ESO Undaunted Coffer Math

13 June 2026

TL;DR. I’m writing this on a weekend, mostly for fun — to play with the probability math. You probably don’t need any of it. If you play ESO and just want the answer for your own collection, the ESO coffer calculator does the work for you.

I love RPGs, especially the ones with real math behind the mechanics. This post is about a small probability problem that came up in The Elder Scrolls Online (ESO).

Quick context. ESO has a faction called the Undaunted with three vendors known as Pledge Masters: Glirion the Redbeard, Maj al-Ragath, and Urgarlag Chief-bane. You earn keys by completing daily dungeon quests and spend those keys at the Pledge Masters on loot boxes (“coffers”) that drop shoulder armor pieces. Different vendors sell different sets.

Each Pledge Master offers two kinds of coffer:

• Mystery coffer — costs 1 key, rolls one random shoulder from any of the vendor’s sets.
• Curated coffer — costs 8 keys, rolls one random shoulder from a single set you choose.

Neither has duplicate protection: you can roll a piece you already own.

The question: per 8 keys spent, which option gives me more new pieces on average?

Setup. A Pledge Master sells $k$ sets. Set $i$ contains $t_i$ shoulder pieces, of which $u_i$ are still unknown to you. Define:

so $N$ is the total pool and $U$ is everything you still need.

Mystery. A single mystery coffer rolls uniformly over all $N$ pieces, so:

Eight mystery coffers cost the same as one curated. For the expected number of distinct new pieces from 8 pulls we use linearity of expectation: each of the $U$ unknown pieces is rolled at least once with probability $1 - ((N-1)/N)^8$, so

(The linear shortcut $8U/N$ overcounts when the same unknown is rolled twice in 8 pulls — the formula above is the strict version.)

Curated. A curated coffer of set $i$ rolls uniformly over the $t_i$ pieces in that set. Of those, $u_i$ are still unknown. So:

You pick the set that maximizes this:

The rule. Mystery wins when:

A useful observation. Per-key analysis is the cleanest way to see why mystery is so dominant for the standard ESO structure.

• Mystery, per key: one pull, probability of a new piece $U/N$.
• Curated of set $i$, per key: $1/8$ of a pull, probability of a new piece per key is therefore $u_i / (8 t_i)$.

Mystery wins per key when:

For the standard Pledge Master structure ($k = 6$ sets, $t_i = 6$ each, $N = 36$), this simplifies to $8 U > 6 \, u_i$, i.e. $U > 0.75 \, u_i$. Since $u_i \le U$ always, this holds whenever $U > 0$. In other words, for a standard 6×6 vendor, mystery coffers are mathematically the better buy as long as you still need anything at all — even when only one piece is left in the entire 36-item pool.

The threshold flips only when the vendor has more than 8 sets — then the mystery pool is diluted enough that targeting via curated becomes worthwhile.

Glirion the Redbeard, worked example. Glirion sells 6 curated coffers, 6 pieces each, so $N = 36$. My current state across the six sets:

Best curated bet is Frigid Crucible at $5/6 \approx 0.833$. So 8 keys spent on mystery coffers yields about $4.24$ new pieces in expectation; 8 keys spent on the best curated yields about $0.83$. Mystery wins by a factor of ~$5.1$.

Edge case sanity check — when only $U = 1$ piece is left anywhere on the vendor:

vs. best curated (the set containing that one piece, $1/6 \approx 0.167$). Mystery still wins by ~3.5 percentage points.

Urgarlag Chief-bane, the exception. As of June 2026 Urgarlag’s lineup is 17 sets at 6 pieces each, so $N = 102$. With $k = 17 \gt 8$, the per-key rule no longer holds automatically. Plugging into $U \gt (k/8) \cdot u_{\max}$:

Read that as: mystery is the better buy only while your unknowns are spread thinly across many sets. The moment most of what you’re missing concentrates in a single set — i.e. $u_{\max}$ creeps close to $U$ — switch to a curated of that set.

Concrete example: if I still need 10 pieces total and 7 of them sit inside one set, $u_{\max} = 7$ and the threshold demands $U > 14.875$. With $U = 10$, mystery loses; the curated of that one rich set is the better play. If those same 10 unknowns are spread one-per-set across 10 different sets, $u_{\max} = 1$, the threshold drops to $U > 2.125$, and mystery comfortably wins.

How many keys to fully clear a vendor? This is the classic coupon collector’s problem. Starting from zero pieces, drawing one per mystery coffer, the expected number of pulls (= keys) to collect all $N$ pieces is:

where $H_N = \sum_{k=1}^{N} 1/k$ is the $N$-th harmonic number and $\gamma \approx 0.5772$ is the Euler–Mascheroni constant. Plugging in:

These numbers assume pure mystery-only play. For Glirion and Maj al-Ragath that’s also the optimal strategy — the per-key rule above proves mystery dominates everywhere. For Urgarlag the optimum cuts in some curated purchases toward the end (once one set carries most of the remaining unknowns), so the real total comes in somewhat below 531; coupon collector still gives the right order of magnitude.

Takeaways.

• Mystery wins almost always. For the two standard 6-set vendors (Glirion the Redbeard, Maj al-Ragath), it’s the mathematically better buy as long as you still need anything at all — even down to one missing piece out of 36.
• Urgarlag Chief-bane is the exception. With 17 sets, mystery still wins when your unknowns are spread thinly, but the moment most of what you need concentrates in a single set, the curated of that set beats mystery.
• Curated feels targeted, but it isn’t a guarantee. It still rolls randomly inside the chosen set (1/6 per pull). One curated for 8 keys rarely beats 8 mystery pulls.
• Clearing a vendor from scratch costs ~150 keys for Glirion or Maj al-Ragath, and ~531 keys for Urgarlag — by the coupon-collector formula above.

I made a calculator that does the table and rule above for arbitrary vendor configurations: ESO coffer calculator →