Excess zeros are encountered in many empirical count data applications. We provide a new explanation of extra zeros, related to the underlying stochastic process that generates events. The process has two rates, a lower rate until the first event, and a higher one thereafter. We derive the corresponding distribution of the number of events during a fixed period and extend it to account for observed and unobserved heterogeneity. An application to the socio-economic determinants of the individual number of doctor visits in Germany illustrates the usefulness of the new approach.