On lower bounds for hypergeometric tails

Let $n,k$ be positive integers such that $n\ge k$, and let $H$ be a hypergeometric random variable counting the number of black marbles in a sample without replacement of size $k$ from an urn that contains $i\in \{1,\dots ,n\}$ black and $n-i$ white marbles. It is shown that $$\mathbb{P}(H\ge \mathbb{E}(H))\ge k\mathrm{/}n,\text{when}n\ge 8k\mathrm{.}$$ Furthermore, provided that $1\le \mathbb{E}(H)\le \min \{i,k\}-1$ as well as that $\frac{(n-i)(n-k)}{n}>1$, it is shown that $$\mathbb{P}(H\ge \mathbb{E}(H))\ge \frac{e^{-1\mathrm{/}12}}{4\sqrt{2}}\cdot \sqrt{\frac{n-1}{n}}\cdot \frac{\sqrt{\text{Var}(H)}}{1+\sqrt{1+\frac{n-1}{n-k}\cdot \text{Var}(H)}}\mathrm{.}$$ Auxiliary results which may be of independent interest include an upper bound on the tail conditional expectation and a lower bound on the mean absolute deviation of the hypergeometric distribution.