What is the formula for average damage for an Avenger against an Oath target?

Question

I am creating a spreadsheet to compute average damage for my party. One of the issues that I am having is calculating average damage for the target of our Hybrid Avenger-Assassin's Avenger attacks against his oath of enmity. I have the same issue with our Ranger's split the tree power.

I have poked around a bit and think I have the correct formula, but I want to be sure. Here is the formula I am using for his attacks:

      (1-((A)/20)^2 - 39/400)*(B) + ((A/20)^2)*C +(39/400*(D))

Where

• A Roll needed to hit
• B Average Damage
• C Miss damage if any (for Dailies)
• D Max Damage

Is this the correct formula?

Answer 1

(1-((A)/20)^2 - 39/400)*(B) + ((A/20)^2)*C +(39/400*(D))
(Hit%         - Crit%)*B    +  Miss% * C   +  Crit% * D

Your Hit % calc is flawed. Replace it with this:

  1-(1-(21-A)/20)^2

The Miss % should be:

(1-(21-A)/20)^2

Answer 2

If the roll needed to hit something is A, the chance to miss isn't A/20, but (A-1)/20.

Thus, you need to subtract 1 from A for your hit chance:

(1-((A-1)/20)^2 - 39/400)

and subtract 1 from A for your miss chance:

(((A-1)/20)^2)

Also, here's an alternate method of coming up with the formula.

The critical chance is independent of the roll needed to hit, and will never change. So let's get that out of the way first.

A. Chance to not crit either roll = (19/20)*(19/20) = 361/400
B. Overall chance to crit         = 1 - A           = 39/400
                                                    = .0975

All we need now is the chance to miss with both rolls. The rest is easily inferred.

C. Roll needed to hit                 = X            = X
D. Chance to miss both rolls          = [(X-1)/20]^2 = (X-1)^2/400
E. Chance to normal hit at least once = A - D        = 361/400 - (X-1)^2/400
                                                     = .9025 - (x-1)^2/400

Thus, in simplest (read: fewest references to X) form, the formula is

Let Q = Average swing damage
Let R = Max swing damage
Let T = Miss swing damage

Q*(.9025-(X-1)^2/400) + R*.0975 + T*(X-1)^2/400