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A Monte Carlo simulation exploring uncertainties in standard stellarevolution theory on the red giant branch of metal-poor globular clusters hasbeen conducted. Confidence limits are derived on the absolute V-band magnitudeof the bump in the red giant branch luminosity function (M_v,b) and the excessnumber of stars in thebump, R_b. The analysis takes into account uncertaintiesin the primordial helium abundance, abundance of alpha-capture elements,radiative and conductive opacities, nuclear reaction rates, neutrino energylosses, the treatments of diffusion and convection, the surface boundaryconditions, and color transformations.  The uncertainty in theoretical values for the red giant bump magnitude varieswith metallicity between +0.13/-0.12 mag at [Fe/H] = -2.4 and +0.23/-0.21 magat [Fe/H] = -1.0$. The dominant sources of uncertainty are the abundance of thealpha-capture elements, the mixing length, and the low-temperature opacities.The theoretical values of M_v,b are in good agreement with observations. Theuncertainty in the theoretical value of R_b is +/-0.01 at all metallicitiesstudied. The dominant sources of uncertainty are the abundance of thealpha-capture elements, the mixing length, and the high-temperature opacities.The median value of R_b varies from 0.44 at [Fe/H] = -2.4$ to 0.50 at [Fe/H] =-1.0. These theoretical values for R_b are in agreement with observations.Comment: 30 pages, 6 figures. To appear in Ap

Theoretical Uncertainties in Red Giant Branch Evolution: The Red Giant Branch Bump