The time-dependent probabilities for the output of the second stage of the Borsellino and Fuortes model for explaining the random kinetics of single photon responses (quantum bumps) observed in Limulus photoreceptors are derived. The expressions differ from those of Borsellino and Fuortes, who incorporated an incorrect assumption into their mathematical treatment. The expressions for the second stage are complex and suggest that formulating exact mathematical predictions from the model is difficult. Therefore, the model was tested by computer simulation of quantum bump kinetics. When gain and time scale were distributed equally among the ten stages, simulated bumps did not resemble quantum bumps recorded from voltage-clamped ventral photoreceptors of Limulus. The distribution of gain and time scale among the stages was varied, as well as the number of stages, but improvement of the match between simulated and real bumps was not achieved. It is concluded that a simple series of gain-producing reactions of the sort modeled by Borsellino and Fuortes is unlikely to underlie the generation of quantum bumps. Alternate possibilities are discussed.