Through Major League Baseball's first 134 years, 1876-2009, some of its most interesting and uncommon events have been the 260 no-hitters (18 of which have been perfect games (1,2)). In 2010, pitchers threw six no-hitters, two of which (and almost a third) were perfect. In this paper, we investigate whether simple mathematical models can explain the frequency of perfect games and no-hitters over the years. We also investigate whether the pitchers who actually pitched the perfect games were those who "should have been expected" to do so.
From 1876 through 2009, pitchers threw 18 perfect games. Each was achieved by a different pitcher and only once before 2010 (way back in 1880) did two perfect games occur in the same year (see Table 1). Of these perfect games, 17 came during the regular season. In this paper, we only consider regular-season events.
Possibly the simplest approach to modeling the occurrence of perfect games is to treat all seasons, all pitchers, and all batters alike. Given this seemingly unrealistic assumption, one may ask, how many perfect games should have been pitched?
Over the first 134 years of Major League Baseball history, the overall on-base percentage (OBP) has been approximately 0.3279, (3) meaning that in about 1/3 of plate appearances, the batter reached base. Yet, in order to pitch a perfect game, a starting pitcher must retire the 27 consecutive hitters he faces. The probability of pitching an out is (1-OBP), and so the probability of pitching a perfect game is [(1-OBP).sup.27].
In general, therefore, the number of perfect games to be expected according to this analysis is:
Perfect games = 2 x Number of games x [(1 - OBP).sup.27] (1)
The reason for the "2" is that either team in a game may pitch a perfect game. 195,177 regular season games were played from 1876-2009, so the number of perfect games to be expected from 1876-2009 is 195,177 x 2 x [(1-.3279).sup.27] = 8.55, just half of the 17 observed.
One can approach this matter in the opposite way and compute the OBP needed in order to obtain the result of 17 perfect games. Solving equation (1) for OBP, we have
OB = 1 - [(Perfect Games / 2 x Number of games).sup.1/27]
This leads to a 0.3106 OBP. From the perspective of the OBP, a difference of 0.0173 (that is, .3279 - .3106), or about 5 % of OBP value, can account for the difference between the observed number of perfect games (17) and the number expected from this simple model (8.55). This demonstrates the sensitivity of the expected number of perfect games to variations in OBP. We present in Graph 1 the relationship between OBP and the expected number of perfect games. As OBP increases, more batters get on base and the likelihood of a perfect game shrinks.
We note that OBP has ranged from a low of 0.267 in 1880 to a high of 0.379 in 1894. If these values persisted through the 134...