## Length of extra inning games

Posted by Andy on November 6, 2007

A few days ago when I posted about run-scoring by inning, there was a cool blog post written about the length of games in extra innings. Read that one first, then come back here for some more data. To get a nice pool of data, I followed up by summing up all the extra inning games over the last 10 years (1998-2007):

Inning Games Fraction ------------------------------- 10 1998 0.468 11 1062 0.484 12 548 0.489 13 280 0.461 14 151 0.517 15 73 0.425 16 42 0.429 17 24 0.500 18 12 0.750 19 3 0.667 20 1 1.000

I got that data from the PI Team Innings Summary pages. Let's talk about the first two columns. The first one is the inning number, and the second tells you the number of games to reach a given inning. In other words, a total of 1,998 games went to extra innings from 1998 to 2007. Of those 1,998, a total of 1,062 went at least 11 innings, and 548 of those went at least 12 innings, etc. (That's a bit confusing due to the coincidence of talking about 1,988 different games and also the calendar year 1998. Also, just to be clear, the data does NOT say that 1,988 games went exactly 10 innings. It says that a total of 1.988 games went to extra innings since they went at least 10.)

Now, using data from consecutive lines means that we can determine the fraction of games that end after each inning. For example, 1,998 games went at least 10 innings, but only 1,062 went at least 11 innings. That means that of those 1,998 games that went at least 10, 936 ended in 10 innings (or during the 10th inning on a walk-off.) Dividing 936 into 1,998 tells us that 46.8% of games that went to the 10th ended in the 10th.

The third column shown us all those percentages. Looking all the way down the list, you can see that in the last 10 years, there was a 100% chance that a game that went to the 20th inning would end in the 20th inning, as no game has gone to a 21st inning.

Now, if you did your homework and read that external blog post I linked to, you'll know what I mean when I say that the length of extra innings is pretty close to being a geometric phenomenon. You can see that the likelihood of a game going one further inning is pretty constant. Taking a weighted average of the above data yields a factor of 48.4%. This means that at the start of any extra inning, there is a 48.4% chance that the game will end in that inning, and thus a 51.6% chance that the game will continue to the next inning. (These are, of course, average figures for the last 10 years' of game data.)

For a series to be truly geometric, it's necessary for the events of the series to be independent. In this case, it means that the likelihood for a team to win the game in any given inning must be independent of the likelihood for that team to win the game in the next inning. We know that in baseball, this isn't totally true. Let's suppose that a team uses its closer in the 10th inning. There's a reasonably good chance that the opposing team won't score. However, in the 11th they may be forced to bring in a middle reliever, against whom the likelihood of scoring is a lot higher. This means that the likelihood of winning from inning to inning is NOT independent, since the decision of which pitcher to use in one inning affects the decision of which pitcher to use in the next inning. It's also true that different batters will hit in consecutive extra innings, and the likelihood of scoring is not the same.

What does this all mean? Well nothing much, really. Managers still need to make good decisions about use of pitchers, pinch-hitters, pinch-runners, and possible use of sacrifices.

November 6th, 2007 at 1:39 pm

Also using PI's inning summary data and a few basic calcs we can see that on average a team coming to bat during 2007 was held scoreless in its half-inning at the plate 71% of the time. Generally speaking then, both teams will be held scoreless in an inning about 71% * 71% of the time, which comes out to 50.5% of the time. That presumably is the reason that in each inning of extra innings the chances are roughly 50-50 that the game will end. Of course there is the small possibility that the teams will both score and both score exactly the same number of runs, thus continuing the game, but that probability is pretty low (chances of both teams scoring exactly one run in an inning are about 2.5% and the chances of both teams scoring exactly the same number of runs at a number of runs above one is less than 1%) and will not have a major effect on the overall likelihood of games ending in each inning of extra innings.

November 6th, 2007 at 1:43 pm

Nice quickie analysis there, and it's consistent with my more detailed one. Of course, there are other factors in extra innings, including the fact that you're almost always going to play for one run, which can actually increase the chances of scoring 1 run (while decreasing the chances of scoring more than 1 run), plus the fact that you might not have any more pinch-hitters available in extra innings, etc.

November 6th, 2007 at 3:04 pm

Good points about those items that might affect the likelihood of teams being held scoreless in extra innings as opposed to ordinary innings. Based on PI's 2007 numbers, in extra innings the percentage of half innings a team was held scoreless went up a bit compared to the numbers for all innings combined. Teams were held scoreless in 74.6% of half-innings after inning 9, compared to 71% for all half-innings in general. Squaring 74.6% to get the chance of a full inning remaining scoreless results in a 55.5% likelihood of a full scoreless inning. Exactly one run was scored in 15.4% of half-innings overall in 2007 and 14.9% of half innings in extra innings. More than one run was scored in 13.5% of half-innings overall in 2007 and 10.5% of half innings in extra innings. So in 2007, essentially all of the small decline in half-innings with at least one run scored that occurred after the ninth inning was attributable to the decline in multi-run innings, while the portion of exactly one-run innings stayed the same after nine innings.

November 6th, 2007 at 4:45 pm

Nice. I would guess that the odds of one team in an extra inning scoring the exact same number of runs as the other team in that extra inning (including zero) is 71.8%. Squaring 0.718 yields 0.516, which are the odds that I calculated that any given extra inning would end still tied.

I love stuff like this--totally independent analysis yielding the same results. You might just be the man (or woman), birtelcom.