This paper develops Bayesian estimation and prediction, for a mixture of Weibull and Lomax distributions, in the context of the new life test plan called progressive first failure censored samples. Maximum likelihood estimation and Bayes estimation, under informative and non-informative priors, are obtained using Markov Chain Monte Carlo methods, based on the symmetric square error Loss function and the asymmetric linear exponential (LINEX) and general entropy loss functions. The maximum likelihood estimates and the different Bayes estimates are compared via a Monte Carlo simulation study. Finally, Bayesian prediction intervals for future observations are obtained using a numerical example.