The question below is a part of a more complex question in probability theory. I can't get my head around why is my solution incorrect...
So here it goes. Distribution of number of boys in a family with $n$ children is binomial with parameters $n$, $\frac{1}{2}$ (my guess is that it means $\mathbb{P}(X=k)=\binom{n}{k}(\frac{1}{2})^k(\frac{1}{2})^{n-k}=\binom{n}{k}(\frac{1}{2})^n$). What is $\mathbb{P}(X=2)$?
I thought it would be $$\sum_{n=2}^{\infty}\binom{n}{2}\left(\frac{1}{2}\right)^n=\frac{1}{2}\sum_{n=2}^{\infty}n(n-1)\left(\frac{1}{2}\right)^n=\frac{1}{8}\left(\sum_{n=2}^{\infty}\left(\frac{1}{2}\right)^n\right)''=\frac{1}{8}\left(\frac{2}{(1-\frac{1}{2})^3}\right)$$
But it definitely is not true, since the above-mentioned sum is $2$...
There is no reason to sum. For families with $n$ children, the probability, as you observed, is $\binom{n}{2}\cdot\frac{1}{2^n}$.
In order to find the probability that a randomly chosen family has exactly $2$ boys, you would have to know the distribution of family sizes.
If the probability that a family has $n$ children is $p_n$, then the probability a randomly chosen family has $2$ boys is $$\sum_0^\infty p_n \binom{n}{2}\frac{1}{2^n}.\tag{1}$$
Without knowing the $p_n$, we cannot go beyond (1).
Remark: We have used some assumptions that are in fact false. The probability of a boy is not exactly $\frac{1}{2}$. And the implicit assumption of independence is quite dubious.