Only butchering quality below 0. Above 0 does nothing to drop chances.
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Again, unless it’s changed recently, higher butchering can help offset the fail chance of a value that is too low in another area (dex or survival skill).
I can’t remember who, but the claim was that in the code, it worked like this:
Random numbers:
-0 to (dex - 12)
-0 to (survival - 3)
-0 to butchering quality
Add those up. If it’s positive, you get that item, if not, you don’t. Once you get all of them to 0 or higher (12 dex, 3 survival, positive butchering quality), you always get everything, but if, say, your dex is only 10, then having higher survival and higher butchering quality makes it less likely for the total of those 3 numbers to be negative (but it can still happen, of course).
A really good butchering tool can make it likely to get most stuff, even with low dex and 0 survival - the butcher knife (butchering quality 25), for instance, would statistically make up for 0 dex and 0 survival more than half the time (pretty darn impressive).
Quite possible my info is out of date, though - anybody care to double-check my possibly-dated knowledge?