Let $X_n \sim Bin(n_1,p_1)$ and $Y_n \sim Bin(n_2, p_2)$ with $n_1 + n_2 = n$ and $p_1,p_2 >0$ be independent, binomially distributed random variables. We furthermore assume that $\frac{n_1}{n} \to \pi_1$ and $\frac{n_2}{n} \to \pi_2$ , as $n \to \infty$.
Now let $Z_n = \frac{X_n}{\max(X_n+Y_n,1)}$.
I want to show that $$\mathbb{E}Z_n = \frac{\pi_1p_1}{\pi_1p_1 + \pi_2p_2} + O(\frac{1}{\sqrt{n}})$$
A first handwavy approach to this would go as follows: Observe that $\Pr(X_n + Y_n \leq 1) \to 0 , n \to \infty$. Also with help of the SLLN we get:
$$ \frac{X_n}{n} \to \pi_1p_1 \;\; a.s.$$
$$ \frac{Y_n}{n} \to \pi_2p_2 \;\; a.s.$$
Thus, with the continuous mapping theorem (and independence), it follows that:
$$\frac{X_n}{X_n + Y_n} = \frac{\frac{X_n}{n}}{\frac{X_n}{n} + \frac{Y_n}{n}} \to \frac{\pi_1p_1}{\pi_1p_1 + \pi_2p_2} \; \; a.s. $$
Now, applying dominated convergence yields the limit at least..
Now, an intuitive explanation for the rate $O(\frac{1}{\sqrt{n}})$ seems to be given by using a similar approach with the CLT and the Delta method.
But I am not sure how to proceed for the expectations and also make the proof rigorous.