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Revisiting Expected Damage

So after more thought I have decided to revisit expected damage.

Consider the case with a 20/x2 critical, which is reasonably standard.

Probability of Hit (No Critical) ~=  (20-(AC of target - attack of attacker))/20
Minimum is always 1/20, maximum is always 19/20

The probability of a critical hit is always 1/20

Mean Damage = (Max Damage + Min Damage)/2

Expected Total Damage = Prob of Hit * Mean Damage + Prob of Crit  * Mean Damage

In our 20/x2 case:

Expected Total Damage = Mean Damage * (21-(AC-Attack)/20


Consider 3 undead (+3 attack, 2d6+2 damage) attacking an AC 14 PC

Expected Total Damage = (14+4)/2 *(21-(14-3)/20 = 9*10/20 = 4.5

With 3 undead, this means expected damage of 3* 4.5 = 14 damage per round

If we ignore criticals, the answer would be only 12 damage per round, a 17% difference!


Generically we can expand the formula for any critical hit as follows:

Expected Total Damage = Mean Damage *((20+Critical Rolls*(Critical Damage Multiplier-1))-(AC-Attack))/20

Where Mean Damage = (Max Damage + Min Damage)/2
And Critical Rolls are the number of dice values for a critical (i.e. 18-20 => 3)
And Critical Damage Multiplier is factor that damage is multiplied by for a crit


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