Hi, all. Happy Hanukkah and Merry Christmas and all that.
I recently survived the last installment of a horrendous training course that wasted 80 hours of my time. (6 Sigma management. Joy.) In fact, the last two hours were so dull that I ended up doodling on Microsoft Excel on the laptop computer on my desk. I actually learned something from the Excel doodling: the probabilities that go along with the board game Risk. If you've never played Risk, don't bother reading the rest of this message.
For years, I've wondered what the odds are when you roll the dice. You can attack with 1, 2, or 3 dice, and defend with 1 or 2 dice. It turns out, the probabilities are as follows:
Attack with 3, defend with 2:
- Offense wins both: 2890/7776 (37.2%)
- Offense and defense win one each: 2611/7776 (33.6%)
- Defense wins both: 2275/7776 (29.3%)
Attack with 3, defend with 1:
- Offense wins: 855/1296 (66.0%)
- Defense wins: 441/1296 (34.0%)
Attack with 2, defend with 2:
- Offense wins both: 295/1296 (22.8%)
- Offense and defense win one each: 420/1296 (32.4%)
- Defense wins both: 581/1296 (44.8%)
Attack with 2, defend with 1:
- Offense wins: 125/216 (57.9%)
- Defense wins: 91/216 (42.1%)
Attack with 1, defend with 2:
- Offense wins: 161/216 (74.5%)
- Defense wins: 55/216 (25.5%)
Attack with 1, defend with 1:
- Offense wins: 5/12 (41.7%)
- Defense wins: 7/12 (58.3%)
I then thought about what other weird modifications we made to Risk, and I calculated what the probability would be if we used 20-sided dice:
Attack with 3, defend with 2:
- Offense wins both: 46%
- Offense and defense win one each: 31%
- Defense wins both: 22%
When you have more sides on the dice, you get fewer ties when you roll them (and the tie goes to the defense). If anyone cares what happens when we use infinite-sided dice (although those are kinda expensive):
Attack with 3, defend with 2:
- Offense wins both: 50%
- Offense and defense win one each: 30%
- Defense wins both: 20%
I thought about running the cases of more dice (attack with 16, etc.), but the class ended.
The bottom line: If your pile of armies is attacking another pile of armies, and you keep rolling 3 vs. 2, the offense wins 54% of the time, and the defense wins 46%. It's better to be the *offense*.
This is what I learned from 6 Sigma training.
Ron
