Guaranteeing safe constraint satisfaction in nonlinear control systems with uncertainty remains a major challenge for control. The most successful control method handling constraints under uncertainty has without doubt been model predictive control (MPC). In particular, recent sampling-based MPC methods have shown success in controlling stochastic systems with complex, nonlinear dynamics. The sampling-based schemes are appealing since they do not need strong assumptions on the underlying model, except that it can be forward simulated. At the same time, the lack of major assumptions on the models make the statement of safety or robustness guarantees difficult. However, the samples drawn during the control process inherently contain probabilistic information about these properties. In this paper, we formally describe the problem that results by adding chance constraints to a sampling-based MPC scheme. Furthermore, based on a variant of the Chernoff bound, we derive trust regions, in which the sampling based estimation of the safety constraint satisfies a specified quality. Finally, we present a case study in the navigation domain to demonstrate the applicability of the proposed approach.