Markov’s Inequality: the first non-negotiable bound in probability

and why finance and Machine Learning still quietly rely on it

One of the most underestimated results in probability theory is Markov’s inequality.

It’s usually introduced in a single line, often dismissed as “too loose to be useful”, and then quickly forgotten in favour of flashier concentration inequalities.

That’s a mistake.

Markov’s inequality is the first line of defence in uncertainty quantification — and everyt…