Magnet Toss

A thousand magnets are thrown into the air one by one, such that were they not magnetic, they would have an equal chance of landing in each of two urns. Each urn initially contains one magnet. The more magnets that are in any particular urn the greater the chance of subsequent magnets landing in it. The probability that a magnet will land in urn x rather than urn y is x/(x+y). How many magnets, on average, would expect to find in the less full urn at the end?

Pressing Go lots of times should reveal that all distributions of magnets are equally likely, and thus the average should be 25% in the little urn.