Concentration inequalities are probabilistic bounds — such as Markov’s, Chebyshev’s, Hoeffding’s and Chernoff’s inequalities — that quantify how tightly a random variable, typically a sum or average, concentrates around its expected value. In machine learning they provide the mathematical basis for generalisation bounds, showing how closely empirical risk on a finite sample tracks true risk as sample size grows. They are a core analytical tool within statistical learning theory and underpin bias-variance and PAC-learning arguments.