In an urn containing $n$ white balls and $n$ black balls, we proceed to draw balls without replacement, until the urn is completely empty. The random variable $M_n$ counts the number of equality indices $k$ between $1$ and $2n$, and it has been shown that:
$$\mathbb{E}(M_n) = \sum_{i=0}^{n-1} \frac{\binom{2i}{i} \cdot \binom{2n-2i}{n-i}}{\binom{2n}{n}}.$$
Deduce the equivalent:
$$\mathbb{E}(M_n) \underset{n \to +\infty}{\sim} \sqrt{\pi n}.$$