A method is presented for the study of first-passage-time statistics in one-dimensional disordered media. We introduce a projection-operator formalism that allows the calculation of an averaged backwards master equation. A generalized Dynkin\'s equation is also presented. We apply this approach to the random-trap model. The exact mean first-passage time to leave a finite domain is calculated for weak disorder. To study strong disorder we introduce a mean-field-like approximation that gives exact results at long times. Long-time tails in the survival probability and in the first-passage-time distribution are found, predicting a divergent mean first-passage time for the case of strong disorder. Our findings confirm numerical results found previously.