  This article is a survey of results concerning an inequality, which may be
seen as a versatile tool to solve problems in the domain of Applied
Probability. The inequality, which we call BRS-inequality, gives a convenient
upper bound for the expected maximum number of non-negative random variables
one can sum up without exceeding a given upper bound $s>0.$ One fine property
of the BRS-inequality is that it is valid without any hypothesis
aboutindependence of the random variables. Another welcome feature is that,
once one sees that one can use it in a given problem, its application is often
straightforward or, not very involved.
  This survey is focussed, and we hope that it is pleasant and inspiring to
read. The focus is easy to achieve, given that BRS-inequality and its most
useful versions can be displayed in three Theorems, one Corollary, and their
proofs. We try to do this in an appealing way. The objective to be inspiring is
harder, and the best we can think of is offering a variety of applications. Our
examples include comparisons between sums of iid versus non-identically
distributed and/or dependent random variables, problems of condensing point
processes, subsequence problems, knapsack problems, online algorithms, tiling
policies, Borel-Cantelli type problems, up to applications in the newer theory
of resource dependent branching processes.
