The improbable persuit of perfection

We are all aware that the Rays have been victims of "perfect games" twice in the past few years, and it seems that these events are taking place with an increasing frequency. MLB has gone as far as to call 2010 "the year of the pitcher," and guys like Ulbaldo Jiminez, Roy Halladay, and others are lights out this time around.

FanGraphs has a good article about The year of the pitcher that basically acknowledges that there has been some decline in offense in recent years, most notably in 2010, but offers very lukewarm conclusions as the author notes that the results are a "black box" and thus causation is merely speculative. Regardless, I find it fascinating that these events have occured with greater frequency in recent years:

• 2010 Halladay
• 2010 Braden (ugh!)
• 2009 Buerhle (ugh!!!!)
• 2004 Johnson
• 1999 Cone
• 1998 Wells
• 1994 Rogers

Prior to Rogers' gem, there were 13 previous "perfect games" in baseball history. This makes for a total of 20 perfect games in over 130 years of history, at a frequency of about a perfect game per six or so years, ASSUMING everything here is homogenous - years = years, there are no other factors than pure chance that determine a perfect game, etc.

A tremendous pitcher may post a BAA around .230. If he is a control artist who walks only 1 batter per 9 IP, he has a 1/27 = .037... Walking average against, for an on-base against average of maybe around .240 including hit batsmen and the like (errors, etc). This means that the probability that he records an out should be about .770, which is good for N = 1 trial or even N = 3 trials. However, a perfect game is 27 outs, and thus the chance of a perfect game for this pitcher should be about (0.770)^(27), which comes out as 0.000861. This means that an EXCELLENT pitcher has about a 0.08% chance of throwing a perfect game on any given start.

Continuing with liberal estimates, we may say that perhaps 10 of these types of pitchers exist in baseball every year, and they complete 350 starts total. Multiply 350 by 135 (number of years) by P, we get around 40.7 perfect games in history, which is about double of what has thus transpired. You get the picture.

What does this have to do with the Rays? Well, the Rays were victims of perfect games in consecutive years. Let's say that past performance != future results, meaning that we should expect about 41 perfect games rather than the 20 that have occured to this point.

Now, we test the hypothesis: Is this a mere fluke of variation, or is there something to it?

In two years, an MLB team should expect a mean of: 40.7 games / (30 teams * 77.5 years) = about 0.2 perfect games. Assuming normalness, you have a variance of about 0.2 * (1 - 40/(30*135))  or still pretty close to 0.2 perfect games. Given that 2 = 0.2 * 100, we are so many deviations off that P(t) is about 0.

Does this mean that the Rays suck? Does this mean that the perfect games were a harbringer of things to come?

No, not really at all. Let's examine further.

Although the probability that THE RAYS IN PARTICULAR would be perfect-gamed twice in two years is extremely small, the probability that TWO TEAMS in consecutive years would be perfect-gamed is much better. This is known as the "Birthday paradox" in intro probability theory. This, of course, assumes that there is no covariance between 2009 Rays and 2010 Rays, which we know not to be the case.

Another way of putting it: the chance that a fastball hits a bird mid-pitch on any given day is small; however, the chance that it occurs at all may be much larger. Think about "teams who are victims of perfect games" as outcomes in a hat. The chance that a particular team is chosen is 1/30, but there is 100% chance that some team is chosen given that there is a perfect game. Let's say for the sake of argument that over a 40 year time window we expect about 10 perfect games. We have 30 teams in the hat and 10 perfect games to hand out. Does this mean that every perfect game will be given to a different team?

P(All different) = 1/1 * 29/30 * 28/30 * 27/30 * 26/30 * 25/30 .... = 29....21/30^9 which is about .184, meaning that there is a greater than 80% chance that at least one lucky team will receive 2 or more of these gems. GO Rays!

The point is that the very infrequency of the perfect game makes the distribution of its victims somewhat staggered, rather than flat across the board, since as the saying goes "it has to happen to somebody." Statistics and probability theory are very useful in certain applications in baseball, but still struggle to tell "The whole story" when we deal with events that are subject to such variation, infrequency, and flat-out luck. Which is the truth? That the Rays are HORRIBLE because there is no way that 2 perfect games can be explained due to randomess? Or that the Rays are simply the victim of a likely occurence that would be handed out to somebody twice more often than not?