This post is meant to provide instructions for how to calculate her average damage per attack with a weapon. You need only a firm grasp of algebra to use it.
The formula has six inputs.
A: Armor Class of the target.
B: Attack bonus of the attacker.
R: Critical hit threat range.
N: Additional bonuses to confirm critical hits.
D: Average damage on a normal hit.
C: Average damage on a critical hit.
Note: R is size of the critical hit threat range, so it should be a single natural number value. For example, a 17-20 range gets you R=4.
There are four initial calculations to make.
P(H&K^) = max{ 0, min{ 19, 21-A+B}-R } }
P(H&K) = max{ 1, min{ 19, 21-A+B }-P(H&K^) }
P(S) = max{ 1, min{ 19, 21-A+B+N } }
P(S^) = 20-P(S)
Note: The operation max{ a, b } means you take the highest value in the set. The operation min{ a, b } means you take the lowest value in the set.
Finally, your average damage is the following expression:
D•( P(H&K^)÷20 + P(H&K)•P(S^)÷20 ) + C•( P(H&K)•P(S)÷400 )
***********************
In case you were wondering:
H is the event that you hit.
K is the event that you rolled within your critical threat range, regardless of whether this roll hit.
K^ is the complement (like an opposite) of K, i.e. you did not roll within your critical threat range.
H&K^ should read H&(K^) for clarity, although I think that complement comes before intersection in set algebra.
S is the event that you confirm a critical threat roll, given that one is called for.
S^ is the complement of S.
P(X) returns a value from 0-20, reflecting how many possible rolls out of twenty will create X.
To give credit where it is due, I got started on this formula earlier this evening after reading Kaizoku's post and the subsequent thread. So thank you to those guys, both for the lost sleep and for their discussion of the variables that need to be considered. I think I've covered all of them, but please correct me if there are any I have left out.
Incidentally, I ran my formula with Kaizoku's numbers (which were themselves copied from elsewhere), and got slightly higher results than he did: 38.76, 38.76, 38.76, 34.68, and 24.48, for a total of 175.44 average damage per round. I haven't tried yet to account specifically for the discrepancy.