	99.98%
5 against 6	25.3%	10.1%
10 against 10	48.0%	32.5%
12 against 10	65.4%	66.7%

Three things change. Forces that are equal after the stay-home army is subtracted fight at exactly 50%, at any size. A real numerical edge becomes close to a guarantee: 8 against 4 goes from 83% to 97%. And attacking uphill is punished hard: 5 into 6 falls from one chance in four to one in ten.

Wild Dice forgives a bad attack now and then. Tactical Dice almost never does.

— Design note

A sensible split is Wild Dice with friends and Tactical Dice when you want the better plan to win. Wild keeps the comeback alive, which is half the fun of the board game. Tactical turns every attack into a counting exercise. Both modes are a toggle in the setup screen, and the CONQUEST rules page covers the rest of the options. CONQUEST is our own game and is not affiliated with Hasbro.

The numbers change how a turn is played

Keep four armies on any territory you plan to attack from. Three dice or nothing.

Count the battle before the first roll. If the table says 47%, the attack is a gamble, whatever the first roll looks like.

Budget for the chain. After every capture you must move in at least as many armies as you rolled dice, and one army stays behind each time. A run through five territories spends five armies before the dice take any.

Give every border territory a second army before you give any territory a tenth.

If real-time suits you better than turns, the same habit of checking the numbers first carries over to Frontlines. The same exercise for its economy is in what to build first in Frontlines.