Rain of Arrows (RoA)

Shout-out to someone who helped me get the drive to make this post. She knows who she is.

Going to use the original RoA for the explanation to keep it simple, and then go over the other ones too a bit.
Bosses are ignored at the start.
Also please note that the way RoA works and the numbers as well have changed more than once during the course of TSO, so stuff you read online may be also no longer be true since BB doesn't document well the changes they make to the game.
Furthermore, I could have some things wrong, so treat this cautiously as well...

RoA_min damage is 40 for the regular RoA. It always removes 40 foes if there are at least that many available.
RoA_max is by default twice the RoA_min for that arrow type, thus 80 for the regular RoA.
It will never remove more foes.
On a camp with at least 80 enemies, RoA can be expected to remove around 60 units, the average of RoA_min and RoA_max: ((40+80)/2=60), but if you are not lucky it will only go up to RoA_min, but on the other hand with luck it can go up to the value of RoA_max.
These units are picked in a way to keep the losses equally spread out (more on this later).

When the camp's headcount is more than RoA_min but less then twice of it, the math changes a bit:
RoA_max gets replaced by the headcount of the camp.
Thus, in a camp with 60 enemies, the regular RoA will on average removes only 50 units, the average of RoA_min and Camp_max: ((40+60)/2=50).
On a camp with 40 foes it becomes exactly ((40+40)/2)=40 always removed and the camp is guaranteed to be wiped out.
On a camp with 42 foes it becomes ((40+42)/2)=41 average removed, but it can end up removing only 40, or all 42 wiping out the camp completely.

https://i.imgur.com/3qe0A5s.png

So what about bosses?
If there are minions and one boss, first the arrow will only remove minions. Once half the minions are already gone, the boss is next, and after that the arrow continues to take the remaining minions.
Health of units is ignored during this process, only headcount matters.

For the other types of arrows, the math is probably similar:
They will do at least their minimum damage, if there are at least as many foes available

Their theoretical maximum damage on camps is twice the minimum at most

Their effective maximum damage is the lower of either the number of hostiles in camp, or twice the min damage

The average damage will be the average of the expected minimum and the effective maximum possible for the arrow type and camp.

Bosses I did not test much, but RoA removes the camp with 4 Grumpy and 4 Insane Guards in Beautiful Princess, and it also wipes the bosscamp in Treasure of Knowledge featuring 3 Hungry Blossom, 3 Fierce Blossom, 3 Bulbous Blossom.

(Taking something I read on the internet elsewhere for blacktree and chocolate arrows)
Blacktree Arrow: min 10 hostiles removed (max 20), reusable on the same camps

Chocolate Covered Arrow: min 20 hostiles removed (max 40), locks out all arrows

old Rain of Arrows: min 40 hostiles removed (max 80), locks out all arrows

Rain of Flaming Arrows: min 100 hostiles removed (max 200), locks out all arrows

To support the above on RoA and RoFA, I did two batches of tests over 26 3rd Thief runs:
1) Used RoA on the Mountain Stronghold.
It has 40 minions (500hp*40=20k health), a Shrewd Thief boss (1k health), and a Silly Thief boss (50k health, more than the rest of the camp combined).
If it was just equally distributing damage, first a bunch of minions would get removed, then the Shred one, then the rest of the minions and the silly one last.
However, instead I found that 10 times one Swordsman minion survived, and 8 times two Swordsman were remaining but in both cases both bosses were gone. The remaining 6 times the entire camp got wiped out and removed.
Note: In earlier TSO versions the regular RoA was guaranteed 100% effective, removing the entire camp in all cases!
And here I would like to take a stop to talk about RNG.
The first 3 attempts, two resulted in the camp being wiped out, making it look like the arrow would try to do the theoretical maximum 80 all the time.
After that, in the next 20 attempts only 2 resulted it the camp being wiped out, removing all 42 hostiles only 10% of the time.
Random generator is random, and has no memory of what happened previously.

2) Used RoFA on the BatCave (guess I am a DC super-villain now)
It has 100-100 of two different types of minions, and one boss.
Got really lucky on the 10th attempt, RoFA removed 100 minions exactly, leaving the boss in peace. Could be called really unlucky as well, but for the testing this was brilliant. In all the other 25 attempts, RoFA inflicted higher losses and thus the boss was removed each time.
What was more interesting and odd, that the surviving number of minions were always the exact same for both types. No RoFA managed to inflict more loss on one type than the other, always removing the same number of units of each type of units plus the boss (making the total removal being an odd number) except the one where the boss survived, only there was the number of losses inflicted even (50+50).
Number of inflicted losses in total: 173, 107, 125, 111, 119, 105, 137, 105, 131, 100(!), 187, 117, 113, 151, 167, 127, 137, 147, 171, 123, 115, 103, 171, 161, 187, 107
Overall 3497 hostiles removed in 26 attempts averaging 134.5, well under the expected 150 but that's probably just my bad luck, had the nice 187s in there as well, one right after the 100 lowest possible.

Hope this helps, and kudos to those who managed to read the whole thing.