Here is a slightly different problem that might make it easier to see what's
going on:

Suppose, in a country with two families, that each family has children
until it has either a boy or two children, then stops.  What fraction
of the births are female?

There are nine ways the history of this country could play out.

1) B/B  (that is, each family has one boy and then stops).  This happens
with probability 1/4.  In this case the fraction of girls is 0.

2) B/GB  (that is, the first family has  aboy and then stops; the second
has a girl and then a boy).  This happens with probability 1/8.  In
this case the fraction of girls is 1/3.

3) B/GG.  This happens with probability 1/8.  In this case, the fraction of
girls is 2/3.

And so forth.  Here are the nine scenarios, their probabilities and the
fraction of girls in each case:

1)  B/B   probability 1/4  fraction of girls 0
2)  B/GB  probability 1/8  fraction of girls 1/3
3)  B/GG  probability 1/8  fraction of girls 2/3
4)  GB/B  probability 1/8  fraction of girls 1/3
5)  GB/GB probability 1/16 fraction of girls 1/2
6)  GB/GG probability 1/16 fraction of girls 3/4
7)  GG/B  probability 1/8  fraction of girls 2/3
8)  GG/GB probability 1/16 fraction of girls 3/4
9)  GG/GG probability 1/16 fraction of girls 1

The expected fraction of girls is therefore

(1/4 x 0) + (1/8 x 1/3) + (1/8 x 2/3) + (1/8 x 1/3) + (1/16 x 1/2)

+ (1/16 x 3/4) + (1/8 x 2/3) + (1/16 x 3/4) + (1/16 x 1)

which adds up to 7/16, not 1/2.  

So even though the expected number of girls equals the expected number of 
boys, the expected fraction of female births need not be 1/2.