With the new baseball season upon us, mathematicians have another chance to predict the future. By trying to pinpoint the apparently unpredictable in baseball, from the incidence of no-hitters to record-breaking home runs, mathematicians make use of old mathematics: the Poisson distribution.
The subject is one of a range of mathematical, scientific, and technological topics covered in the syndicated TV series Discoveries and Breakthroughs Inside Science (DBIS). The American Institute of Physics produces these science news programs, with the MAA as a contributing partner. The NSF-funded DBIS project delivers twelve 90-second segments each month for showing on local TV stations across the country.
One episode focused on baseball and the Poisson distribution. This distribution is at the heart of the effort to predict a rare baseball event: the no-hitter. It comes into play because mathematicians use it to identify the number of times an event occurs at random, independently in time, and at a constant rate.
The Poisson distribution goes back a long way, to the French mathematician Simeon-Denis Poisson. In 1837, Poisson developed it for a serious—and surprising—reason: as a model for the deliberations of juries. The distribution was rediscovered in 1898, when it was used to identify the number of soldiers in the Prussian cavalry who were likely to be kicked to death by horses.
Today, the Poisson distribution is used, for instance, in epidemiology, to determine how diseases spread through populations; in publishing, to predict the number of typos likely to be found on a printed page; and in manufacturing, to determine the number of defects one would expect in an automobile.
In baseball, the distribution can help baseball fans look forward to the rarer occurrences in the game of inches: the no-hitter, the triple play, and the number of batters who will hit for the cycle (a single, double, triple, and home run).
In the analysis of baseball statistics, the Poisson distribution allows us to predict the following. For 2007, in this era of the long ball, fans can expect that the season will bring us four triple plays, four players who hit for the cycle, and two no-hitters. Indeed, the statistics show that the first no-hitter could occur on or about game 730 of the season. DBIS Video.
But we don't know where these events will take place.
We do know, however, where the most home runs will be hit, thanks to mathematics and physics. At Denver's Coors Field, baseballs carry 10% farther than they do in other Major League ballparks. The reason is in the air, or the lack of. The stadium is a mile above sea level, where the air pressure is less, making the air "thinner."
At that elevation, baseballs will be hit farther. Home runs will be more frequent, and pitchers, unfortunately, will be less effective because their pitches will "break" less. DBIS Video.
To mitigate the air's impact at Coors Field, all baseballs are placed in a humidor before the games. Still, sluggers will reap the rewards of playing at Coors Field, as their teams hit 60% of their home runs at the ballpark.
At this stadium, the chances of a no-hitter or a triple play may be reduced, but the odds of hitting for the cycle increase.—H. Waldman
Huber, M., and A. Glen. 2007. Modeling rare baseball events—Are they memoryless? Journal of Statistics Education 15(1).
Schaffer, J.R., and E. Heiny. 2006. The effects of elevation on slugging percentage in Major League Baseball. Chance 19(1):28-34.