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The performance of quantum key distribution (QKD) heavily depends on statistical inference. For a broad class of protocols, the central statistical task is a random sampling problem, customarily addressed using exponential tail bounds on the hypergeometric distribution. Here we devise a strikingly simple exponential bound for this task, of unprecedented tightness among QKD security analyses. As a by-product, confidence intervals for the average of non-identical Bernoulli parameters follow too. These naturally fit in statistical analyses of decoy-state QKD and also outperform standard tools. Lastly, we show that, in a vast parameter regime, the use of tail bounds is not enforced because the cumulative mass function of the hypergeometric distribution is accurately computable. This sharply decreases the minimum block sizes necessary for QKD, and reveals the tightness of our simple analytical bounds when moderate-to-large blocks are considered.