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.

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 |