what kind of distribution does this lead to?

[imfatandugly]

I have a bag with n balls labeled 1,2,...n in a bag. Every time you take out a ball, you make a copy of that ball and stick the two balls back in the bag. Repeat. What kind of distribution will this lead to?

For instance in the bag there are balls labeled {1,2,3}, i reach in and pick the 2 ball. I take it out, make a copy of it put them into the bag. now in the bag there are 4 balls two of which are labeled 2, so now when you pick a ball out of the back theres a 50/50 shot that it will be another two ball.

[pzhon]

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I have a bag with n balls labeled 1,2,...n in a bag. Every time you take out a ball, you make a copy of that ball and stick the two balls back in the bag. Repeat. What kind of distribution will this lead to?

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For n=2, there is a limiting proprotion with probability 1, and this proportion is uniformly distributed on [0,1].

For larger values, there is a limiting proportion with probability 1, and this proportion is uniformly distributed on a simplex. This follows from the case n=2.

To prove the case n=2, compare with the following: Imagine sampling repeated flips from a coin whose probability of heads is uniformly distributed on [0,1]. After h heads and t tails, the probobability that the next toss is a head is (h+1)/(h+t+2). That is the same as the conditional probability in your model, so the almost sure convergence and uniform distribution apply to yours, too.