--- /dev/null
+---
+author: admin
+categories:
+- Non-Technical
+- Technical
+date: 2025-12-20 18:00:01
+tags:
+- minecraft
+- math
+title: "Minecraft Villager Trading Halls: Probability Math"
+---
+Recently I was making a villager trading hall in minecraft.
+
+
+
+One of the main goals of a trading hall is to collect all villager trades. One of the trickiest is books, provided by a librarian. I got to wondering -- how long is this going to take?
+
+Well, we can do some math to find out.
+
+There are currently (as of Minecraft Java Edition 1.21.11) 40 trade-able books. 36 of them are available from the enchanting table, treasure chests\[1\], or trading.
+
+4 are available only from treasure chests and trading. These are called *Treasure* enchantments.
+
+- Curse of Binding
+- Curse of Vanishing
+- Frost Walker I and II
+- Mending
+
+There's also three books, which can be found only from treasure chests. We don't care about them for trading halls:
+
+- Soul Speed
+- Swift Sneak
+- Wind Burst
+
+There are no books available only from enchanting and not trading.
+
+---
+
+The core mechanic of searching for book trades is *resetting*. If we look at a librarian and find it has a trade we don't want, we reset it.
+
+Villagers remember their profession and trades forever after trading. But if we haven't traded with a villager, we simply remove its profession, and then give it a profession again. Then we can see if we like the new starting trades better.
+
+This is very useful for librarians, because they have every book available as a starting trade, so there's no need to investigate later trades for books.
+
+These are the options to make the villager forget their profession I'm aware of:
+
+- Ignore the villager and get a new one (for example with a breeder), moving or killing the old one. This isn't a "reset" per se, but it acts similarly.
+- Break the profession block manually. In the case of a librarian, the lectern. When breaking the block, the villager loses their profession instantly.
+- Block the path between the villager and their profession block. I haven't seen this documented, but they reset at the same time as trades reset (twice per minecraft day). I did this by dropping the villager 1 block using a piston.
+- Move the villager at least 48 taxicab blocks from their profession block. (Not tested)
+- Move the profession block with a piston. This is an instant reset, but you can't do it for a lectern in Java edition. (Not tested)
+
+Some of these are instant, some take longer. Once villagers are shown a profession block, it only takes them a couple seconds to get their new profession, so that part is easy.
+
+I found breaking and re-placing blocks to be annoying, so I settled on moving librarians up and down with pistons. It takes about 5 real-time minutes for them to reset, so I used about 50 librarians to counteract that. By the time I finished checking all 50 librarians, they were ready to reset again because 5 minutes had passed.
+
+Then the question is: How many librarians do we need to look at, to get every book?
+
+---
+
+Well, the first question is: what are we interested in? Let's say we're interested in getting each of the 40 enchantments.
+
+Well, it turns out each enchantment is equally likely: there's a 1/40 chance of getting it. Well actually, 1/60 -- there's a chance that no book trade is offered at all.
+
+Then this is the [coupon collector's problem](https://en.wikipedia.org/wiki/Coupon_collector%27s_problem), a classic math problem.
+
+The number of trades to look at turns out to be: `3/2 x n x H(n)` where `H(n)` is the *n*-th harmonic number. For n=40, `H(40) = 1/1 + 1/2 + 1/3 + ... + 1/39 + 1/40 = 4.2785`. So we need to check **257 librarians** on average to get every enchantment.
+
+---
+
+But, are we really okay with that result? Given that Efficiency V is available as a starting trade, I want a librarian with Efficiency V, not Efficiency I!
