Summary: A simple general framework for deriving explicit deterministic approximations of probability inequalities of the form P(\(\xi\geq a)\leq \alpha\) is presented. These approximations are based on limited parametric information about the involved random variables (such as their mean, variance, range or upper bound values). First the case of a single random variable \(\xi\) is analyzed, followed by the cases of independent and dependent summands \(\xi =\sum^{n}_{1}\xi_ i.\) As examples of possible applications, a stochastic extension of the ``knapsack problem'' and the stochastic linear programming problem with separate chance-constraints are investigated: we provide approximate deterministic surrogates for these problems.