# How rare are eighteen-inning games, really?

What’s More Improbable: An 18-Inning Playoff Game Or A 13-Inch Penis?, from Deadspin’s Regressing blog, on sports statistics. Ross Benes points out that postseason baseball games are on average 9.22 innings with a standard deviation of 0.79, so the 18-inning Nationals-Giants game is about eleven standard deviations from the mean, or as rare as the title phallus.

But this should raise a red flag – an eleven sigma event, in a normally distributed population, should never happen. (Doing the arithmetic in my head, roughly one time in 1027, and there haven’t been more than a couple thousand playoff games.). Of course the culprit is that game lengths are not normally distributed. As Darren Glass and Philip Lowry have written, game lengths are actually modeled well by a quasi-geometric distribution. They claim that the probability that a game is still tied after $n$ is $Tk^{n-9}$, where T is the probability of a game being tied after nine innings (about 0.103) and k is the probability of both teams scoring the same number of runs in a given inning (about 0.556). The basic idea is that once you finish nine innings, each inning is an independent Bernoulli trial. (Think “weighted coin flip” except that weighted coins don’t exist.). Under this model, the probability of a game being tied after seventeen innings (and therefore going at least eighteen) is $0.103 (0.556)^8$, or about 0.00094. Just under one in a thousand. There have been perhaps fifteen hundred postseason games in history, so the fact that it’s taken this long for a one in a thousand event to occur is not all that surprising.