High Programmer > Alan De Smet > Games > Video Games > Dicewars probabilities

Dicewars probabilities

The Flash game Dicewars is amusing. It's very similar to Risk.

A location in Dicewars can not have more than 6 dice. In an attack, the attacker rolls and totals all of the dice in the attacking location while the defender rolls and totals all of the dice in the defending location. If the attacker has a higher total, all of the defenders dice are removed and the attacker moves all but 1 of the attacking dice in. If the defender has the higher total or there is a tie, all but 1 of the attacking dice are removed. An attacker must have 2 dice in the attacking location.

The following were all calculated by myself using software I developed.

Here is the attacker's chance of success graphically. The top line is for an attacker with 6 dice, below that 5 dice, and so on down through 2 dice.

Graph of probabilities of the game dicewars. The data in tabular form follows.

In this table, probabilities are approximate. In particular it is possible to lose with 6 dice attacking 1, but it's extremely unlikely. Below this table is another giving exact probabilities.

		Defense Dice
		1		2		3		4		5		6
Atk Dice	2	83.8	%	44.4	%	15.2	%	3.6	%	0.6	%	0.1	%
	3	97.3		77.9		45.4		19.2		6.1		1.5
	4	99.7		93.9		74.3		46.0		22.0		8.3
	5	100.0		98.8		90.9		71.8		46.4		24.2
	6	100.0		99.8		97.5		88.4		70.0		46.7

Here are the exact probabilities expressed as fractions:

		Defense Dice
		1		2		3		4		5		6
Atk Dice	2	181/	216	575/	1,296	197/	1,296	31/	864	1,709/	279,936	143/	186,624
	3	1,261/	12,96	1,009/	12,96	3,527/	7,776	559/	2,916	50,987/	839,808	74,971/	5,038,848
	4	2,585/	2,592	541/	576	23,105/	31,104	85,759/	186,624	246,839/	1,119,744	420,355/	5,038,848
	5	46,649/	46,656	17,285/	17,496	84,853/	93,312	25,127/	34,992	97,345/	209,952	73,29,985/	30,233,088
	6	279,935/	279,936	1,676,621/	1,679,616	9,828,775/	10,077,696	26,724,643/	30,233,088	126,909,383/	181,398,528	507,985,465/	1,088,391,168

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