+
+There are:
+
+| Enchantment | Level |
+|-----------------------|-------|
+|Aqua Affinity|I|
+|Channeling|I|
+|Curse of Binding|I|
+|Curse of Vanishing|I|
+|Flame|I|
+|Infinity|I|
+|Mending|I|
+|Multishot|I|
+|Silk Touch|I|
+|Fire Aspect|II|
+|Frost Walker|II|
+|Knockback|II|
+|Punch|II|
+|Depth Strider|III|
+|Fortune|III|
+|Looting|III|
+|Loyalty|III|
+|Luck of the Sea|III|
+|Lunge|III|
+|Lure|III|
+|Quick Charge|III|
+|Respiration|III|
+|Riptide|III|
+|Sweeping Edge|III|
+|Thorns|III|
+|Unbreaking|III|
+|Blast Protection|IV|
+|Breach|IV|
+|Feather Falling|IV|
+|Fire Protection|IV|
+|Piercing|IV|
+|Projectile Protection|IV|
+|Protection|IV|
+|Bane of Arthropods|V|
+|Density|V|
+|Efficiency|V|
+|Impaling|V|
+|Power|V|
+|Sharpness|V|
+|Smite|V|
+
+- 9 tradable enchantments with a max level of I
+- 4 with a max level of II
+- 13 with a max level of III
+- 7 with a max level of IV
+- 7 with a max level of V
+
+What's the chance of getting each level of enchantment? It's equal. So for Mending, there's a 1/60 chance to get Mending I, because it's the only choice. For Efficiency, there's a `2/3 * 1/40 x 1/5 = 1/200` chance to get Efficiency I, Efficiency II, or Efficiency V.
+
+How do we calculate the coupon collector's problem for un-equal probabilities? Well... it's really complicated\[2\].
+
+But the answer is that we will have to talk to an average of **933 librarians** to get all enchants at max level.
+
+---
+
+But hey. We can buy Efficiency V for 17 emeralds, if we get the right trade. Are we really okay getting a 64 emerald trade? What if we want only the best trades?
+
+| Enchantment | Level | Cost |
+|-----------------------|-------|------|
+|Aqua Affinity|I|5-19|
+|Bane of Arthropods|V|17-71|
+|Blast Protection|IV|14-58|
+|Breach|IV|14-58|
+|Channeling|I|5-19|
+|**Curse of Binding**|I|10-38|
+|**Curse of Vanishing**|I|10-38|
+|Depth Strider|III|11-45|
+|Density|V|17-71|
+|Efficiency|V|17-71|
+|Feather Falling|IV|14-58|
+|Fire Aspect|II|8-32|
+|Fire Protection|IV|14-58|
+|Flame|I|5-19|
+|Fortune|III|11-45|
+|**Frost Walker**|II|16-64|
+|Impaling|V|17-71|
+|Infinity|I|5-19|
+|Knockback|II|8-32|
+|Looting|III|11-45|
+|Loyalty|III|11-45|
+|Luck of the Sea|III|11-45|
+|Lunge|III|11-45|
+|Lure|III|11-45|
+|**Mending**|I|10-38|
+|Multishot|I|5-19|
+|Piercing|IV|14-58|
+|Power|V|17-71|
+|Projectile Protection|IV|14-58|
+|Protection|IV|14-58|
+|Punch|II|8-32|
+|Quick Charge|III|11-45|
+|Respiration|III|11-45|
+|Riptide|III|11-45|
+|Sharpness|V|17-71|
+|Silk Touch|I|5-19|
+|Smite|V|17-71|
+|Sweeping Edge|III|11-45|
+|Thorns|III|11-45|
+|Unbreaking|III|11-45|
+
+Mostly, the price range is based only on the level, but there are a few minor complications:
+
+- Some price ranges go above 64! In the game, these get capped. For this reason, you're 8 times more likely to get Efficiency V for 64 emeralds than any other number.
+- Treasure enchantments (in bold above) are double the price of any other enchantment. This is actually a double -- they're never offered for odd numbers of emeralds. Interesting!
+
+The chance of getting an Efficiency V book at the best possible price is: `1/16,500 = 2/3 x 1/40 x 1/5 x 1/55` (because there are 55 possible different prices -- counting ones above 64).
+
+To get every book at the best price, we'd need to talk to **45,594 librarians**\[2\] to get every max-level enchant at the best price.
+
+\[1\]: I think
+\[2\]: Source code [here](https://gist.github.com/za3k/d695594048ab8eb6239cb1dafdb97413). This uses the inclusion-exclusion principle to estimate set sizes, together with optimizations to take care of repeat probabilities.
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