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Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number ofedges 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 $X$ isfar from its mean. It is possible to apply these results to discrete structuressuch 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 $B\simB_p$ is generated by including each vertex independently with probability $p$.In this setting our result on arithmetic progressions extends a result ofBhattacharya, Ganguly, Shao and Zhao \cite{BGSZ}. We also mention connectionsto related central limit theorems.

Deviation probabilities for arithmetic progressions and irregular discrete structures