The Forward Search Experiment (FASER) at CERN's Large Hadron Collider (LHC) has detected the first collider neutrinos, opening a new field of TeV laboratory neutrino physics. This study updates previous simulations and presents theoretical predictions for neutrino fluxes and cross sections, along with their uncertainties. The neutrino fluxes are simulated using a fast neutrino flux simulation that models the production of forward particles from light and charm hadrons. The neutrino interactions are described using the Bodek-Yang model implemented in GENIE, with uncertainties defined by the spread in predictions from different event generators. The expected neutrino event rates for electron, muon, and tau neutrinos in LHC Run 3 and Run 4 are presented, with uncertainties dominated by the uncertainty in charm hadron production. The results show that Run 3 will provide sufficient statistics to distinguish contributions from different parent hadrons, and Run 3 + Run 4 will significantly enhance the number of tau neutrino events observed. The study also discusses the potential for future measurements, including the study of very high-energy neutrinos, forward hadron production, and related topics.The Forward Search Experiment (FASER) at CERN's Large Hadron Collider (LHC) has detected the first collider neutrinos, opening a new field of TeV laboratory neutrino physics. This study updates previous simulations and presents theoretical predictions for neutrino fluxes and cross sections, along with their uncertainties. The neutrino fluxes are simulated using a fast neutrino flux simulation that models the production of forward particles from light and charm hadrons. The neutrino interactions are described using the Bodek-Yang model implemented in GENIE, with uncertainties defined by the spread in predictions from different event generators. The expected neutrino event rates for electron, muon, and tau neutrinos in LHC Run 3 and Run 4 are presented, with uncertainties dominated by the uncertainty in charm hadron production. The results show that Run 3 will provide sufficient statistics to distinguish contributions from different parent hadrons, and Run 3 + Run 4 will significantly enhance the number of tau neutrino events observed. The study also discusses the potential for future measurements, including the study of very high-energy neutrinos, forward hadron production, and related topics.