Ratio of Boys and Girls In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of having a boy or a girl is the same)

The expected number of boys per family is clearly 1. The probability that a family will have exactly $k$ girls is $(1/2)^k \cdot (1/2) = (1/2)^{k+1}$, as the first $k$ "draws" need to be a girl, while the $(k+1)$'th draw needs to be a boy (if not for the latter, the family would have more than $k$ girls).
Hence, the expected number of girls in a family is
$$(1/2) \cdot 0 + (1/4) \cdot 1 + (1/8) \cdot 2 + (1/16) \cdot 3 +\dots = \sum_{k=0}^{\infty}(1/2)^{k+1} k$$
Due to a well known property of geometric distributions, this expection is equal to 1. The ratio of boys to girls will thus be 1.
