Risk Battle Odds Calculator

How it works

single clash: compare highest dice pairwise, ties to the defender; battle: P(win | a, d) = Σ P(outcome) × P(win | a − attacker losses, d − defender losses), computed by dynamic programming

A Risk battle is a chain of dice clashes in which the attacker rolls up to three dice and the defender up to two, the highest dice are compared pairwise and the defender wins ties. Listing all 7,776 combinations of 3 v 2 gives the classic figures — the attacker takes both armies 37.2% of the time, loses both 29.3% and splits 33.6% — and every smaller configuration has its own table. The whole battle is then a random walk down from the starting army counts, and working backwards from the states where one side is eliminated gives the exact probability of taking the territory and the expected survivors. The per-die tie rule favours the defender, but the extra attacking die more than compensates, so large stacks attacking equal stacks are favoured.

Worked example

Ten committed attackers against six defenders win the territory about 86% of the time and finish with roughly 5.2 armies on average. Against equal numbers of ten, the attacker is a modest favourite at about 57%; at 5 v 5 it is a coin flip (51%) and with 3 v 3 the odds drop to 47%, because the small-stack endgame is fought with fewer attacking dice.

Frequently asked questions