Risk Battle Probability
Monte-Carlo probability of an attacker beating a defender in classic Risk.
Overview
The Risk Battle Probability calculator uses Monte-Carlo simulation to estimate the chance of an attacker eliminating a defender in classic Risk. Tell it how many attacking armies and defending armies are stacked, and it runs thousands of dice-by-dice battles to produce a winning probability for each side along with the expected losses.
Risk battle math is famously counter-intuitive: the defender wins ties on the dice, which tilts even-numbered confrontations more in the defender's favour than most players expect. The calculator removes the guesswork and helps you decide whether to commit a stack to an attack or to fortify and wait.
How it works
Each battle round, the attacker rolls up to 3 dice (one per attacking army, capped at 3 minus 1 reserve) and the defender rolls up to 2 dice. The highest die from each side is compared; the defender wins ties, so the attacker must strictly beat the defender's top die. The second-highest dice are compared the same way if both sides have at least two dice. Each lost comparison removes one army.
The simulator repeats this until one side has too few armies to continue. Running tens of thousands of independent battles produces a tight estimate of the overall victory probability. Closed-form analytical solutions exist for small army counts; Monte-Carlo is more flexible and handles the larger stacks players actually use without rebuilding state tables.
Examples
- 3 attackers vs 2 defenders: attacker wins about 37 percent of the time.
- 5 attackers vs 3 defenders: attacker wins about 64 percent.
- 10 attackers vs 5 defenders: attacker wins about 90 percent.
- 20 attackers vs 20 defenders: defender wins about 73 percent — defender ties strongly favour the defence.
FAQ
Why does the defender win ties?
That's the published Risk rule; it gives the defender an inherent edge that scales as battles get longer.
How many dice can the attacker roll?
Up to 3, equal to the number of attacking armies minus one (one army must occupy the territory), capped at 3.
Should I attack at 2:1 odds?
Roughly even chance at 2:1 (attacker:defender). 3:1 starts to look reliably favourable.
Are repeated attacks across multiple turns more efficient?
Splitting attacks across turns doesn't change individual battle math, but reinforcements between turns can shift the stack ratio.
Does the calculator handle the cards-trade bonus armies?
No — it only simulates the dice battle. Card trade-ins are a strategic layer above the dice math.