  Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of
edges of a hypergraph $\mathcal{H}$ induced by a random $m$-element subset $B$
of its vertex set. Focussing on the case that the degrees of vertices in
$\mathcal{H}$ vary significantly we prove bounds on the probability that $X$ is
far from its mean. It is possible to apply these results to discrete structures
such as the set of $k$-term arithmetic progressions in the $\{1,\dots, N\}$.
Furthermore, our main theorem allows us to deduce results for the case $B\sim
B_p$ is generated by including each vertex independently with probability $p$.
In this setting our result on arithmetic progressions extends a result of
Bhattacharya, Ganguly, Shao and Zhao \cite{BGSZ}. We also mention connections
to related central limit theorems.
