I don't have time to read this entire long thread (dating from 2006!), so apologies if this link has been referenced already. But everyone should read the following website, in full:
Dice Odds in the Board Game RiskSimply put, the odds are not as great as some of you seem to think they should be. It seems to me that most people fail to recognize the sheer power of the defender's ability to win ties.
For example, the very best odds an attacker can have are 3 dice versus 1 die. Note: that is not 3v1, it's 4v1 or better! 3v1 is 2 dice versus 1 die, and I've seen this mistaken notion referenced many times on these boards already. Anyway, back to the example. 3 dice versus 1 die (4v1 or better) is only a 66% win chance for the attacker. That means, on average, you'll win about 6 or 7 out of every 10 rolls against 1 defender. That's pretty good, but it's far from a sure thing.
And that's only on average. In real life, random doesn't conform to average, it follows a bell curve, which means that sometimes you'll get "average" results, but you'll also get extremes at either end, including well above average and well below average. Play enough and roll enough dice, and you too will experience the 30v3 loss - the more you play, the more you approach certainty of experiencing such an event. Those of you who have played thousands of games should not be surprised by this at all.
There is also a lot of selective memory going on here. You tend to remember the bad rolls - they really suck and they stick out in your mind. You tend to forget all the average rolls, because they're simply not notable. And you might remember the great rolls, but probably not as well as you remember the pain of the bad rolls. So, there's that.
Note that 2 dice versus 1 die is very nearly 50/50, and 1 die versus 1 die is in favor of the defender. So all those 3v1 and 2v1 battles you keep losing? Don't complain - if you knew the odds were even or against you, maybe you'd choose to fight those less often.
And 3v3? 2 dice versus 2 dice? That's suicide, almost a 50% chance for the defender to win both rolls, and another 30% to win one. Not a recommended attack.
The most common situation is multiple armies versus multiple armies, so 3 dice versus 2 dice. The average stats? Only a 37% chance the attacker will win both rolls. The defender has nearly that good a chance of winning both himself, almost 30%. And there's a 33% chance you'll each win one. That means on 2 out of every 3 rolls, on average, the attacker will lose at least one army. A good rule of thumb, if you want to approach guaranteed victory in any conquest, is to take 3 times as many men as what you're attacking. Even that's no guarantee, but it hedges the odds about as much as possible. Don't expect to win many 12v12 battles - the cumulative odds of taking such battles are not much better than 50/50 (actually a little worse once you take into account what happens when you're reduced to 3 men and only 2 attacking dice, and then 2 men and only 1 attacking die).
So it just seems to me that people are expecting too much from the dice if they're the attacker. Perhaps they've forgotten what it's like to play real Risk, where the same kinds of crappy rolls happened all the time (more selective memory, me thinks). And the defender's ability to win all ties is very, very powerful - it can change games. If you don't know all these odds already, I suggest you learn them by heart and play accordingly. For your convenience, I'll repost the chart from the above-referenced website right here:
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 %)
Memorize this, it will help, I promise. Knowing the odds by heart has won me games, because I know which attacks are worth attempting and which ones are best avoided, and not knowing the odds has lost games for some of my opponents, as I've watched them sometimes make utterly ill-advised attacks and then act surprised when they lose. And read the website, it's very illuminating. (It's not my website, so this isn't a shameless plug.)
I've taken notes on over 100 games I've played in, and the dice I've rolled approximately conform to these averages. I would expect them to get even closer as I keep playing, and I've experienced a 25v4 loss as well as a 10v30 defensive win. I've read of other people doing similar analysis and coming to the same conclusion. My understanding is that CC's numbers come from random.org, which is about as truly random as you can get. (And for crying out loud, don't listen to Klobber - he simply doesn't know what he's talking about and needs to take some remedial math classes.)
So my recommendation: stop complaining and play the game. The dice are random - really!
