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Community gathered data on drop rates of the three loot items in the game. Crates, Barrels, and Treasure Chests.

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Version 0.1.122+Edit

The chances are as follows (as found in the game file):

Quality Without Luck With Luck
legendary 0.2% (1 every 500 crates) 0.5% (1 every 200 crates)
scarce 2.4% (1 every 42 crates) 4.5% (1 every 22 crates)
rare 16% (1 every 6.3 crates) 24% (1 every 4 crates)
uncommon 25% (1 every 4 crates) 25% (1 every 4 crates)
common 56.4% (1 every 1.8 crate) 46% (1 every 2.2 crates)

Versions 0.1.94-0.1.97Edit

These numbers were collected side by side over the same timeframe, one player with Luck, the other without.

Crates
Luck Normal
Common 46% 60%
Uncommon 25% 24%
Rare 24% 14%
Scarce 4.9% 2.2%
Legendary 0.13% 0.00%
(total opened) 790

535

Chests
Luck Normal
Common 27% 41%
Uncommon 51% 48%
Rare 15% 9%
Scarce 5.6% 2.0%
Legendary 1.10% 0.13%
(total opened) 819 747
Barrels
Luck Normal
Common 0% 0%
Uncommon 40% 55%
Rare 48% 39%
Scarce 11% 5.7%
Legendary 0.77% 1.14%
(total opened) 391 352

After July 20 update, without LuckEdit

Treasure Chest
Common Uncommon Rare Scarce Legendary Total
299 328 60 11 2 700
42.7% 46.8% 8.6% 1.6% 0.29% 100%


Floating Crate
Common Uncommon Rare Scarce Legendary Total
280 127 77 15 1 500
56% 25.4% 15.4% 3% 0.2% 100%



Bobbing Barrel
Common Uncommon Rare Scarce Legendary Total
0 172 109 16 3 300
0% 57.33% 36.33% 5.33% 1% 100%


After July 20 update, WITH LuckEdit

Treasure Chest
Common Uncommon Rare Scarce Legendary Total
140 253 72 27 8 500
28% 50.6% 14.4% 5.4% 1.6% 100%


Floating Crate
Common Uncommon Rare Scarce Legendary Total
221 124 133 20 2 500
44.2% 24.8% 26.6% 4% 0.4% 100%


Bobbing Barrel
Common Uncommon Rare Scarce Legendary Total
0 95 88 16 1 200
0% 47.5% 44% 8% 0.5% 100%

Before July 20 update, without LuckEdit

Treasure Chest total
Common Uncommon Rare Scarce Legendary Total
138 37 22 3 0 200
69% 19% 11% 2% 0% 100%
Floating Crate total
Common Uncommon Rare Scarce Legendary Total
150
26
21
3
0
200
75%
13%
11%
2%
0% 100%

The person organizing the original collection posted the following (but I don't have the raw data):

OK - here are the stats from before the update (averages) over 639 collected by players without 'luck':
common 67, std dev. 7
uncommon 16, std dev. 5
rare 14, std, dev. 4
scarce 2, std. dev. 2
legendary 0

The two people who did have luck researched and who posted results for 126 crates reported these averages:
common 65, uncommon 13, rare 15, and scarce 7.

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