Average proportion of boys out of the whole population

Question

I passed by this question today, in which I'd like to have some help or hints regarding part b)

So the story is as follows:

The residents of a certain country value boys over girls, and every couple makes sure a boy is born in the family. So, if the first child is a boy, they stop there. If the first child is a girl, they have another child, and keep on having children until the first boy. So, the progression of children for each family ends with a boy. Some examples would be B, GB, GGB, GGGGB. At birth, the probability of a child being a girl or a boy is equal (½). (a) What is the expected number of children of a family? (b) What is the expected value of the proportion of males to the total population in this country?

Part a) is solved easily, as it's equal to the $\sum_{i\ge1}{ip^i}$ which leads to $2.$

In part b) I solved according to the ratio between the average of boys and the average number of family children. I'm not sure if it shall be solved as follows : $$\frac{E(\#\text{boys} )}{E(\#\text{boys})+E(\#\text{girls})}$$

I'm not sure if the parents shall be ignored in part b) as well

Can someone guide me if that's true ?

Thanks!

Answer 1

Expected number of girls${}=\sum_{i=0}^\infty ip^{i+1}=1$ for $p=\frac{1}{2}$ so the ratio is $1:2$.

Answer 2

$\newcommand{\e}{\operatorname E}$ Let $B$ be the total number of births, a random variable.

Let $I$ be a random variable whose conditional distribution given $B$ is uniform in the set $\{ 1, \ldots, B\},$ so that each of those numbers has probability $1/B$ of being the value of $I.$

The conditional expected value $$ \e \left( \dfrac{\#\text{boys in the $I$th birth}}{\#\text{boys in the $I$th birth} + \#\text{girls in the $I$th birth}} \mid B, I \right)$$ is clearly $1/2,$ i.e. for each birth separately it's $1/2.$

So the marginal (or "unconditional") expected value of that fraction is the expected value of $1/2,$ i.e. it's $1/2.$

The fact that large families have more girls than boys is countered by the fact that there would be few of those, plus the fact that half the families have only one child and that one is a boy, and in the other half of the families there are also boys.