Consider a country with $n$ families, each of which continues having children until they have a boy and then stop. In the end, there are $G$ girls and $B=n$ boys.

Douglas Zare's highly upvoted answer to this question computes the expected fraction of girls in a population formed of complete families and explains why we shouldn't expect it to equal $1/2$. My current question concerns a different statistic, namely the probability that there are more boys than girls (after all families have finished reproducing). This probability turns out to be exactly $1/2$, and I'm looking for an intuitive explanation of why.

Indeed, for fixed $n$, it's not hard to see that

$$Prob(G=k)=\binom{n+k-1}{k}\cdot {1\over 2^{n+k}}$$
(The binomial coefficient is the number of ways to assign $k$ indistinguishable girls to $n$ distinguishable families.)

Therefore $$Prob(G < B)=\sum_{k=0}^{n-1}\binom{n+k-1}{k}\cdot {1\over 2^{n+k}}$$ It's not hard to check that this sum is exactly equal to $1/2$ (and therefore, in particular, independent of $n$).

That is, regardless of the number of families, we always have the surprisingly (to me) simple formula

$$Prob(G < B )=1/2$$

(Note that this implies $Prob(G>B)$ is strictly less than $1/2$ --- because there's always some probability weight on the event $G=B$ --- though an application of Stirling's formula shows that $Prob(G>B)$ converges to $1/2$ as $n$ gets large.)

My question is:

Is there some simple intuitive reason I should have expected this result?