Let Å be an arbitrary matrix and let A be a slight random perturbation of Å. We prove that it is unlikely that A has a large condition number. Using this result, we prove that it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we show that the smoothed precision necessary to solve Ax = b, for any b, using Gaussian elimination without pivoting is logarithmic. Moreover, when Å is an all-zero square matrix, our results significantly improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 18 (1997), pp. 499-517).