The answer suggested that even the expected value is never going to be 1/2 but instead something similar to 1/2-1/(4k) for k families which stop having children

The answer suggested that even the expected value is never going to be 1/2 but instead something similar to 1/2-1/(4k) for k families which stop having children after the first boy. So, it's a bit more sophisticated than that involving situations where you test the average percentage of an infinite number of islands with k families each.