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Disclaimer: Even with a medicine of 1000 and a listed save % of 100, it's still possible to "roll a 1" and fail once in a while.
tl;dr Medicine's survival chance description is wrong, "1.0%" is way more than just adding 1%, going from "1.0%" to "2.0%" means less than going from "0.0%" to "1.0%".
This forum post (https://forums.taleworlds.com/index.php?threads/tactical-decisions-on-the-battlefield-should-play-a-bigger-role-in-deciding-the-tide-of-battle.426052/post-9474705) talks about someone else on the TW Discord going and crunching the numbers from the code. I'm not on that Discord though, so this is more secondhand.
If you don't feel like visiting the link, the big point is that having 100 medicine is enough to have a +50% survival rate for any troop.
I myself did go and look at the code to verify, and the numbers from that post do seem to be accurate.
There are a few different variables that affect survival chances, but this topic is focusing on the first:
1) Surgeon's Medicine skill
2) Downed character's level
3) Enemy Surgeon's Medicine skill if they have Doctor's Oath
4) Heroes start off with a 10% to die instead of the usual 100%
The big thing is that the more of 1-3 that you have, the less the next point of any of those affects survival chances. This is especially true with 4). Going from 0 Medicine to 330 Medicine only brings a Level 1 hero from a ~90.9% chance of survival to a ~93% chance, but that doesn't matter right now. Getting more medicine is great, but just grabbing a decent surgeon wanderer could be enough.
In other words:
-For a level 26 unit (any T5 unit), with 0 medicine the survival rate is about 44%.
-For a level 26 unit, with 100 medicine the survival rate is about 64%.
-For a level 26 unit, with 300 medicine the survival rate is about 79%.
The gap is even more noticeable for lower tier units:
-For a level 6 unit (any T1 unit), with 0 medicine the survival rate is about 15%.
-For a level 6 unit, with 100 medicine the survival rate is about 54%.
Note that the way the equation works, the early levels in medicine are worth significantly more than the later levels.