In the Monte Carlo (MC) method, statistical noise is usually present, and it may become dominant in the calculation of a distribution, usually by iteration, but it is less important in calculating integrals. The subject of the present work is the role of statistical noise in iterations involving stochastic simulation (the MC method). Convergence is checked by comparing two consecutive solutions in the iteration. The statistical noise may randomize or pervert the convergence. We study the probability of convergence and the correct estimation of the variance in a simplified model problem. We also study the statistical properties of the solution to a deterministic problem with a stochastic source obtained from a stochastic calculation. There are iteration strategies resulting in nonconvergence or a randomly stopped iteration.