Risk is still one of the most popular strategic board games in the world. The basic principle is simple: Create an army, attack countries occupied by other players and fulfill your goal.

When it comes to a fight, dices are rolled . The attacker must attack with at least one unit , but can also attack with as any many soldiers as he wants, as long as he has sufficient resources. The attacker must roll one dice when attacking with one a unit . He may roll with two dice when attacking with two units. And he may roll with three dice when attacking with 3 or more units.

The defender can defend with a maximum of 2 dice . After the dices have been thrown, the two highest eye are compared. If both show the same number (for example, both 6) the defender wins . If the number of the attacker is higher, the attacker wins.

This principle makes it easy to choose the best strategy using probability theory. The following graph shows on the x axis the number of aggressor (1 to 9) and on the y axis the probability that the attacker wins . The colored lines show how many defenders are there in each case.

Look at the blue line on the left. If there is only one defender which is attacked by one unit, the probability that the attacker wins is about 42 %. If you attack with two units , the probability of a win increases to approximately 76 % and with 3 units, you win with a probability of about 92%.

Here is a list of of many attackers are necessary in comparrison to the number of defenders to win the battle with a probability over 80%:

Number of defenders | Necessary number of attackers |

1 | 3 |

2 | 5 |

3 | 6 |

4 | 7 |

5 | 8 |

6 | 9 |

7 | 10 |

8 | 12 |

9 | 12 |

And there’s another interesting observation: If your army is as big as the army of the defender: Does it matter HOW big it is? In other words: Does the probability change when your army if 5 units strong (as well as the defender) or if your army is 50 units string (as well as the defender)?

The perhaps a bit surprising answer is, that it makes a difference! The larger the army, the greater is the advantage of the attacker! An example: If the attacker has 3 units and the defender also 3 units, the probability for the attacker to win is only 49%.

If the attacker has 50 units and the defender also 50 units, the probability for the attacker to win is now 73%!

Why? The reason is that the attacker is allowed to use 3 dice, the defender only 2. This offsets the advantage of the defender that he wins when both dices show the same number.

This means for the practical game: If two players get armed up and the army gets large (over 30 units) you should attack first!