What are you enchanting
The table
Every player carries a hidden enchantment seed that changes each time they enchant something. You cannot read yours in game, so treat this as any one player's table: the offers below are exactly what the game would put in front of you for this seed.
The three offers
The top row of each offer is what the game lets you see before you commit. The chips under it are the full list it applies, which is why a level 30 offer that reads as Unbreaking III can hand you four enchantments.
Your odds on Diamond Pickaxe at 15 bookshelves
Ten thousand seeds through the same routine, counted. "Any offer" is the chance that enchantment and level appears somewhere in the three, which is what matters when you are deciding whether to spend the levels. The columns after it split that by which offer it turned up in. Read from the game's own files for 26.3 Pre-Release 1.
The cost you see is not the enchantment power. The game takes that cost, adds two random rolls scaled by the item's enchantability, nudges the total by up to 15 percent either way, and only then asks which enchantments are in range. A gold pickaxe and a diamond one at the same 30 levels are not playing the same game.
Enchantability is why gold gear enchants so well: gold tools carry 22 and diamond only 10, so the same 30 level offer lands much higher up the cost curve on gold. It is also why a book, at enchantability 1, reaches so few high level enchantments on its own.
Once the pool is known, the game picks one by weight, then keeps picking while a fresh roll under 50 comes out at or below the adjusted cost, halving that cost each time and throwing out everything incompatible with what it just chose. That loop is the reason a level 30 offer can hand you four enchantments and a level 5 offer almost never hands you two.
The routine is a line by line port of EnchantmentMenu and EnchantmentHelper out of the unobfuscated client, running real java.util.Random rather than a stand-in, so a seed here means what a seed means in game. The weights, cost curves, maximum levels and exclusive sets are read out of the game's own data files, and the enchantability values come from ToolMaterial and ArmorMaterials. No wiki was involved, which matters here: the wiki's weight tables are correct and still will not tell you your odds, because the odds live in the selection loop rather than the weights.
Because it is meant to be a gamble. The game picks the whole list up front and then shows you one of them at random, in an alphabet you cannot read without a resource pack. This page runs the same code and prints the list, so you can see what a given offer would really apply.
Fifteen, and nothing less will do it. The cost formula is a random 1 to 8, plus half your bookshelf count, plus a second random roll up to your bookshelf count, and the bottom offer is the larger of that total and twice your bookshelves. At fifteen shelves twice your bookshelves is 30 and the total can never beat it, so the bottom slot is always exactly 30.
It depends on the item and the shelf count, and the odds table on this page is the honest answer for both: ten thousand seeds through the real routine, counted. Weights alone will not tell you, because the selection loop keeps drawing while a random roll under 50 stays below the adjusted cost, and each draw removes everything incompatible with what it just picked.
Not in vanilla. Every player carries one, it changes each time you enchant, and nothing in the interface shows it. Use this page as any one player's table: the mechanics, the costs and the odds are the same for everyone, only the particular three offers move.
Because the game will not either. An item can only go in an enchanting table if it carries an enchantability value, and elytra, shears, shields, flint and steel and brushes carry none. They take enchantments from books on an anvil instead. The 77 items listed here are exactly the ones the game accepts.
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