There is a country where every family keeps having children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. Boys are girls are equally likely to be born.
What is the proportion of boys to girls in this country?
The ratio of boys to girls is 1:1.
Proof
Let’s assume for now that we don’t know the strategy that the families use to decide whether to have another child. However, the births are statistically independent.
For each birth in the country, we have a \(\frac{1}{2}\) chance of having a girl, and a \(\frac{1}{2}\) chance of having a boy. Thus after \(n\) births, we expect to have on average \(\frac{n}{2}\) girls and \(\frac{n}{2}\) boys.
This gives a ratio of 1:1. Further this is true regardless of what strategy the families use to decide when to have children.
In this specific case we can also determine the expected number of girls per family as:
\[E(g) = \frac{1}{2}(1 + \frac{1}{2}(1 + \frac{1}{2}( 1 + ... ))) = \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + ... = 1\]This represents that there is a 50% chance of having at least one girl. Given one girl, there is a 50% chance of having at least 2 girls, and so on. Since each family stops at one boy, there is on average 1 boy and 1 girl in each family.