A new formulation aimed at quantifying uncertainties of Monte Carlo (MC) tallies such as keff and the microscopic reaction rates as well as nuclide number density estimates in MC depletion analysis is presented. It is shown that when the two major MC inputs - the microscopic cross sections and nuclide number densities - are assumed to have uncertainties, the variance of a given MC tally used as a measure of its uncertainty in this formulation arises from four sources: the statistical uncertainty of the MC tally, uncertainties of microscopic cross sections and nuclide number densities, and the cross correlations between them and the latter three contributions can be determined by computing correlation coefficients between uncertain variables. It is also shown that the variance of any given nuclide number density at the end of each depletion time step (DTS) stems from uncertainties of the nuclide number densities and microscopic reaction rates of nuclides at the beginning of each DTS, and they are determined by computing correlation coefficients between these two uncertain variables. The new formulation is incorporated into the Monte Carlo Code for Advanced Reactor Design (McCARD) of Seoul National University, and a McCARD depletion analysis for a U-TRU-Zr fuel assembly is performed to examine quantitatively the uncertainty propagation behavior of MC tallies such as k and the number densities of actinides as a function of DTS. The results demonstrate that the formulation is useful not only for quantifying the uncertainty propagation analysis in MC depletion analysis but also for identifying the types of nuclear cross-section data that need to be improved to obtain a more reliable incineration physics analysis of the transuranium fuel.