As I understand it, dumping 30 levels into a Diamond Helmet actually gives me a non-optimal chance to get Aqua Affinity or Respiration specifically. Assuming I'm not mistaken, what level of enchantment should I be applying to maximize my chances of receiving these specific enchantments?
Minecraft – What level of enchantment gives me the best chance for Aqua Affinity or Respiration
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Aqua Affinity allows the wearer of the enchanted helm to mine (break blocks) faster while underwater. Normally, being underwater slows down this action considerably. It also works while you're on ladders or otherwise suspended.
It seems like you are actually looking for enchantment probabilities:
Note that the lines of Lure I and Luck of the Sea I overlap (same for II and III). This means that the probability of getting Lure is the same as the probability of getting Luck of the Sea.
To answer you questions:
1) The lure enchantment is available on level 12, but the probability of getting it is less than 1%
2) The answer to the second question depends on what you actually want to achieve. Assuming you want the best chance of getting Lure and/or Luck of the Sea in a single enchantment, it is best to enchant on level 30. The combined probability of getting any Lure (I, II, or III) is 53% at level 30. Same for Luck of the Sea:
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Best Answer
Actually, you are mistaken. I have run 1 million enchantments for each level from 1 to 30 and counted the number of times I got each particular enchantment. Here are the results for diamond helmet:
You can clearly see that 30 level enchanting gives the maximum chance of getting Respiration or Aqua Affinity.
Running the same test for a golden helmet (which has better enchantability) gives almost the same picture - Aqua affinity actually peaks on level 28, but Respiration III is rising significantly on levels 29 and 30, so still a better strategy is to enchant on level 30:
Here is the raw data including all diamond helmet enchantments if you'd like to play around with it. I have used MCP 9.03 (Minecraft 1.7.2) to generate the results, but I'm not aware of any changes in the enchantment probabilities since then. Even in the upcoming 1.8 where the enchanting mechanic is changed, the probabilities and enchantment selection were not changed (according to Jeb). It is possible however that the enchanting will be revisited once more before the final 1.8 release.
I guess I could include the book enchantment probabilities for completeness, but they are lower than the helmet ones by an order of magnitude: