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Miscellaneous: Character Builder Summary and Ring of the Radiant Storm Damage Math

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A particularly thorny calculation this build has to make is its "roll damage twice" feature via the Ring of the Radiant Storm: this skews the damage dice probability distribution and thus the expected damage result on a damage die. The effect this feature has is to make the dice more "top-heavy"; that means that it makes it more likely for the higher damage values to show up than the lower ones (whereas before each value had the same chance as the others). If you look at a distribution on what damage rolls will net you what result, you come up with the following result:

A damage roll result of 1 will only happen if you roll 1 twice on the die.
A damage roll result of 2 will happen if you roll 2 and 1, 1 and 2, or 2 and 2.
A damage roll result of 3 will happen if you roll 3  and 1, 3 and 2, 1 and 3, 2 and 3, or 3 and 3.

And so on. If you extend the distribution to infinity, you will notice that it forms a succession following the pattern of a = 2(i-1) + 1, where i represents any given damage roll value and a is the number of different ways that event can occur. This succession is easily applicable for the purposes of calculating expected damage, as one can simply truncate the succession at a given roll of n (4 for a d4, for instance), and use the following formula:

Expected damage value of a dn = ∑(0 --> n) (2[i-1] + 1)/n²

Using this formula, we can attain the following results:

d4 = 3.13 expected damage = +0.63 over the average of 2.5
d6 = 4.47 expected damage = +0.97 over the average of 3.5
d8 = 5.81 expected damage = +1.31 over the average of 4.5
d10 = 7.15 expected damage = +1.65 over the average of 5.5
d12 = 8.49 expected damage = +1.99 over the average of 6.5

An interesting note is that the damage contribution from the Ring of the Radiant Storm is not only variable by die, but also more significant the larger the die is. This is due to the fact that it makes the distribution top-heavy, and so the highest damage value is the likeliest one to occur, so dice with the larger damage values benefit more.

Another note is that dice with the brutal property behave slightly different for this purpose: since they never roll 1's (or in some cases 2's), the way to most accurately represent them in this calculation is to subtract the brutal rating from the range of possibilities, then add it as a static number later. For example, the proper way to consider a craghammer's expected damage (d10 brutal 2) is as 1d8+2, whereas a khopesh (d8 brutal 1) would be most accurately fit by 1d7+1.

NOTE: As some posters in this thread have pointed out, the Ring's effect only lets you choose the best set of dice rolled, which means that the figures presented above are at their most accurate when there is only one die involved. As more dice are added to the situation, the average number should diminish somewhat.

Jul 19, 2010 -- 6:04AM, lordduskblade wrote:

@Zamsz: The increased [W] doesn't come up as much as the feat bonus to attack rolls [...]

Weapon Focus is giving damage rolls feat bonus.

It was just an example.

Busy day today, wanted to give him a chance for extra comments.