Consider a stochastic variable $X$ taking positive real values and the events $P(X\geq a)\leq\frac{1}{3}$ and $P(X \leq b) \leq \frac{1}{2.9}$. We define $X_m$ as the median of $k$ independent outcomes of $X$.

I would like to prove that $P(b \leq X_m\leq a) \geq 1 - \frac{C}{k}$, where $C$ is some constant.

I've been trying to look at the complementary event $1 - P(X_m \leq b) - P(X_m \geq a)$ and to think of the median as a sum of Bernoulli trials of the events, being greater or equal to $k/2$. And then perhaps apply a Chernoff bound, but I can't really get it to work out.