Right, I hear you. The number of outs per inning is always 3. So why ask how the number of outs per inning affects run production?
Well there was likely a time, somewhere in the 1800’s, when the number of outs per inning wasn’t decided for sure. What if those crafting the game of baseball had settled on 4-out innings instead of 3-out innings?
What if those changing the game of baseball today were to settle on 4-out innings instead of 3-out innings? It may not be much more drastic than some of the changes that have been proposed.
Would 4-out innings have 4/3 the number of runs scored as 3-out innings? If you think about it, you may find some reasons why the answer is no. But what do those reasons amount to, in terms of actual numbers of runs?
There is a run scoring model that provides a prediction of what they would amount to. It’s called Expected Binomial Production (EBP). It’s nothing special – it’s middle-of-the-pack in terms of run-predicting accuracy, compared to the most well known run scoring models such as Base runs, Extrapolated Runs, and Estimated Runs Produced. But there is a key difference. Those other models were constructed with the help of empirical data, and all that empirical data is from games played with three-out innings. They therefore cannot make predictions about games played with any other number of outs per inning. EBP, on the other hand, is 100% derived, using no empirical data, and thus can be adjusted for any number of outs.
EBP makes some simplifying assumptions. Baserunners never advance, except by a hit. Baserunners never make outs, either. It turns out that this run-decreasing assumption pretty much neutralizes this run-increasing assumption, leaving EBP with little overall bias when applied to major league baseball games. This may not be the case for amateur or minor league baseball.
Because of these simplifying assumptions, EBP will not reflect how more or less conservative baserunning, or increased numbers of double plays, will affect runs per inning when the number of outs per inning is changed. It solely reflects the change in runs scored due to fewer runners being stranded because they had extra opportunities to score.
The EBP formulas take as inputs team on base percentage, the team’s “power profile” (ratios of home runs to triples to doubles, etc.), and percentages of baserunners who take an extra base on a hit. When using modern numbers for those things, EBP predicts that in a 4-out inning, instead of increasing by a factor of 4/3 over 3-out innings, runs per inning increases by an additional 16%. Put another way, runs scored per out is increased by 16% when you switch from 3-out innings to 4-out innings, due to some runners that would have been stranded instead scoring.
Here is a full introduction to Expected Binomial Production. However, more relevant to this post is this discussion of how the formula changes when numbers of outs per innings are changed, and this discussion of the meaning of the three factors that comprise the basic EBP formula.