# Risk Probabilities

##### January 1, 2014

## Introduction

In the board game Risk, in a turn a player chooses to attack other players' armies, and has to defend his own armies from attack. This problem attempts to quantify the probabilities to justify how many dice to roll in certain situations.

The attacking player rolls up to 3 dice, where rolling n dice requires the player to have at least n+1 armies attacking (but player may roll less dice if desired).

The defending player rolls up to 2 dice, where rolling n dice requires the player to have at least n armies defending (may roll less dice if desired).

Only as many armies can be defeated per turn as the least number of dice rolled by either player. So in one sense, for a defender it is preferred to roll less dice to reduce the effects of defeat. But in another sense, if the attacker chooses to roll more dice, then with less dice the defender will have a higher probability of defeat. There should be a compromise for how many dice to roll in each situation, and because of the limit on the number of dice these situations are easily enumerable. A code was written to check all dice combinations and find probability of defeat or victory in each case.

## Results

The table below represents the probability that on the next roll of dice either the defender wins (no loss of defending troops), the attacker wins (no loss of attacking troops), or a 'tie' occurs in which case both sides lose 1 troop.

Attacker Dice | Defender Dice | Attacker Wins | Defender Wins | Lose 1-1 |

1 | 1 | 42% | 58% | 0% |

2 | 1 | 58% | 42% | 0% |

3 | 1 | 66% | 34% | 0% |

1 | 2 | 25% | 75% | 0% |

2 | 2 | 23% | 45% | 32% |

3 | 2 | 37% | 29% | 34% |