I've been stuck on this problem for ages now and don't even know where to start.
The problem states that women are supposed to give birth to children until they give birth to a boy. The probability of either a boy or a girl is 0.5. We have 100 discrete stochastic variables for the amount of children a woman has given birth to (So $X_1$ could be 4 children, 3 girls, 1 boy)
Then according to the central limit theorem we can say:
$$ \frac{X_1+X_2+X_3 + \cdots + X_{100}}{100} = N(\mu,\sigma^2) $$
Now what is $\sigma^2$ ?
It seems to me that $X_i$ has a geometric districution, which has variance $\frac{1-p}{p^2}=2$. For independent $X_i$, the sum has variance $2\times 100$ and $\sum_i X_i/100$ has variance $\frac{200}{100^2}=0.02$.