a tail bound

Most of a distribution's probability sits near its centre, but the interesting risks live in the tails — the far-out values, the rare large deviations. A tail bound is any guarantee that caps how much probability can hide out there: a ceiling on P(X >= a) (the upper tail), on P(X <= a) (the lower tail), or on P(|X - mean| >= a) (a two-sided tail). It answers 'how unlikely is an extreme outcome?' with a provable upper limit.

The 'tail' is literally the part of the distribution beyond a threshold — the area under the density curve out past a, or the sum of probabilities of all values that extreme. A tail bound replaces the often-uncomputable exact tail probability with a clean, guaranteed-larger expression you can actually use. The bounds in this field form a ladder of sharpness using progressively more information: Markov uses only the mean and gives a slow 1/a decay; Chebyshev adds the variance for 1/a^2; Chernoff, Hoeffding, and Bernstein use the moment generating function or boundedness to deliver exponentially small tails like e^(-c a^2). More information bought, tighter ceiling delivered.

Tail bounds are the practical face of concentration: they are what you cite when you need a finite-sample guarantee — 'with probability at least 0.99, the error is below epsilon'. Translating between a confidence level and an interval is exactly inverting a tail bound. Two honest cautions. First, a bound is one-directional: it says the tail is NO BIGGER than something, never that it is that big — a loose bound can vastly overstate the true risk. Second, the right tool depends on what you know: with only a mean you are stuck with Markov; with light tails you can claim exponential decay; with heavy tails (infinite variance) the strong exponential bounds simply do not hold.