During a round of the game Risk, a player is frequently faced with the problem of exactly how many troops it will take to expect a victory against the opponents 10 soldiers or how many players they should leave at the border in order to have good odds of fending off the incoming onslaught of a cannon and a horse. In order to take the guessing out of the picture and make the game play more theoretically sound, I generated the following table. The top row indicates how many soldiers the attacker has and the left column counts the number of defenders. The corresponding square in the table contains the odds of the attacker winning, assuming that both sides always roll as many dice as they can (it can be shown that this is optimal). For example, if 12 players attack 15 defenders then the attackers will be victorious only 97 out of 1000 times.
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | ||
1 | 0.6 | 0.7 | 0.6 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | |
2 | 0.9 | 1.4 | 1.4 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | |
3 | 1.0 | 1.7 | 1.9 | 2.1 | 2.2 | 2.2 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | |
4 | 1.0 | 1.9 | 2.3 | 2.7 | 2.9 | 3.0 | 3.1 | 3.1 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | 3.2 | |
5 | 1.0 | 1.9 | 2.5 | 3.0 | 3.4 | 3.6 | 3.8 | 3.9 | 3.9 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | |
6 | 1.0 | 2.0 | 2.7 | 3.3 | 3.8 | 4.1 | 4.4 | 4.6 | 4.7 | 4.8 | 4.8 | 4.8 | 4.9 | 4.9 | 4.9 | 4.9 | 4.9 | 4.9 | 4.9 | 4.9 | 4.9 | 4.9 | 4.9 | 4.9 | |
7 | 1.0 | 2.0 | 2.8 | 3.5 | 4.1 | 4.6 | 4.9 | 5.2 | 5.3 | 5.5 | 5.6 | 5.6 | 5.7 | 5.7 | 5.7 | 5.7 | 5.7 | 5.7 | 5.7 | 5.7 | 5.7 | 5.7 | 5.7 | 5.7 | |
8 | 1.0 | 2.0 | 2.9 | 3.7 | 4.4 | 4.9 | 5.4 | 5.7 | 6.0 | 6.2 | 6.3 | 6.4 | 6.5 | 6.5 | 6.5 | 6.6 | 6.6 | 6.6 | 6.6 | 6.6 | 6.6 | 6.6 | 6.6 | 6.6 | |
9 | 1.0 | 2.0 | 2.9 | 3.8 | 4.5 | 5.2 | 5.7 | 6.2 | 6.5 | 6.8 | 7.0 | 7.1 | 7.2 | 7.3 | 7.3 | 7.4 | 7.4 | 7.4 | 7.4 | 7.4 | 7.4 | 7.5 | 7.5 | 7.5 | |
10 | 1.0 | 2.0 | 3.0 | 3.9 | 4.7 | 5.4 | 6.0 | 6.5 | 7.0 | 7.3 | 7.6 | 7.8 | 7.9 | 8.0 | 8.1 | 8.2 | 8.2 | 8.2 | 8.3 | 8.3 | 8.3 | 8.3 | 8.3 | 8.3 | |
11 | 1.0 | 2.0 | 3.0 | 3.9 | 4.8 | 5.6 | 6.3 | 6.9 | 7.4 | 7.8 | 8.1 | 8.4 | 8.6 | 8.7 | 8.9 | 8.9 | 9.0 | 9.1 | 9.1 | 9.1 | 9.1 | 9.1 | 9.1 | 9.2 | |
12 | 1.0 | 2.0 | 3.0 | 3.9 | 4.8 | 5.7 | 6.4 | 7.1 | 7.7 | 8.2 | 8.6 | 8.9 | 9.2 | 9.4 | 9.6 | 9.7 | 9.8 | 9.8 | 9.9 | 9.9 | 10.0 | 10.0 | 10.0 | 10.0 | |
13 | 1.0 | 2.0 | 3.0 | 4.0 | 4.9 | 5.8 | 6.6 | 7.3 | 8.0 | 8.5 | 9.0 | 9.4 | 9.8 | 10.0 | 10.2 | 10.4 | 10.5 | 10.6 | 10.7 | 10.7 | 10.8 | 10.8 | 10.8 | 10.8 | |
14 | 1.0 | 2.0 | 3.0 | 4.0 | 4.9 | 5.8 | 6.7 | 7.5 | 8.2 | 8.8 | 9.4 | 9.9 | 10.2 | 10.6 | 10.8 | 11.0 | 11.2 | 11.3 | 11.4 | 11.5 | 11.6 | 11.6 | 11.6 | 11.7 | |
