  In this article, we consider a stochastic numerical simulator to assess the
impact of some factors on a phenomenon. The simulator is seen as a black box
with inputs and outputs. The quality of a simulation, hereafter referred to as
fidelity, is assumed to be tunable by means of an additional input of the
simulator (e.g., a mesh size parameter): high-fidelity simulations provide more
accurate results, but are time-consuming. Using a limited computation-time
budget, we want to estimate, for any value of the physical inputs, the
probability that a certain scalar output of the simulator will exceed a given
critical threshold at the highest fidelity level. The problem is addressed in a
Bayesian framework, using a Gaussian process model of the multi-fidelity
simulator. We consider a Bayesian estimator of the probability, together with
an associated measure of uncertainty, and propose a new multi-fidelity
sequential design strategy, called Maximum Speed of Uncertainty Reduction
(MSUR), to select the value of physical inputs and the fidelity level of new
simulations. The MSUR strategy is tested on an example.
