We use recent proper motion measurements of the tangential velocity of M31, along with its radial velocity and distance, to derive the likelihood of the sum of halo masses of the Milky Way and M31. This is done using a sample of halo pairs in the Bolshoi cosmological simulation of Lambda CDM cosmology selected to match the properties and the environment of the Local Group. The resulting likelihood gives an estimate of the sum of the masses of M-MW,M-200c + M-M31,M-200c = 2.40(-1.05)(+1.95) x 10(12) M-circle dot (90% confidence interval). This estimate is consistent with individual mass estimates for the Milky Way and M31 and is consistent, albeit somewhat on the low side, with the mass estimated using the timing argument. We show that although the timing argument is unbiased on average for all pairs, for pairs constrained to have radial and tangential velocities similar to that of the Local Group the argument overestimates the sum of masses by a factor of 1.6. Using similar technique, we estimate the total dark matter mass enclosed within 1 Mpc from the Local Group barycenter to be M-LG(r < 1 Mpc) = 4.2(-2.0)(+3.4) x 10(12) M-circle dot (90% confidence interval).