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All-Girl World Simulation
I saw this post on Twitter. It was almost perfectly designed to trigger nerd arguments:
In a primitive society, every couple prefers to have a baby girl. There is a 50% chance that each child they have is a girl, and the genders of their children are mutually independent. If each couple insists on having more children until they get a girl and once they have a girl they will stop having more children, what will eventually happen to the fraction of girls in this society?
Your gut says “more girls, obviously” — every family is guaranteed a girl but boys are just a byproduct of getting there. But the stopping rule doesn't actually change the odds. Every individual coin flip is still 50/50, and the decision to stop flipping can't retroactively change the results of the flips you already made. On average, each couple has one girl and one boy. The fraction of girls stays right around 0.5.
That's the true spirit of the puzzle, and it's the level of analysis where it actually sheds useful insight into probability for most people. Stopping rules feel like they should matter, but they don't change the expected ratio. If you walk away with that intuition, you've gotten the point.
Of course, if you want to out-nerd the other nerds on Twitter, you can go deeper. There are at least two more layers to this thing.
The Fraction Paradox
The total number of boys born equals the total number of girls born, on average. But the average of the fraction of girls in each generation isn't exactly 0.5. It's subtly higher. This is because fractions are nonlinear — a small population that's 60% girls contributes disproportionately to the average fraction compared to a large population that's 49% girls. For large populations the effect is negligible, but for small ones you can see it clearly.
Inevitable Collapse
The question asks what will eventually happen to the fraction of girls. The pedantic answer is that the fraction becomes undefined — because the population goes to zero. Each couple produces on average 2 children, but random variation means each generation probably won't have a perfectly equal number of males and females. The surplus members of the more numerous sex can't pair up, so the effective breeding population shrinks over time. Given enough generations, every society in this model collapses. The question is just how long it takes.
Or you can just mess around with the parameters yourself: