Braden, Oddly

This post will run on my website tomorrow. I thought Athletics Nation would be particularly interested since it concerns a recent feat by a pitcher named Braden, so I'm previewing it here.

Dallas Braden threw a perfect game in Oakland on Sunday. Within a few hours, analysis started to pop up around the web. Jack Moore of FanGraphs was one of the first on the topic, looking into the chance of Braden's feat and finding it to be about a one in 66,000. Probability is a tricky thing, and with so many variables like park factors, the defense, the batting team, and slight changes in a pitcher's skill over the years, it is not really feasible to come up with accurate numbers. However, with some very imprecise estimates, I will check Moore's work and add additional thoughts on the general probability of perfect games.

Moore used approximations in his calculation, and he left out hit batsmen. A perfect game involves 27 straight outs; to calculate its probability, it is necessary at some point to raise to the 27th power, so tiny differences really add up. I added up the hits, walks, hit batsmen, and bases reached on error that Braden had allowed in his major league career before Sunday, for a total of 467. Before Sunday, he had faced 1371 batters. Divide the first by the second, and this results in a career .34 probability of a batter reaching base against Braden. This means there was about a .66 chance that he would retire each batter, and raising .66 to the 27th power gives the probability that he would, on command, do it 27 times in a row.

I kept the fractions intact before raising to the 27th, and the calculation yielded a one in 76,000 chance that Braden would throw a no-hitter on any particular day. What does this mean? If you showed up at the game on Sunday thinking to yourself, "I wonder how likely it is that Braden will throw a no-hitter today," then one in 76,000 would give you a very rough answer. There were of course other factors at play -- the game was at the Oakland Coliseum, a pitcher-friendly park; the opposing team was the Rays, not a pitcher-friendly club; and so on. We do know that the chances were pretty slim.

However, asking the question differently results in a very different answer. If you wondered on opening day about the chances that Braden would throw a perfect game at some point during the season, the probability would have been much higher. Most of Braden's preseason projections pegged him for somewhere between 20 and 30 starts, so I will go with about 25 starts, which accounts for the possibility of injury. I will also assume that Braden is capable of facing 27 batters every time he takes the mound. He may get the quick hook once in a while due to bad performance, but he at least has the capability of lasting 27 batters. In each of these 25 starts he has a one in 76,000 chance of a perfect game. We need to subtract that fraction from one to get the odds of a perfect game not happening on a particular day; raise the result to the 25th power to get the odds of no perfect game all season; subtract from one again to get the odds that a perfect game will occur. The final result is a one in 3,000 chance of a perfect game at some point over the course of the whole season -- still a long shot, but much more likely due to the lack of specificity.

This demonstrates a problem with probability calculations that occur after the fact, as we tend to ask questions regarding the chances of the event happening at the specific time that it occurred. But a perfect game would be impressive at any time, not just this past Sunday, so stretching out the inquiry to a fixed interval gives us a better intuitive idea of the likeliness of the feat. It would be ideal to pinpoint the chances of it occurring at any point in Braden's career. However, pitcher skill changes greatly over time, and career length is also difficult to predict. These factors would need to be part of any career-based calculation.

Braden's odds clearly were long, but how does he measure up to the rest of the league? Since 2010 is just getting started, I gathered the MLB-wide numbers from 2009. They are available at Baseball Reference: 43,524 hits, 16,620 walks, 1,590 hit batsmen, and 1679 bases reached on error, in 187,079 plate appearances. The same calculation as above -- adding up the bases reached, dividing by plate appearances, and raising to the 27th power -- yields one in 71,000 odds. This is the approximate probability that, for a starter who performs at league average, a perfect game will happen on a given day. Again, I am assuming that all starters have the capability of facing 27 hitters. I took the 4,860 starts that occur in a season and adjusted down by a factor of .95 to account for the chance the pitcher leaves early due to injury. I used these two numbers -- one in 71,000 odds and 4617 starts -- performed the same calculation that I did with Braden, and found a one in 16 chance of a perfect game by some pitcher in 2009.

This probability does not look quite right, since there have been 10 perfect games in the past 30 years alone, meaning we should expect the underlying probability to be close to one in three (using the past 30 years as a sample and going by the standard error, it is fairly likely that the probability is somewhere between one in two and one in six). However, as I mentioned earlier, all of these calculations are extremely rough, and missing by a factor of five is not bad. The distribution of hitting and pitching talent may also have an effect. Some teams are easier to shut down than others, and some pitchers stand far above the rest of the league in their ability to make outs. A more sophisticated analysis could probably get closer to true perfect game probability.

One more question. Casting aside actual occurrences, who in history had the best chance of a perfect game? A good guess might be Pedro Martinez, who from 1999 through 2002 decimated hitters to a ridiculous degree. He faced 2,895 batters over that span, striking out more than 34 percent of them and allowing just over 25 percent to reach base, including via the error. Running the original calculation on his numbers results in a one in 2,600 chance of a perfect game in any particular start. He started 106 games during that period, but considering he was injured a few times, I will round down to 100. The final result is about a one in 27 chance that Pedro would have thrown a perfect game at some point over those 100 starts. He did not, of course ... except that technically, he did a few years earlier.

Pedro's 1999-2002 seasons, and particularly his 2000 season, represented one of the greatest runs of pitching dominance of all time. Despite that, his chances of pitching a perfect game in that period were no better than one's chances of picking the World Series champions out of a hat. To repeat the obvious, it simply is not an easy feat to accomplish. Whether it is one in 76,000 for a single start, one in 3,000 for a whole season, or one in whatever for a career, before Sunday, Braden was incredibly unlikely ever to put a perfect game on his resume. Now, after that magical day, he will have a well-deserved place in the Hall of Fame.

Thanks for reading. I also write about baseball in general at Ball Your Base.