As supplement to question 1.: To look at individual items and multiply them just seems very counter-intuitive. The same goes for looking at it from the perspective of opening a trap chest (= not getting lucky). This is probably a stupid question, but... are you sure that it works this way?
Is there actually an 80% cap on luck? If there is:
How does it work? Since you apparently do not add up the individual percentages, how or where is that cap enforced?
What exactly is capped, anyway? Is it the sum of item percentages? Is it perhaps the whole factor, as in (1 - 0.13)*(1 - ...) that cannot go below 0.8? Or 0.2?
What would be the best combination of item luck perks to get the highest possible luck?
Assuming there is a luck cap somehow.
There are six items... it seems like having six times 13.33% leads to less luck than, say, two times 20% and four times 10%. (This assumes that the total cannot be over 80%, which might be completely wrong... which is why I'm asking)
Could you clear things up? It seems like there is some interest in this topic...
First of all, the question is not stupid.
There is no artificial "80% cap" in the implementation of the vault luck perk.
While there is no explicit capping, there is a limit induced by the maximum value of the perk and the number of Hero Items the hero can wear.
Now as described in the previous post, the perk lowers the chance to get a trap with a base chance of getting the trap in the first chest of 25%.
In other words, without the perk the chance to get a reward in the first chest is 75%.
The theoretical maximum value of the perk itself is 17%.
Now if all of your 9 equipped hero items has the maxed out vault luck perk in both perk slots, this will result in the following probability to get a trap in the first chest:
0.25 * (1 - 0.17)^18 = 0.25 * 0.034946659 = 0.008736665 => 0.873666476%.
This leads to a probability to get a reward in the first chest of (100 - 0.873666476) = 99.126333524%.
This is a theoretical value, as you cannot have the perk on all of the slots and items.
To answer your question:
6 items with a 13.333333333% perk on a single perk slot, will result in
0.25 * (1 - 0.13333333333)^6 = 0.25 * 0.423752779 = 0.105938195 (~10.6% trap probability) which means a probability of a reward in the first chest of 89.4061805%.
With 4 items of 10% and 2 items of 20%, this will result in
0.25 * (1 - 0.1)^4 * (1 - 0.2)^2 = 0.25 * 0.419904 = 0.104976 which means a probability of a reward in first chest of 89.5024%.
So yes, you cannot just add the perk values.
We hope that the explanation help to understand the maths behind the luck perk.