We present new expressions for the cumulative reaction probability (N(E)), cast in terms of time-correlation functions of reactant and product wave packets. The derivation begins with a standard trace expression for the cumulative reaction probability, expressed in terms of the reactive scattering matrix elements in an asymptotic internal basis. By combining the property of invariance of the trace with a wave packet correlation function formulation of reactive scattering, we obtain an expression for N( E) in terms of the correlation matrices of incoming and outgoing wave packets which are arbitrary in the internal coordinates. This formulation, like other recent formulations of N(E), allows calculation of the quantum dynamics just in the interaction region of the potential, and removes the need for knowledge of the asymptotic eigenstates. However, unlike earlier formulations, the present formulation is fully compatible with both exact and approximate methods of wave packet propagation. We illustrate this by calculating N(E) for the collinear hydrogen exchange reaction, both quantally and semiclassically. These results indicate that the use of wave packet cross-correlation functions, as opposed to a coordinate basis and flux operators, regularizes the semiclassical calculation, suggesting that the semiclassical implementation described here may be applied fruitfully to systems with more degrees of freedom.