In many physical processes, there is uncertainty in the parameters which define the process and this input uncertainty is propagated through the equations of the process to its output. Experimental design is essential to quantify the uncertainty of the input parameters. If the process is simulated by a computer code, propagation of uncertainties is carried out through the Monte Carlo method by sampling in the input parameter distribution and running the code for each sample. It is then important to obtain information about the way in which the parameters are influential on the output of the process. This is useful in order to decide how to sample in the input space when propagating uncertainties and on which parameters experimental effort should be more concentrated. Here, we use dimensional and similarity analyses to reduce the dimension of the input variable space with no loss of information and profit from this reduction when propagating uncertainties by Monte Carlo. Using dimensional analysis, the output is expressed in terms of the inputs through a series of dimensionless numbers, a dimension reduction is achieved since there are less dimensionless numbers than original parameters. In order to minimize the uncertainty of the estimation of the output, propagation of uncertainties should be carried out by sampling on the space of the dimensionless numbers and not on the space of the original parameters. The purpose of this paper is an application of propagation of uncertainties to a code which simulates the interaction of metal drilling with a laser beam, where there exists uncertainty in the absorbed intensity of the beam and the density of the medium. By sampling in the reduced input space, a substantial variance reduction is achieved for the estimators of the mean, variance and distribution function of the output. Moreover, the output is found to depend on the intensity and the density through their quotient.