We use the very large Millennium Simulation of the concordance Λ cold dark matter cosmogony to calibrate the bias and error distribution of Timing Argument estimators of the masses of the Local Group and of the Milky Way. From a large number of isolated spiral–spiral pairs similar to the Milky Way/Andromeda system, we find the interquartile range of the ratio of timing mass to true mass to be a factor of 1.8, while the 5 and 95 per cent points of the distribution of this ratio are separated by a factor of 5.7. Here, we define true mass as the sum of the ‘virial’ masses, M200, of the two dominant galaxies. For present best values of the distance and approach velocity of Andromeda, this leads to a median likelihood estimate of the true mass of the Local Group of 5.27 × 1012 M⊙ or log MLG/M⊙= 12.72, with an interquartile range of [12.58, 12.83] and a 5–95 per cent range of [12.26, 13.01]. Thus, a 95 per cent lower confidence limit on the true mass of the Local Group is 1.81 × 1012 M⊙. A timing estimate of the Milky Way's mass based on the large recession velocity observed for the distant satellite Leo I works equally well, although with larger systematic uncertainties. It gives an estimated virial mass for the Milky Way of 2.43 × 1012 M⊙ with a 95 per cent lower confidence limit of 0.80 × 1012 M⊙.