The two-mode dye laser exhibits a quasibistable behavior wherein the lasing switches alternately between the modes at random intervals. This behavior can be described in terms of a first-passage-time (FPT) formalism. A Langevin model of the system has been the basis for most analytical approximations to the problem and here the results of detailed, four-dimensional, Monte Carlo simulations are presented. These results are compared to the various analytical and experimental results available. Attention is paid to the differences in alternative FPT problems and especially to problems involved in comparison with experimental data. One recent major advance in modeling the dye laser has been the introduction of pumping fluctuations into the Langevin equations. Effects of the introduction of pumping fluctuations on the FPT problem are explored and comments are made on previous attempts to account for their presence by the use of averaging techniques.