I have been working on a diceless combat mechanic for Pathfinder (it could be applied to any system in which combat involves a roll to hit and a roll for damage, but Pathfinder will be used for examples) which applies basic concepts of probability theory.
Since every dice roll has an expected value (E[1d6]=3.5, E[1d8]=4.5, etc), every weapon has an expected value for damage. By determining the probability of hitting, we can determine the damage of a given attack by multiplying the probability of a hit by a weapon's expected value.
This leads to a diceless attack mechanic. When an attack is declared, rather than rolling for attack and damage, the attacker would automatically deal damage to the defender based on his expected damage and probability of hitting.
For example, a character wielding a weapon that does 1d8 damage (or 4.5 damage) attacks a goblin. He needs to roll an 11 or higher on a d20 to hit the goblin. Thus when attacking, the character would automatically deal 0.5*4.5=2.25 damage to the goblin (rounding up or down depending on preferences).
I know this would remove critical hits from the game. Perhaps characters could hold critical tokens based on their level, traits, etc which they could spend to attain a critical hit (which could mean double damage, the maximum dice value of damage, weapon's expected damage not multiplied by probability, or something along these lines).
I'm also not sure how this could be applied to saving throws vs death, sleep, etc where there is no expected value of damage, but simply hit or miss.
What other side effects, advantages, or downsides would this averaging method have on gameplay? It's likely some game out there has done similar in the past. Does any one have any experiences playing in a system like this and can report on its effect in play?