What are the chances of gathering rare components from resource nodes?

When you gather materials from a resource node, you have a small chance of obtaining a rare item depending on what type of node it is. (For example, Aspen Saplings occasionally drop a few cinnamon sticks when you cut one down.)

In addition, there are several gathering tools that the game describes as:

More efficient at gathering resources and uncovering rare components.

What is the base chance of getting rare materials, and how do these special gathering tools modify that chance?

I haven't seen an answer from Arenanet on this one, but I believe it depends on what type of node you're gathering from. Ore nodes, for example, all appear to have approximately a 20% chance per hit to receive gemstones, while gathering nodes have a much lower chance to yield an unidentified dye (likely to be about 1%).

From my experience, tools labeled as "more efficient" or "extremely efficient" only make a small difference to the total drop rate of rare components.

It is also noteworthy that such tools seem to have a higher chance to preserve the node that was gathered from rather than destroying it, although this chance is still low.