On a certain island each of the inhabitants is a member of one of the two existing clubs.
The membership distribution is such that when two random people meet, the probability of those two belonging to the same club is equal to the probability of them belonging to distinct clubs.
When 100 newcomers arrive on the island and each enrolls in one of the two clubs, the distribution still retains this feature. How many people belong to either club?
We have two possible solutions to the given problem:
(I) The island is initially unpopulated. After the induction of 100 persons into the island, the respective membership of the two clubs is 55 and 45.
(II) The island had a initial population of 576, with the respective membership of the two clubs being 300 and 276. After an increase in the population by 100, the island now has a population of 676, with the respective membership of the two clubs being 351 and 325.