Relaxation and diffusion data are often analyzed using a Laplace inversion algorithm that incorporates regularization. Regularization is used because Laplace inversion with finite and noisy data is an ill-conditioned problem for which many solutions exist for a given data set. This paper reports a different approach. Instead of finding a "best" solution by some ad hoc criterion, we developed an efficient Monte Carlo algorithm that generates thousands of probable solutions from which the statistical properties of the solution can be analyzed. We find that although all of the individual solutions are spiky, the mean solution spectrum is smooth and similar to the regularized solution. From the Monte Carlo solutions we obtain probability distributions for quantities derived from the spectrum, such as porosity and bound fluid. This ability to characterize the uncertainty of such quantities is novel.