Coins In A Bucket

Time for a random new puzzle. I managed to solve this one last week, its not very hard but has a really neat solution. I first learned this one from the forums on xkcd:

You have two buckets that you are throwing coins into. The probability that a coin goes into a particular bucket is proportional to the number of coins currently in that bucket. The buckets are initially seeded each with one coin. You have 1000 coins in total, and after you have thrown all of those coins into the buckets you look into the bucket which has fewer coins in it. On average, how many coins are in that bucket?

In particular, just so you understand the whole "probability that a coin goes into a particular bucket is proportional to the number of coins currently in that bucket" thing, if the two buckets have 5 and 2 coins in them currently, then the next coin will go into the bucket with 5 coins with probability 5/7 and the one with 2 coins with probability 2/7.

The solution itself isn't actually all that neat, but there is something about this situation that you find along the way that is really cool.