## Distribution of the batting order slot that ends a baseball game

The God Plays Dice blog has a nice piece attempting to model a baseball question. Baseball is wonderful for all kinds of mathematics questions, partly because the game has since its creation kept data about the plays made, partly because the game breaks its action neatly into discrete units with well-defined outcomes.

Here, Dr Michael Lugo ponders whether games are more likely to end at any particular spot in the batting order. Lugo points out that certainly we could just count where games actually end, since baseball records are enough to make an estimate from that route possible. But that’s tedious, and it’s easier to work out a simple model and see what that suggests. Lupo also uses the number of perfect games as a test of whether the model is remotely plausible, and a test like this — a simple check to whether the scheme could possibly tell us something meaningful — is worth doing whenever one builds a model of something interesting.

Tom Tango, while writing about lineup construction in baseball, pointed out that batters batting closer to the top of the batting order have a greater chance of setting records that are based on counting something – for example, Chris Davis’ chase for 62 home runs. (It’s interesting that enough people see Roger Maris’ 61 as the “real” record that 62 is a big deal.) He observes that over a 162-game season, each slot further down in the batting order (of 9) means 18 fewer plate appearances.

Implicitly this means that every slot in the batting order is equally likely to end the game — that is, that the number of plate appearances for a team in a game, mod 9, is uniformly distributed over {0, 1, …, 8}.

Can we check this? There are two ways to check it:

- 1. find the number of plate appearances in every game…

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