Nonlinear model predictive control is a popular control approach for highly nonlinear and unsteady state processes, which however can fail due to unaccounted uncertainties. This paper proposes to apply a sample-average approach to solve the general stochastic nonlinear model predictive control problem to handle probabilistic uncertainties. Each sample represents a nonlinear simulation, which is expensive. Therefore, variance-reduction methods were systematically compared to lower the necessary number of samples. The method was shown to perform well on a semi-batch bioreactor case-study compared to a nominal nonlinear model predictive controller. Expectation constraints were employed to deal with state constraints in this case-study, which take into account both magnitude and probability of deviations.