## Justin Verlander, Second No-Hitter of 2011May 8, 2011

Posted by tomflesher in Baseball.
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Yesterday, the Tigers’ Justin Verlander threw 2011’s second no-hitter, blanking the Blue Jays on a phenomenal 9-0 win. Verlander’s line was 9.0 innings pitched, 0 runs, 0 hits, 0 errors, 1 walk, 4 strikeouts, 108 pitches (74 for strikes) and a game score of 90. The Jays’ Ricky Romero took the loss.

It was Verlander’s second no-hitter. Although Bleacher Report’s Josh Rosenblat suggests that this is the start of a second “Year of the Pitcher,” I would be careful about making that conclusion. Remember that no-hitters are probably poisson distributed, so we should expect clumps of them. If pitchers keep it up, I’ll start crunching more numbers, but for now it’s just an odd coincidence.

## Matt Garza, Fifth No-Hitter of 2010July 26, 2010

Posted by tomflesher in Baseball.
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Tonight, Matt Garza pitched the fifth no-hitter of 2010. He joins Edwin Jackson, Roy Halladay, Dallas Braden, and Ubaldo Jimenez in the Year of the Pitcher club.

As I pointed out when Jackson hit his no-hitter, no-hit games are probably Poisson distributed. Let’s update the chart.

The Poisson distribution has probability density function

$f(n; \lambda)=\frac{\lambda^n e^{-\lambda}}{n!}$

Maintaining our prior rate of 2.45 no-hitters per season, that means $\lambda = 2.45$. Our function is then

$f(n; \lambda = 2.5)=\frac{2.45^n (.0864)}{n!}$

The probabilities remain the same:

 n p cumulative 0 0.0863 0.0863 1 0.2114 0.2977 2 0.2590 0.5567 3 0.2115 0.7683 4 0.1296 0.8978 5 0.0635 0.9613 6 0.0259 0.9872 7 0.0091 0.9963 8 0.0028 0.9991 9 0.0008 0.9998 10 0.0002 1.0000

And though the expectation (E(49)) and cumulative expectation (C(49)) remain the same, the observed values shift slightly:

 E(49) Observed C(49) Total 4.23 5 4.23 5 10.36 11 14.59 16 12.69 8 27.28 24 10.36 17 37.65 41 6.35 1 43.99 42 3.11 5 47.10 47 1.27 1 48.37 48 0.44 0 48.82 48 0.14 1 48.95 49 0.04 0 48.99 49 0.01 0 49.00 49

The tailing observations (say, for 4+ no-hitters) don’t quite match the expected frequencies, but the cumulative values match quite nicely. There might be some unobserved variables that explain the weirdness in the upper tail. Still, cumulatively, we have 47 seasons with 5 or fewer no-hitters, which is almost exactly what’s expected. This is unusual, but not outside the realm of statistical expectation.

## Edwin Jackson, Fourth No-Hitter of 2010June 25, 2010

Posted by tomflesher in Baseball, Economics.
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Tonight, Edwin Jackson of the Arizona Diamondbacks pitched a no-hitter against the Tampa Bay Rays. That’s the fourth no-hitter of this year, following Ubaldo Jimenez and the perfect games by Dallas Braden and Roy Halladay.

Two questions come to mind immediately:

1. How likely is a season with 4 no-hitters?
2. Does this mean we’re on pace for a lot more?

The second question is pretty easy to dispense with. Taking a look at the list of all no-hitters (which interestingly enough includes several losses), it’s hard to predict a pattern. No-hitters aren’t uniformly distributed over time, so saying that we’ve had 4 no-hitters in x games doesn’t tell us anything meaningful about a pace.

The first is a bit more interesting. I’m interested in the frequency of no-hitters, so I’m going to take a look at the list of frequencies here and take a page from Martin over at BayesBall in using the Poisson distribution to figure out whether this is something we can expect.

The Poisson distribution takes the form

$f(n; \lambda)=\frac{\lambda^n e^{-\lambda}}{n!}$

where $\lambda$ is the expected number of occurrences and we want to know how likely it would be to have $n$ occurrences based on that.

Using Martin’s numbers – 201506 opportunities for no-hitters and an average of 4112 games per season from 1961 to 2009 – I looked at the number of no-hitters since 1961 (120) and determined that an average season should return about 2.44876 no-hitters. That means

$\lambda = 2.44876$

and

$f(n; \lambda = 2.44876)=\frac{2.44876^n (.0864)}{n!}$

Above is the distribution. p is the probability of exactly n no-hitters being thrown in a single season of 4112 games; cdf is the cumulative probability, or the probability of n or fewer no-hitters; p49 is the predicted number of seasons out of 49 (1961-2009) that we would expect to have n no-hitters; obs is the observed number of seasons with n no-hitters; cp49 is the predicted number of seasons with n or fewer no-hitters; and cobs is the observed number of seasons with n or fewer no-hitters.

It’s clear that 4 or even 5 no-hitters is a perfectly reasonable number to expect.

 2.44876