What is the expected value of $2n$ such that a couple have to make $2n$ children to have as many sons as daughters?
I calculated the probability that after $2n$ children there are as many boys as girls: it should be
$\frac{{2n}\choose{n}}{2^{2n}}.$
However, we need the probability that this is the first time that it happens, and apart from a nasty inductive formula I don't know how to deal with that…
Best Answer
It's a trick question: There is no expectation!
Without loss of generality the first child is a girl. So the expectation we're looking for is (if it exists) $1+T$ where
Let's try to compute $T$. With probability $\frac12$ the first child will be a boy, and then we're done. But otherwise -- that is again with probability $\frac12$ -- our boy deficit has just increased by one, and we need to breed $T+T$ children before we're back to zero. ($T+T$ is because expectation is additive; we can count the time it takes to make up for each missing boy separately).
But this means that $$ T = \frac12\cdot 1 + \frac12(1+2T) $$ which is the same as $$ T = 1 + T $$ which has no solution. So the expectation cannot exist.