We study the integration problem for the generalized Hölder class \(B(H_{\Omega }^{k}([0,1]^{d}))\) , which is determined by a generalized modulus of smoothness \(\Omega \) , in the restricted Monte Carlo setting. We obtain the exact order of the minimal randomized error for this class by using n function values. Moreover, we derive the exact order \(d\log _{2}n\) of the minimal number of the random bits, achieving this error. We apply our general results to some important cases. In particular, we discuss the case \(\Omega (t)=t^{a}\left( \log _{2}\left( 2+\frac{1}{t}\right) \right) ^{b}\) with \(0<a<k\) and \(b\in \mathbb {R}\) and obtain the corresponding error and complexity of the restricted Monte Carlo method.
