Linearity of Expectation


The answer is immediately clear by Linearity of Expectation. If the probability of having a boy is 1/2, then the expected number of boys is 1/2 of all children. Same for the number of girls. So the ratio is 1:1.
It doesn’t matter what scheme is used to decide when each couple stops having children, we expect half of the children to be boys and half to be girls.