For this analysis, hit by pitch totals have been added to walks. This yields the following table of expected future runs using the OERA calculation. The same events files show the following distribution of inning states. That is, the listed state occurred that many times during the analyzed seasons. The careful reader will observe that the number of 0 out bases empty states exceeds the number of innings played.
Any state can occur more than once.
A home run with no outs creates another instance of the no out bases empty state. Besides the inning states, runs scored following each state to the end of the inning were also tabulated. Dividing the these runs totals by the number of times the inning state occurred provides the observed expected future runs from each inning state. Comparing this set of ratios to the ones computed from the Palmer EFR data shows considerable similarity. Interestingly, the numerical solution can easily be generalized to an arbitrary number of outs per inning.