In this paper the influence of stochasticity (i.e. a Gaussian white noise forcing) on the dynamic behavior of a selfsustained oscillator coupled to a non-linear energy sink is investigated. To this end, the standard stochastic averaging is used to compute the slow flow dynamics of the system. Preliminary results show that the reasoning which allows to predict the system behavior in the deterministic case can be contradicted in presence of stochasticity. Then, by means of the Monte Carlo method, the stochastic averaging procedure is validated. Finally, two quantities are introduced to highlight more precisely the special features of the stochastic system behavior compared to that of the deterministic system. These are the probability of being in a harmless regime and the First-Passage Time to reach a harmful regime which are computed and investigated combining again the Monte Carlo approach with numerical integrations of the slow flow dynamics. The results obtained show afresh that the stochastic forcing can modify significantly the dynamic behavior of the corresponding deterministic system. Indeed, when they are computed on the latter, the two quantities aforementioned have a discontinuity at the mitigation limit (i.e. the value of the bifurcation parameter under consideration below which the NES acts and above which it no longer acts) revealing an abrupt change of behavior of the coupled system. The paper shows that this typical characteristic of the deterministic system is lost in the presence of stochasticity, the stochastic system becoming smooth at the mitigation limit.