We investigate the reaction kinetics of hard-sphere reactants that undergo reaction upon collision. When the reaction probability at a given collision is unity, the Noyes rate theory provides an exact expression of the rate coefficient. For the general case with the reaction probability less than unity, Noyes assumed that successive recollision times between a tagged pair of reactants are decorrelated. We show that with this renewal assumption, the rate theory of Wilemski and Fixman yields the same rate coefficient expression as the Noyes theory. To evaluate the validity of the renewal assumption, we carry out molecular dynamics simulations. Contrary to the usual expectation, we find that the renewal assumption works better at higher particle densities. The present study shows that the rate coefficient for collision-induced hard-sphere reactions can be estimated with great accuracy by using the first recollision time distribution alone, regardless of the magnitude of the reaction probability at a given collision.