This appendix presents elastic-rebound-theory (ERT) motivated time-dependent probabilities, conditioned on the date of last earthquake, for the segmented type-A fault models of the 2007 Working Group on California Earthquake Probabilities (WGCEP). These probabilities are included as one option in the WGCEP?s Uniform California Earthquake Rupture Forecast 2 (UCERF 2), with the other options being time-independent Poisson probabilities and an ?Empirical? model based on observed seismicity rate changes. A more general discussion of the pros and cons of all methods for computing time-dependent probabilities, as well as the justification of those chosen for UCERF 2, are given in the main body of this report (and the 'Empirical' model is also discussed in Appendix M). What this appendix addresses is the computation of conditional, time-dependent probabilities when both single- and multi-segment ruptures are included in the model. Computing conditional probabilities is relatively straightforward when a fault is assumed to obey strict segmentation in the sense that no multi-segment ruptures occur (e.g., WGCEP (1988, 1990) or see Field (2007) for a review of all previous WGCEPs; from here we assume basic familiarity with conditional probability calculations). However, and as we?ll see below, the calculation is not straightforward when multi-segment ruptures are included, in essence because we are attempting to apply a point-process model to a non point process. The next section gives a review and evaluation of the single- and multi-segment rupture probability-calculation methods used in the most recent statewide forecast for California (WGCEP UCERF 1; Petersen et al., 2007). We then present results for the methodology adopted here for UCERF 2. We finish with a discussion of issues and possible alternative approaches that could be explored and perhaps applied in the future. A fault-by-fault comparison of UCERF 2 probabilities with those of previous studies is given in the main part of this report.