We know that the probabilty of having boy is $\frac{3}{4}$ and every family has kids until they have boy, after boy they have no children, what is the ratio between boys and girls in population?
I suppose we want to calcualte this using expected value.
Every family has 1 boy so $E(boys) = 1 $
However i am not sure how to calculate expected value of girls. The boys have exponential distribution. So we can calculate probabilty of 0..1... inf girls by calculating "what is the probability that xth child is boy" e.g
$\sum_{i = 1}^{\inf} (\frac{1}{4})^{i-1}*\frac{3}{4}$
and expected value by
$\sum_{i = 1}^{\inf} i* (\frac{1}{4})^{i-1}*\frac{3}{4}$
my question is, this formula basicly tell us expected value of of boys e.g $x *P(x)$ where x is on what attempt family had boys * probability that it was on that attempt.
This does not tell us nothing about girls. So if i modified it to
$\sum_{i = 1}^{\inf} i* (\frac{1}{4})^{i}*\frac{3}{4}$
This should tell us $x * P(x)$ where x = number of girls and P(x) = probability that family had boys on i + 1 attempt = probability that family had i girls.
And just adjusting it to
$\frac{1}{4}*\sum_{i = 1}^{\inf} i* (\frac{1}{4})^{i-1}*\frac{3}{4} = \frac{1}{4} * E(boys) = \frac{1}{4} * \frac{1}{\frac{3}{4}} = \frac{1}{4} * \frac{4}{3} =\frac{1}{3}$
So boys:girls should be $1:\frac{1}{3} = 3:1$
But i am not sure if logic behind this is correct. Can i adjust the formula this way for it to make sense?
Thanks for answers.