The model is as follows (Wikipedia): you have groups, each of which starts initially with 1 person. Every second, a new person joins a random group, where their probability of joining a given group is directly proportional to that group’s size. For instance, if and the groups currently have sizes and respectively, a new person will join the first group with probability and the second with probability . Essentially, all the people are sheep and tend to follow the crowd. Now, the problem is the following: after seconds, what is the distribution of possible group sizes? Will it be skewed toward distributions with a single large group?
Solution
It turns out that all possible group sizes are equally likely!
To see why, consider reframing the problem in the following way: there are initially people in a line and divider between each of them, for total. Then, every second, each new person randomly inserts themselves in front of a person already in line. Do you see how this model is equivalent? From here, it’s not to difficult to show that all partitions are equally likely. In a partition with people, any of the people not initially in line could have been the last one added. Therefore, there are exactly ways to reach every partition of people, meaning they are all equiprobable.