  Iterative numerical algorithms are typically equipped with a stopping
criterion, where the iteration process is terminated when some error or misfit
measure is deemed to be below a given tolerance. This is a useful setting for
comparing algorithm performance, among other purposes. However, in practical
applications a precise value for such a tolerance is rarely known; rather, only
some possibly vague idea of the desired quality of the numerical approximation
is at hand. We discuss four case studies from different areas of numerical
computation, where uncertainty in the error tolerance value and in the stopping
criterion is revealed in different ways. This leads us to think of approaches
to relax the notion of exactly satisfying a tolerance value. We then
concentrate on a {\em probabilistic} relaxation of the given tolerance. This
allows, for instance, derivation of proven bounds on the sample size of certain
Monte Carlo methods. We describe an algorithm that becomes more efficient in a
controlled way as the uncertainty in the tolerance increases, and demonstrate
this in the context of some particular applications of inverse problems.
