Time delays between lensed multiple images have been known to provide an interesting probe of the Hubble constant, but such an application is often limited by degeneracies with the shape of lens potentials. We propose a new statistical approach to examine the dependence of time delays on the complexity of lens potentials, such as higher order perturbations, nonisothermality, and substructures. Specifically, we introduce a dimensionless reduced time delay and explore its behavior analytically and numerically as a function of the image configuration, which is characterized by the asymmetry and opening angle of the image pair. In particular, we derive a realistic conditional probability distribution for a given image configuration from Monte Carlo simulations. We find that the probability distribution is sensitive to the image configuration such that more symmetric and/or smaller opening-angle image pairs are more easily affected by perturbations on the primary lens potential. On average time delays of double lenses are less scattered than those of quadruple lenses. Furthermore, the realistic conditional distribution allows a new statistical method to constrain the Hubble constant from observed time delays. We find that 16 published time delay quasars constrain H₀ to be 70 ± 6 km s^-1 Mpc^-1, where the value and its error are estimated using jackknife resampling. Systematic errors coming from the heterogeneous nature of the quasar sample and the uncertainty of the input distribution of lens potentials can be larger than the statistical error. After including rough estimates of important systematic errors, we find H₀ = 68 ± 6(stat.) ± 8(syst.) km s^-1 Mpc^-1. The reasonable agreement of the value of the Hubble constant with other estimates indicates the usefulness of our new approach as a cosmological and astrophysical probe, particularly in the era of large-scale synoptic surveys.