Probability of finding a specific weapon/jewelry drop in a dungeon

2023-Dec-03 | eso, pvp, gaming

Sometimes you want to get a specific weapon or piece of jewelry for an armor set from a dungeon. What are the chances that it will drop for you?

I'll go through the proof below. But assuming that:

• you are playing with people that are willing to give you the piece if they find it
• and that you search for chests and loot them
• and that you play the dungeon four times

Your chance of finding that item would be about 57%.

Solo

Assuming you're going to Dungeons and not Trials (which have a curated system), then the final boss will drop one random weapon or jewelry piece from 3 different sets. If you count the possibilities per set there are 15 choices, and since there are 3 sets, that's a total of 45 choices.

If you were looking for something like an Arkasis Sword and you are in the Stone Garden dungeon. Then your chances are:

$$\begin{equation} \begin{split} P(find) &= \frac{1}{45}\\ &= 0.022\\ &= 2.2\% \end{split} \end{equation}$$

Pretty grim.

In a party

Let's assume that the rest of the party is willing to trade with you. To calculate this chance we need to check what the chance of no-one finding something is.

$$\begin{equation} \begin{split} P(found) &= 1-P(not found by player) ^ 4\\ &= 1-(\frac{44}{45})^4\\ &= 1-0.914\\ &= 8.6\% \end{split} \end{equation}$$

Again this assumes that others that find it are willing to trade with you. So a bit better, but still pretty grim.

What if we attempt multiple runs.

Multiple runs

$$\begin{equation} \begin{split} P(found) &= 1 - P(not found per run) ^8\\ &= 1 - 0.914^8\\ &= 1 - 0.487\\ &= 51.3\% \end{split} \end{equation}$$

So if you do eight runs, and the other players are willing to trade with you each time, your chance of finding it is 51.3%

That's a lot of work, and we're still only a coin toss ... can we improve our odds?

What about chests?

What about chests you might ask. Well, with chests you might get any piece from the set. So assuming it's a regular set there would be 22 pieces. And a max of 2 chests per dungeon.

Let's assume:

• 4 players that are willing to trade
• 2 chests
• Dungeon is completed - killed final boss

$$\begin{equation} \begin{split} P(find) &= 1 - (\frac{44}{45}*\frac{65}{66}*\frac{65}{66})^4\\ &= 1 - (0.978 * 0.985 * 0.985)^4\\ &= 1 - 0.811\\ &= 18.9\% \end{split} \end{equation}$$

Better odds with chests?

So what are our chances if we loot chests and play with friends four times in a row?

$$\begin{equation} \begin{split} P(find) &= 1 - (0.811)^4\\ &= 1 - 0.433\\ &= 1 - 0.811\\ &= 56.7\% \end{split} \end{equation}$$

That's more reasonable, but still only slightly better than a coin toss, lol.