So I play a game called warframe, and there's a currency you can get called ducats. You get this currency by going into special missions (called void fissures [or void storms]) where you crack open "relics", basically schrodinger's gacha, to reveal an item inside.

--- context ---

Each relic has 3 common items, 2 uncommon items, and a rare item that you can get from them. And you can spend a different resource (called reactant) to change the odds of getting each item (it only changes the odds per rarity, so each common has the same chance, and each uncommon has the same chance).

Normally, a relic has a 76% chance of giving a common (25.33% per item), a 22% chance of an uncommon (11% each), and a 2% chance of getting rare. When you refine a relic to "radiant" using the reactant, the odds become 50% for common (16.67% each), 40% uncommon (20% each), and 10% rare. Still not amazing, but much better than normal (called "intact").

The cool thing about these relics, though, is that when you complete one of these missions with other players, you can choose the reward they get if you want. So if you get a common item, but someone else gets a rare, you can both choose to get that rare item as your rewards. Each player is allowed to choose any of the rewards given (and you see what everyone got before you have to make your choice).

The items in the relic can then be sold for ducats, there are some exceptions, but we'll ignore those and just assume the following: common items are sold for 15 ducats, uncommon for 45, and rare for 100.

So very often, players will go in a team of 4 and open relics together. If everyone is bringing radiant relics, this is called a "radshare". It's the best option for people to attempt getting rare items. It's far from guaranteed, but a 34.5% chance is a lot better than 10%.

--- math part ---

So... you do radshares in a party of 4 a bunch to try and farm ducats. In doing so, each relic you open has...

50% chance of common (15 ducats)

40% chance of uncommon (45 ducats)

10% chance of rare (100 ducats)

Each player can choose whichever of the 4 opened rewards they want, so if someone gets a rare, every player can get it. And we'll just assume players will always pick the highest value reward.

On average, how many ducats can you expect for each radshare you do?

... I technically can do this myself, by just listing out all the possible combinations, but I was hoping there would be some formulaic way to determine this answer.