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Not too long ago, I stayed awake until practically daybreak running calculations on the statistics of Risk dice rolls. And because I'm such a great person, I'll share what I learned with you: Always roll as many dice as you can.
Rolling 3 attack dice versus 2 defense dice, the most dice that can be rolled in one attack, the defender will win at least one of the attacker's pieces nearly 73% of the time. (The defender will win both rolls almost 37% of the time to the attacker's 27%.)
In fact, the defender will win at least 1 piece at least 42% of the time, no matter how dice are rolled. The defender will win all of the pieces outright at least 32% of the time. That's almost a full 10% over the attacker in both cases.
If the attacker is worried about attacking and losing any pieces, his opponent must be able to defend with only one die. The attacker has at best a 27% chance of total victory if his opponent is defending with two dice, no matter how many dice the attacker employs. (If the attacker rolls 2 or more dice against his opponent's single die, he'll still win less than 58% of the time.)
So I'd recommend putting at least 2 armies on every territory you control, then play a game of slow and steady expansion, relying on defense rolls and card redemption.
Now, don't you feel smarter? I just feel tired.