15 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 5.9 | 6.8 | 7.6 | 8.4 | 9.1 | 9.7 | 10.2 | 10.7 | 11.1 | 11.4 | 11.7 | 11.9 | 12.0 | 12.2 | 12.3 | 12.4 | 12.4 | 12.5 | 12.5 | |
16 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 5.9 | 6.8 | 7.7 | 8.5 | 9.3 | 9.9 | 10.6 | 11.1 | 11.5 | 11.9 | 12.2 | 12.5 | 12.7 | 12.9 | 13.0 | 13.1 | 13.2 | 13.2 | 13.3 | |
17 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 5.9 | 6.9 | 7.8 | 8.6 | 9.4 | 10.2 | 10.8 | 11.4 | 11.9 | 12.4 | 12.7 | 13.1 | 13.3 | 13.5 | 13.7 | 13.8 | 13.9 | 14.0 | 14.1 | |
18 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 6.9 | 7.8 | 8.7 | 9.6 | 10.3 | 11.1 | 11.7 | 12.3 | 12.8 | 13.2 | 13.6 | 13.9 | 14.1 | 14.4 | 14.5 | 14.7 | 14.8 | 14.9 | |
19 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 6.9 | 7.9 | 8.8 | 9.7 | 10.5 | 11.2 | 11.9 | 12.6 | 13.1 | 13.6 | 14.1 | 14.4 | 14.7 | 15.0 | 15.2 | 15.4 | 15.5 | 15.6 | |
20 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 7.9 | 8.8 | 9.7 | 10.6 | 11.4 | 12.1 | 12.8 | 13.4 | 14.0 | 14.5 | 14.9 | 15.2 | 15.6 | 15.8 | 16.0 | 16.2 | 16.3 | |
21 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 7.9 | 8.9 | 9.8 | 10.7 | 11.5 | 12.3 | 13.0 | 13.7 | 14.3 | 14.9 | 15.3 | 15.7 | 16.1 | 16.4 | 16.6 | 16.9 | 17.0 | |
22 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 8.9 | 9.9 | 10.8 | 11.6 | 12.5 | 13.2 | 13.9 | 14.6 | 15.2 | 15.7 | 16.2 | 16.6 | 16.9 | 17.2 | 17.5 | 17.7 | |
23 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 8.9 | 9.9 | 10.8 | 11.7 | 12.6 | 13.4 | 14.1 | 14.8 | 15.5 | 16.1 | 16.6 | 17.0 | 17.4 | 17.8 | 18.1 | 18.3 | |
24 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 9.9 | 10.9 | 11.8 | 12.7 | 13.5 | 14.3 | 15.0 | 15.7 | 16.4 | 16.9 | 17.4 | 17.9 | 18.3 | 18.6 | 18.9 | |
25 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 9.9 | 10.9 | 11.8 | 12.7 | 13.6 | 14.4 | 15.2 | 15.9 | 16.6 | 17.2 | 17.8 | 18.3 | 18.7 | 19.1 | 19.5 |
In order to know how much you should trust this table in any given instance, here are the standard deviations:
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | ||
1 | 0.5 | 0.8 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
2 | 0.3 | 0.8 | 1.3 | 1.5 | 1.6 | 1.7 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | |
3 | 0.2 | 0.7 | 1.3 | 1.6 | 1.9 | 2 | 2.1 | 2.2 | 2.2 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | |
4 | 0.1 | 0.5 | 1.1 | 1.6 | 1.9 | 2.1 | 2.3 | 2.5 | 2.5 | 2.6 | 2.6 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | 2.7 | |
5 | 0 | 0.4 | 1 | 1.4 | 1.8 | 2.2 | 2.4 | 2.6 | 2.8 | 2.9 | 2.9 | 3 | 3 | 3 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 | 3.1 | |
6 | 0 | 0.2 | 0.8 | 1.2 | 1.7 | 2.1 | 2.4 | 2.6 | 2.8 | 3 | 3.1 | 3.2 | 3.3 | 3.3 | 3.3 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | |
