The urn where the rich get richer

Question

An urn contains one red ball and one blue ball. Repeatedly: draw a ball uniformly at random, look at its colour, and return it to the urn together with a new ball of the same colour. So the urn grows by one ball per step, and colours that are already common tend to get drawn and reinforced. 1. After the first n draws, what is the probability that exactly k of them were red? 2. What does the proportion of red balls in the urn look like after a very large number of draws? Does it settle down, and to what?

Accepted Answer

After n draws, every count of reds from 0 to n is equally likely, with probability 1/(n + 1). The proportion of red balls converges, but to a random limit that is uniformly distributed between 0 and 1: the urn "picks" a proportion at random early on and then sticks to it. A specific sequence. Take red, red, blue, red. The probabilities of each draw are 1/2 (one red of two), then 2/3 (two reds of three), then 1/4 (one blue of four), then 3/5 (three reds of five). Product: (1 × 2 × 1 × 3) / (2 × 3 × 4 × 5) = 6/120 = 1/20. Any order is equally likely. For a sequence with k reds and n minus k blu…

The urn where the rich get richer

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