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Majorly Perfect Baseball

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It is almost here. Can you hear the crowds assembling? Can you taste the peanuts and cracker jacks? Can you see the pitchers warming up in the bullpen? The umpires are about to yell "Play ball!"

There is always anticipation in the air as opening day of the baseball season approaches. "Could this be the year my team wins it all?" "How is the bullpen going to look this year?" "Will this year's starting rotation be as good as last year's group?"

There is one statistic from the 2012 season that may be difficult to improve upon in 2013: the number of perfect games pitched in the major leagues. A perfect game is one in which a pitcher (or group of pitchers) does not allow any batter from the opposing team to reach base (27 batters up, 27 batters down). Perfect games, as you might expect, are rare. Of all the major league games played since 1900, a mere 21 of them have been perfect games. And this is where it gets interesting. The 2012 season alone witnessed three perfect games! A full one-seventh of all perfect games since 1900 took place in that single season.

Baseball great Mickey Mantle, who had many big moments in his career, knew how special perfect games were. He said, "The biggest game I ever played in was probably Don Larsen's perfect game." Mantle was referring to pitcher Larsen's perfect game in Game 5 of the 1956 World Series. Famed pitcher Cy Young, who himself tossed a perfect game in 1905, knew that keeping the other team from hitting required special circumstances: "A pitcher's got to be good, and he's got to be lucky."

Considering that luck is involved in these rare perfect games, how special was it that three such games occurred in the same season? To answer this question, we turn to the field of probability to help us quantify luck.

Let's make a few simplifying assumptions. First, let us assume that the probability of a perfect game occurring is fixed for each game. In reality, this is most certainly not the case since abilities of pitchers and the batters they face vary significantly from game to game. For the purpose of estimating probability, though, this simplifying assumption is helpful.

Legendary baseball sage Yogi Berra (who incidentally was Larsen's catcher in that perfect World Series game) famously said, "Baseball is 90 percent mental, and the other half is physical." Faulty math aside, Yogi's point was that a player's ability to perform well is related to his mental state at that point in the season. For our analysis, let's pretend that Yogi was wrong -- that is, let's say that the games are independent of one another (what happens in one game has no effect on what happens in other games).

Now, what probability should we assign to pitching a perfect game? Pitcher David Cone pitched a perfect game in 1999. There have been approximately 32,000 games played since that game in 1999, and seven of them have been perfect games. So it seems reasonable that we can think of the probability of a perfect game as p = 7/32000 (or about 0.00022). Using this value of p, we can calculate that the probability of seeing three or more perfect games in a single season is around 0.017. That is, there is a 1.7 percent chance of seeing three or more perfect games in a single season (under our assumptions). Pretty special, we'd say!

Each major league regular season includes n = 2430 games (over all teams). Using n = 2430 and p=.00022, we can compute the expected number of perfect games in a single season to be np = (2430)(.00022) = .5346. This would translate to about one perfect game every two seasons. This number is a bit high, since, on average, there has been one perfect game every five years since 1900.

Perhaps there is a reason for this, though. Maybe the game is evolving in an interesting way. If the 2013 season is a multiple perfect game season, it will be interesting indeed.

There is certainly something special about perfect games. They are part of the magical lore of the game. David Cone's 1999 perfect game took place on a special "Yogi Berra" day at Yankee Stadium. The ceremonial first pitch was thrown out that day by none other than Don Larsen himself, with Yogi catching it behind the plate. After the game, Cone said, "You probably have a better chance of winning the lottery than this happening, but what an honor."

Now, let's play ball!

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