A Monte Carlo simulation exploring uncertainties in standard stellar evolution theory on the red giant branch of metal-poor globular clusters has been conducted. Confidence limits are derived on the absolute V-band magnitude of the bump in the red giant branch luminosity function (M_v,b) and the excess number of stars in thebump, R_b. The analysis takes into account uncertainties in the primordial helium abundance, abundance of alpha-capture elements, radiative and conductive opacities, nuclear reaction rates, neutrino energy losses, the treatments of diffusion and convection, the surface boundary conditions, and color transformations. The uncertainty in theoretical values for the red giant bump magnitude varies with metallicity between +0.13/-0.12 mag at [Fe/H] = -2.4 and +0.23/-0.21 mag at [Fe/H] = -1.0$. The dominant sources of uncertainty are the abundance of the alpha-capture elements, the mixing length, and the low-temperature opacities. The theoretical values of M_v,b are in good agreement with observations. The uncertainty in the theoretical value of R_b is +/-0.01 at all metallicities studied. The dominant sources of uncertainty are the abundance of the alpha-capture elements, the mixing length, and the high-temperature opacities. The median value of R_b varies from 0.44 at [Fe/H] = -2.4$ to 0.50 at [Fe/H] = -1.0. These theoretical values for R_b are in agreement with observations.

30 pages, 6 figures. To appear in ApJ