[OscarCunningham]

Bear in mind the default behaviour if there were just two balls and you never added any more. Then the proportion of red picks vs blue picks would tend to 1/2. So there's naturally a tendency for the proportion to concentrate in the middle.

As you say, the way in which new balls are added tends to push the proportion towards the extremes.

The uniform distribution is the result of these two tendencies exactly cancelling out.

[cmoog]

This is not correct as stated. It does not “tend towards extremes” as you might expect intuitively. That would be the case if half the time it approached 100% blue and half the time it approached 100% red, which is precisely not what happens.