  We show that the probability of the exceptional set decays exponentially for
a broad class of randomized algorithms approximating solutions of ODEs,
admitting a certain error decomposition. This class includes randomized
explicit and implicit Euler schemes, and the randomized two-stage Runge-Kutta
scheme (under inexact information). We design a confidence interval for the
exact solution of an IVP and perform numerical experiments to illustrate the
theoretical results.
