I haven’t seen any posts about how these chests work, but I figured some forum users might find this information useful.

The chests I’m referring to are the red chests from DLC4 that have two little D20s on top of them. When you approach one you’ll be given the option to “Roll One” for free or Roll Two" for 5 Eridium. When you “roll,” the die will spin, then a little number will appear over the chest. The number Tells you what kind of loot you are getting (by rarity). Say you roll one. Here’s a list of the loot you’ll get with each number rolled.

1: One Eridium + A Live Grenade

2 - 14: White or Green Rarity Loot

15 - 18: Blue Rarity Loot

19: 1 Blue + 1 Purple

20: 2 Purples

Simple Enough. Here’s How the “Roll Two” Option works. The RNG basically rolls two D20s and displays the higher roll - unless you get double 1s or double 20s. There is no special result for any other doubles (a double-18 gets you the same result as an 18). If you roll two:

Double 1s: 4 Eridium + 3 Live Grenades

2 - 14: Whites or Greens

15 - 18: Blues

19: 1 Purple + 1 Blue

20 (Single): 2 Purples

Double 20: 1 Legendary + 1Blue

Now for the part of the discussion that has (Gasp) math. Paying 5 E to roll two doesn’t just get you the chance for a legendary, it also increases the probability that you’ll get a higher number on your roll. When you roll one, you have an equal 5% chance of getting any result. So in effect rolling one means a 70% chance of garbage (1 - 14), a 20% chance to get blues (15 - 18), a 5% chance to get 1 blue and 1 purple, and a 5% chance to get 2 purples.

When You roll two, this changes because you always get the higher of the two rolls, except for the double-1 and double-20 outcomes.

For these special cases, since each of the two rolls are (presumably) independent, the probability of occurrence is the probability of rolling a 1 on the first die times the probability of rolling a 1 on the second. So, 1/20 * 1/20. In effect this means there’s only a 1/400 (0.25%) chance of getting double-1s, and similarly there’s only a 1/400 chance of getting double-20s.

Now for the rest of the outcomes. The probability of a single 20 when you roll two is the probability of getting one die that’s a 20 and one die that is not a 20. The probability of getting a 20 on the first die and a non-20 on the second is 1/20 * 19/20. That’s also the probability of getting a non-20 on the first Die and a 20 on the second. Add the probabilities of these two events and you get your chance of a single-20. So the probability of getting a single-20 when you roll two is 2*1/20*19/20 = 38/400, or 9.5%.

Similarly, the probability of getting a 19 when rolling two is twice the probability of the first die being a 19 times the probability that the second is 18 or below, PLUS the probability of rolling double 19s. That is 2*1/20*18/20 + 1/400 = 37/400, or 9.25%.

The pattern should start to be clear at this point. Probability of an 18 rolling two = 2*1/20*17/20 +1/400 = 35/400 , or 8.75%. Probability of 17 = 2*1/20*16/20+1/400 = 33/400 = 8.25%. Of 16 = 2*1/20*15/20 + 1/400 = 31/400 = 7.75%. Et Cetera.

Short Version:

Double-1s: 0.25%

2 - 14: 48.75%

15 - 18: 32%

19: 9.25%

Single-20: 9.5%

Double-20: 0.25%

A few final comments:

“Purple” includes E-tech rarity.

Any drops from the chests must be non-uniques in the global loot pool - you won’t get a Norfleet, you won’t get E-tech relics, you won’t get any of the “____ of Terramorphous” items, etc.

I think these chests may not drop relics, but that may just be my memory failing.

I am fairly certain they don’t ever do the “eridium and cash” drops that some other red chests can produce.

Exception: When a rocket launcher drops, you will get only the one item. As best as I can tell the launcher will be the higher of the rarities corresponding to the roll - e.g. a launcher dropped when you rolled 19 will be purple, a launcher on a double-20 should be a legendary (I got a pyrophobia this way).