Forward and inner probabilities not only subsume the prefix and string probabilities, they are also straightforward to compute during a run of Earley's algorithm. In fact, if it weren't for left-recursive and unit productions their computation would be trivial. For the purpose of exposition we will therefore ignore the technical complications introduced by these productions for a moment, and then return to them once the overall picture has become clear.
During a run of the parser both forward and inner probabilities will
be attached to each state, and updated incrementally as new
states are created through one of the three types of transitions.
Both probabilities are set to unity for the initial state
.
This is consistent with the interpretation that the initial state is
derived from a dummy production 
for which no alternatives exist.
Parsing then proceeds as usual, with the probabilistic computations detailed below. The probabilities associated with new states will be computed as sums of various combinations of old probabilities. As new states are generated by prediction, scanning and completion, certain probabilities have to be accumulated, corresponding to the multiple paths leading to a state. That is, if the same state is generated multiple times, the previous probability associated with it has to be incremented by the new contribution just computed. States and probability contributions can be generated in any order, as long as the summation for one state is finished before its probability enters into the computation of some successor state. Appendix B.2 suggests a way to implement this incremental summation.