Following the same approach as in this thread: So which do you think will be harder to get - #16 by Iria-1342 I will use a Markov chain to compute the probability of a perfect primal Cold/ChC/ChD/Socket FoT.
The main difference in that math I used in that last thread is that a socket is 100% guaranteed and there are only 2 random primaries instead of 3 such that the probability is certainly higher, BUT getting the FoT to be primal in the first place kills that benefit!
Here are the preliminaries:
- A primal FoT rolls with socket and two random primaries which to be ideal must be 2 of ChC, ChD, and correct element (Cold for GoD DH).
- Each affix has a weight and the total weights for all possible primary affixes is 37500 (taken from this data). Since a socket is 100%, then it is removed from the pool so the weight total is 36500.
- The weights for ChC, ChD, and each element are 1000 each, but only one element may roll on an amulet (once the element roll appears, it removes the other elements from the pool). For example, if ChC rolls first, then subtract 1000 from the total remaining weight; but if Cold% rolls first, then subtract 4000 from the total remaining weight (since Physical, Fire, and Lightning were also removed).
Case 1: Correct element rolls first, then one of ChC or ChD
Case 2: One of ChC or ChD rolls first, then correct element
Case 3: Both ChC and ChD roll (element does not roll)
Note: cases 2 and 3 will have the same probability since the element roll (with its special rule) was the last affix rolled (or not present).
Case 1 (elem 1st): (1000/36500) * (2000/32500) = 0.0016859852
Case 2 (elem 2nd): (2000/36500) * (1000/35500) = 0.0015435076
Case 3 (no elem): (2000/36500) * (1000/35500) = 0.0015435076
Total = 0.0016859852 + 0.0015435076 + 0.0015435076 = 0.0047730004
Thus, if a FoT is primal it has about a 1 in 210 chance of being perfect (as it can get correct element, ChC, and ChD rolls with the natural socket and cooldown).
However, since a primal is about 1 in 500 of all legendaries then this number becomes 1 in 104,756 chance from any FoT! So my initial guess was VERY accurate!