Using the central limit theorem on a binomial random variable

Question

I need to show that the limit as n goes to infinity of $P(X_n \geq \frac{n}{2})$ is 1/2 if the expected value is equal to 1. The random variables are binomially partitioned but it's not given that they are independent.

This would be straightforward if $X_n$ were less than or equal to $\frac{n}{2}$, but since it isn't I think I would need to use Markov's inequality somewhere to get an estimate. A short explanation or hint on how to strategize would suffice.

Answer 1

$X_n\sim B(m_n,p_n)$ with $m_np_n=1$. $P(X_n\geq \frac n 2)\leq 4\frac {EX_n^{2}}{n^{2}}=4\frac {m_np_n(1-p_n)+(m_np_n)^{2}} {n^{2}}=4\frac {2-p_n}{n^{2}} \to 0$.