How do I improve my chances of getting silk touch when enchanting?
[Current status: Unfortunately, as of now (2015) these tables have not been updated for Minecraft 1.8, only 1.6.2 (2013). Version 1.8 changed enchanting so that when you enchant an item it only consumes up to three of your levels, not all of the levels required to permit the enchantment. This means that the XP cost tables are inaccurate.]
Stuart Pernsteiner's enchantment strategy guide is, or at least was, an excellent resource. It provides tables and charts which list the conditions for the minimum XP or item cost for a given enchantment. According to the table for Silk Touch, you want to be at level 30 to get the best chance per single enchantment, you should still expect to enchant an average of 6 tools before getting Silk Touch rather than another enchantment.
However, if you have plenty of materials you may wish to use levels which are more efficient in terms of XP required rather than tools enchanted, since higher levels require gathering more XP per level.
@ Kevin Reid - Just so I understand. If this lines up with SaintWacko's calculation above, then enchanting 4 Iron Pickaxes with level 11 enchants (using 44 levels, giving 2 in 9 chance), gives better odds than enchanting 1 Gold pickaxe at level 49?
Acording to the strategy table I linked, Silk Touch on a gold pickaxe at 49 has a 14.3% chance. According to Iron Tool, Level 11, you have a 0.7% chance of Silk Touch in that case, and probability theory tells us that doing that 4 times is `1-(1-(0.007))^4 = 0.028 = 2.8%` chance. So, no, that would not be an improvement. You can also see this in the Item Cost graphs: the line is off the scale at level 11
@ Kevin Reid - Ah, okay so SaintWacko's numbers are off? Although according to that table it would still be better to attempt 2 level 25 enchants than 1 level 50 enchant. And the actual best use of experience is to enchant at level 26. Is that correct?
For “best use of experience”, you want the “XP Cost Local Minima” table in the “Summary” section, which indicates that the best level depends greatly on the material (26 is that value for *stone* tools, but lower levels are better for other materials).
I'm hesitant to pick answers that are essentially links, but I think what I really wanted came out in the comments. Thanks.
Well, you originally asked for “the best odds”, which I can only interpret as for a single enchanting — you didn't say you wanted the XP cost minimum. I agree the answer should be in the answer, but in this case I would be just copying his tables to give a sufficiently general example.