Odds of winning various dice combinations when playing online Risk.
Attacker: one die; Defender: one die: Attacker wins 15 out of 36 (41.67 %) Defender wins 21 out of 36 (58.33 %) Attacker: two dice; Defender: one die: Attacker wins 125 out of 216 (57.87 %) Defender wins 91 out of 216 (42.13 %) Attacker: three dice; Defender: one die: Attacker wins 855 out of 1296 (65.97 %) Defender wins 441 out of 1296 (34.03 %) Attacker: one die; Defender: two dice: Attacker wins 55 out of 216 (25.46 %) Defender wins 161 out of 216 (74.54 %) Attacker: two dice; Defender: two dice: Attacker wins both: 295 out of 1296 (22.76 %) Defender wins both: 581 out of 1296 (44.83 %) Both win one: 420 out of 1296 (32.41 %) Attacker: three dice; Defender: two dice: Attacker wins both: 2890 out of 7776 (37.17 %) Defender wins both: 2275 out of 7776 (29.26 %) Both win one: 2611 out of 7776 (33.58 %)
Sample interpretation of the online game data above (three vs. two). If an attacker starts with 1000 armies and a defender starts with 1000 armies and a 3 vs. 2 attack is ensued, the results should be (given fair dice): after 100 rolls, each side will have lost 1 army about 34 times. The defender will have lost 2 armies about 37 times, and the attacker will have lost 2 armies 29 times. Therefore, after 100 rolls, the attacker should have 908 armies left, and the defender should have 892 armies left.
Conclusion: heads up with three dice versus 2 dice, the attacker has an advantage in the long run. Similar interpretations can be made for the remainder of the data, which can be summarized as follows:
- Attacker 1 versus defender 1: defender has the advantage, winning about 4 out of 7 battles
- Attacker 2 versus defender 1: attacker has the advantage, winning about 4 out of 7 battles
- Attacker 3 versus defender 1: attacker has the advantage, winning about 2 out of 3 battles
- Attacker 1 versus defender 2: defender has the advantage, winning about 3 out of 4 battles
- Attacker 2 versus defender 2: defender has the advantage, winning about 3 out of 5 battles
- Attacker 3 versus defender 2: attacker has the advantage,
but the advantage is much more narrow than any of the battles
described above. The attacker's advantage is such that he will
win about 7 out of 13 battles on average.