Calculations of mathematical expectations of the type MV(w) are analyzed for the use of the Monte Carlo method. V is a functional in the space of continuous functions, w a Wiener process [cf. I. M. Gel’fand and A. M. Yaglom, Usp. Mat. Nauk 11, No.1(67), 77-114 (1956; Zbl 0071.224)]. It is possible to calculate MV(w) with the given accuracy without using the same degree of accuracy for the realization of w. The idea of more crude but specially selected approximation for w is discussed. For the approximate calculation the functionals V are taken in the space [0,T] of functions continuous from right and the piecewise constant function \(w^ h(t)\) is used (0\(\leq t\leq T)\) so that the accuracy at \(h\to 0\) \(MV(w)-MV(w^ h)=O(h^ 2)\). Two approaches were used to attain this accuracy: (a) by integration of stochastic differential equations by the Runge-Kutta method and (b) by Taylor expansion of smooth functionals of the form \(V(x)=\phi (x(T),\int^{T}_{0}f(t,x(t))dt.\) Proofs of (a) and (b) procedures are given.