7 | 0 | 0.2 | 0.6 | 1.1 | 1.5 | 1.9 | 2.3 | 2.6 | 2.9 | 3.1 | 3.2 | 3.4 | 3.5 | 3.5 | 3.6 | 3.6 | 3.6 | 3.7 | 3.7 | 3.7 | 3.7 | 3.7 | 3.7 | 3.7 | |
8 | 0 | 0.1 | 0.5 | 0.9 | 1.3 | 1.7 | 2.1 | 2.5 | 2.8 | 3.1 | 3.3 | 3.4 | 3.6 | 3.7 | 3.8 | 3.8 | 3.9 | 3.9 | 3.9 | 3.9 | 3.9 | 4 | 4 | 4 | |
9 | 0 | 0.1 | 0.4 | 0.7 | 1.1 | 1.5 | 1.9 | 2.3 | 2.7 | 3 | 3.2 | 3.5 | 3.6 | 3.8 | 3.9 | 4 | 4.1 | 4.1 | 4.1 | 4.2 | 4.2 | 4.2 | 4.2 | 4.2 | |
10 | 0 | 0.1 | 0.3 | 0.6 | 1 | 1.3 | 1.7 | 2.1 | 2.5 | 2.9 | 3.2 | 3.4 | 3.6 | 3.8 | 4 | 4.1 | 4.2 | 4.3 | 4.3 | 4.4 | 4.4 | 4.4 | 4.4 | 4.4 | |
11 | 0 | 0 | 0.3 | 0.5 | 0.8 | 1.2 | 1.5 | 1.9 | 2.3 | 2.7 | 3 | 3.3 | 3.6 | 3.8 | 4 | 4.2 | 4.3 | 4.4 | 4.5 | 4.5 | 4.6 | 4.6 | 4.6 | 4.7 | |
12 | 0 | 0 | 0.2 | 0.4 | 0.7 | 1 | 1.3 | 1.7 | 2.1 | 2.5 | 2.9 | 3.2 | 3.5 | 3.8 | 4 | 4.2 | 4.3 | 4.5 | 4.6 | 4.7 | 4.7 | 4.8 | 4.8 | 4.8 | |
13 | 0 | 0 | 0.2 | 0.3 | 0.6 | 0.8 | 1.2 | 1.5 | 1.9 | 2.3 | 2.7 | 3 | 3.4 | 3.7 | 3.9 | 4.2 | 4.4 | 4.5 | 4.7 | 4.8 | 4.8 | 4.9 | 5 | 5 | |
14 | 0 | 0 | 0.1 | 0.3 | 0.5 | 0.7 | 1 | 1.3 | 1.7 | 2.1 | 2.5 | 2.8 | 3.2 | 3.5 | 3.8 | 4.1 | 4.3 | 4.5 | 4.7 | 4.8 | 4.9 | 5 | 5.1 | 5.2 | |
15 | 0 | 0 | 0.1 | 0.2 | 0.4 | 0.6 | 0.9 | 1.2 | 1.5 | 1.9 | 2.3 | 2.6 | 3 | 3.4 | 3.7 | 4 | 4.2 | 4.5 | 4.7 | 4.8 | 5 | 5.1 | 5.2 | 5.3 | |
16 | 0 | 0 | 0.1 | 0.2 | 0.3 | 0.5 | 0.7 | 1 | 1.3 | 1.7 | 2.1 | 2.4 | 2.8 | 3.2 | 3.5 | 3.8 | 4.1 | 4.4 | 4.6 | 4.8 | 5 | 5.1 | 5.3 | 5.4 | |
17 | 0 | 0 | 0.1 | 0.1 | 0.3 | 0.4 | 0.6 | 0.9 | 1.2 | 1.5 | 1.8 | 2.2 | 2.6 | 3 | 3.3 | 3.7 | 4 | 4.3 | 4.5 | 4.8 | 5 | 5.1 | 5.3 | 5.4 | |
18 | 0 | 0 | 0 | 0.1 | 0.2 | 0.3 | 0.5 | 0.7 | 1 | 1.3 | 1.6 | 2 | 2.4 | 2.7 | 3.1 | 3.5 | 3.8 | 4.1 | 4.4 | 4.7 | 4.9 | 5.1 | 5.3 | 5.4 | |
19 | 0 | 0 | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.6 | 0.9 | 1.2 | 1.5 | 1.8 | 2.2 | 2.5 | 2.9 | 3.3 | 3.6 | 4 | 4.3 | 4.6 | 4.8 | 5.1 | 5.3 | 5.4 | |
20 | 0 | 0 | 0 | 0.1 | 0.1 | 0.2 | 0.4 | 0.5 | 0.8 | 1 | 1.3 | 1.6 | 2 | 2.3 | 2.7 | 3.1 | 3.4 | 3.8 | 4.1 | 4.4 | 4.7 | 5 | 5.2 | 5.4 | |
21 | 0 | 0 | 0 | 0.1 | 0.1 | 0.2 | 0.3 | 0.5 | 0.6 | 0.9 | 1.1 | 1.4 | 1.8 | 2.1 | 2.5 | 2.8 | 3.2 | 3.6 | 3.9 | 4.2 | 4.6 | 4.8 | 5.1 | 5.3 | |
22 | 0 | 0 | 0 | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.6 | 0.8 | 1 | 1.3 | 1.6 | 1.9 | 2.2 | 2.6 | 3 | 3.4 | 3.7 | 4.1 | 4.4 | 4.7 | 5 | 5.2 | |
23 | 0 | 0 | 0 | 0 | 0.1 | 0.1 | 0.2 | 0.3 | 0.5 | 0.6 | 0.9 | 1.1 | 1.4 | 1.7 | 2 | 2.4 | 2.8 | 3.1 | 3.5 | 3.9 | 4.2 | 4.5 | 4.8 | 5.1 | |
24 | 0 | 0 | 0 | 0 | 0.1 | 0.1 | 0.2 | 0.3 | 0.4 | 0.6 | 0.7 | 1 | 1.2 | 1.5 | 1.8 | 2.2 | 2.5 | 2.9 | 3.3 | 3.6 | 4 | 4.3 | 4.7 | 5 | |
25 | 0 | 0 | 0 | 0 | 0 | 0.1 | 0.1 | 0.2 | 0.3 | 0.5 | 0.6 | 0.8 | 1.1 | 1.4 | 1.7 | 2 | 2.3 | 2.7 | 3 | 3.4 | 3.8 | 4.1 | 4.5 | 4.8 |