Forecasts of the probability of a large earthquake occurring on a fault during a specific time interval assume that a probability distribution describes the interevent times between large earthquakes. However, current models have features that we consider unrealistic. In these models, earthquake probabilities remain constant or even decrease after the expected mean recurrence interval, implying that additional accumulated strain does not make an earthquake more likely. Moreover, these models assume that large earthquakes release all accumulated strain, despite evidence for partial strain release in earthquake histories showing clusters and gaps. As an alternative, we derive the necessary equations to calculate earthquake probabilities using the long‐term fault memory (LTFM) model. By accounting for partial strain release, LTFM incorporates the specific timing of past earthquakes, which commonly used probability models cannot do, so it can forecast gaps and clusters. We apply LTFM to the southern San Andreas fault as an example and show how LTFM can produce better forecasts when clusters and gaps are present. LTFM better forecasts the exceptionally short interevent time before the 1857 Fort Tejon earthquake. Although LTFM is more complex than existing models, it is more powerful because (unlike current models) it incorporates fundamental aspects of the strain accumulation and release processes causing earthquakes.