Let S denote a sum of (finitely many) independent non-negative random variables with means not exceeding 1. A remarkable result of Uriel Feige (SIAM Journal on Computing, 2006) states that for every 0<t<1 the inequality P(S<ES+t) > Ct holds true, where C is some universal positive constant (i.e. it does not depend on t, distributions and number of the random variables).

A short and simple proof will be presented, to a large extent along the lines of He, Zhang and Zhang (Mathematics of Operation Research, 2010), as well as some generalizations of the result.