I've taken half an hour just looking through the auction house, and I have taken my own experience and various forum posts into account, and these are my findings:
The maximum number of sockets per equipment type varies. It seems to be independent of item rarity (magic and rare items, legendaries are something special). Instead, it seems to depend entirely on the body type of the item.
These are the maximums I was able to identify:
First of all: Shoulders, belts, gloves, and boots can't have
sockets at all (more on that later).
Helmets, rings, amulets, weapons, and offhands:
Pants
Chest Armor
Legendary items are a special case: They always have a certain fixed amount of sockets per named item, but just for completeness: The maximum number of sockets on legendaries I could find was two.
There's something I have to add about Shoulders, belts, gloves, and boots:
Even though there seem to be no belts with sockets, it is theoretically still possible that belts can have sockets, and simply nobody has found one yet. I come to this conclusion from a simple fact: Shoulders, gloves, and boots have no search option for "has sockets" in the auction house, indicating that it is indeed not possible for them to have one. Belts, however, do have this search option. It is entirely possible that Blizzard just left it in by accident, though.
Another interesting fact about items and sockets:
Items have a fixed amount of random properties (this can differ, even for same rarity and body placement). When an item is dropped, or identified, the contents of these properties are randomly rolled out. Sockets, no matter how many there are in the item slot, take up exactly one of these random properties.
This make for an interesting conclusion: If you're unlucky, sockets may take away important stats from your item. In general, pure item stats will always be higher than you could at to it with gems. For example, perfect strength gems provide +22 strength. In comparison, items of roughly the same level seem to hit an upper cap of around +60 strength. This means that if you, for example, identify a rare chest armor piece and it finds +50 vitality, +20 experience per kill and +1.2% chance to stun on hit, plus one socket, then that last socket could have been a slot for +50 strength, whereas now you can only try to compensate for it with gems.
On the contrary, however, this becomes a nice bonus when you actually want to stack a certain stats. Items can't roll the same bonus stat twice. If we go with the example from above, you now can circumvent this fact by adding another +22 vitality gem into the socket, thus giving you even more life on the item (good for tanks, for example).
This thread discusses all of this (notice Bashiok's post later in the thread, acknowledging these facts).
I can't say I know a particular formula, and I'm sure that an official source would never divulge it, but from what I am noticing it seems that the "gold ceiling" continues to get higher as you progress in difficulty.
For example, in Act I - Normal difficulty you may never see a gold drop higher than 60 gold or so, and they can be as low as 1 gold.
In Act I - Nightmare difficulty, I have seen gold drops as high as 300 gold in just the first half of the act, but have also still seen gold drops in the single digits.
My theory, is that the range expands to include higher potential values, but it is random anywhere within that range, meaning you can still get 1 gold drops later on but you could just as easily get 1000 gold drops.
Best Answer
Your formula is correct: if the chances of getting a ring in one run are 1/10, then the expected number of runs is 10. And if that is the case, then the expected number of runs to get 3 rings is 3*10=30 runs.
Once you have all three keys, you craft them into a portal, which you use to get a chance at one of three organs. Assuming these organs also have a 10% drop rate, that means you'll need an average of 10 portals ( = 30 keys = 300 key-runs) per organ. Since you need 3 organs, your overall expected number of runs is 900 for all the keys you need, plus another 30 to get the organs, for a total of 930 (at MP1).
In general, if
k
is the probability of a key dropping ando
is the probability of an organ dropping, your expected number of runs (including both keys and organs) will be9/po + 3/o
.Note that the error in your logic is that you appear to be assuming the expected value is the same as the point at which you have 50% chance of succeeding - it's not. Take coin-flips for example; you have a 50% chance of getting heads after only one flip, but the expected number of flips to see a heads is 2.