Raids of Different Sizes Cannot be Tuned to Equal Individual Difficulties
This note is in regards to the tuning of raids of different sizes to equal difficulties.
The conclusion of this note is that for raids of different sizes, except in a narrow category of raid mechanics, tuning of the raids to the same difficulty is not possible.
The key statistic which is explored by this note is the change of failure (PF_s) of a raid member in cases where the failure of a single raid member during a boss encounter will very probably lead to a failure of the entire raid during that boss encounter.
Examples of boss encounters where such is the case are numerous. For example, in Blood Queen, Blood Princes, and Thaddius.
There are other examples where a single failure does not necessarily lead to a complete failure, for example, Patchwerk.
PS_s = The chance of success of a single raid member
PF_s = 1.00 - PS_s = The chance of failure of a single raid member
PS_r = The chance of success of the entire raid.
PF_r = 1.00 - PS_r = The chance of failure of the entire raid
n = The number of members of the raid
Raid Failure Formula:
(1) PS_r = (PS_s)^n
(1') PF_r = 1.00 - (1.00 - PF_s)^n
1) The chance of a single raid member succeeding or failing is independent
of the chance of any other raid member succeeding or failing.
2) All raid members must success for the success of the entire raid.
(The failure of a single raid member will fail the entire raid.)
Inverting the raid failure formulas (1) and (1') gives:
(2) PS_s = PS_r^(1/n)
(2') PF_s = 1.00 - (1.00 - PF_r)^(1/n)
Working from a target overall chance of failure of 0.50 (50%), this results to PS_s(n = 10) = 0.933 and PS_s(n = 20) = 0.966, and PF_s(n = 10) = 0.067 and PF_s(n = 20) = 0.034. Note that PS_s(10) = PS_s(20)^2 because of the particular selections for n.
That is, depending on the the particular mechanics of the boss encounter, tuning the fight to provide the same level of difficulty for individual raid members leads to a different chance of success for the entire boss encounter. Conversely, again for particular mechanics, having the same overall chance of success for the encounter
requires that the encounter be set at a different level of difficulty for each raid member.
The selection of raid sizes (n = 10) and (n = 20) does not limit the result: The key relationship is the difference in raid sizes.
The selection of raid mechanics to single point failures is limiting, but is a valid approximation of many current raid mechanics.