We present a simple, analytic description of the main neutrino production reactions in the GRB fireball model, including baryon resonances, direct pion exchange and multipion photo-production, taking into account the effects of pion and muon EM cooling. This analysis generalizes the Waxman Bahcall (WB) analysis, based on the Delta resonance approximation, while retaining its simple analytic description of the dependence on fireball model parameters. Our analysis reproduces the results of detailed numerical calculations performed for specific fireball parameters, and enables us to determine the uncertainties in neutrino production (which are not given by the numerical analyses) due to uncertainties in photo-production differential cross sections. Our approach requires the knowledge of only the photo-production cross section and 4-momentum distribution at photon energies lower than ~10GeV which are experimentally available, and may be generalized to describe other photo-production neutrino sources. We demonstrate that multi pion photoproduction contributes significantly to the neutrino emission, and therefore influences the flavor content, throughout the neutrino spectrum. Significant constraints on the GRB fireball model are expected to be obtained after several years of full IceCube operation.