We investigate a nonequilibrium biochemical network described in terms of transit time distributions. We assume that the network is made up of two subsystems which are connected to each other as well as to the environment. We evaluate the probability distribution of the time necessary for a molecular species to cross the system, i.e., the distribution of the time that elapses from the moment the species enters one subsystem and leaves it by passing through the second subsystem. For this process the total probability of crossing the system is at the same time the probability of net transformation of a chemical in a desired product. The analysis leads to the surprising result that the probability of crossing the system increases with the average time spent by the species in one subsystem. A physical explanation of this apparent paradox is given by taking into account the multiple reflections occurring within the system. The escape from the system takes place after a random number of forward and backward transitions between the two subsystems. Although large transit times for the two subsystems correspond to very small individual crossing probabilities, they also favor a large number of reflections. The total crossing probability is made up by the sum of the contributions of the passage events taking place after different numbers of reflections. In this sum the low values of the different terms are outweighed by the large number of terms, resulting in an amplification of the efficiency of the passage process due to multiple reflections. The increase of the crossing probability takes place at the expense of slowing down the process. The average crossing time of the whole system increases linearly with the number of reflection events within the system. It is shown that such a phenomenon of amplification of the crossing probability may also occur in the case of thermally activated passage over a succession of energy barriers.