Know your no-nos

Were you watching Ubaldo Jimenez’s no-hitter a few weeks ago? At what point did you begin to think that he could actually do it? In the third inning? Fifth? Seventh? Every game starts as a no-hitter. If you’re ESPN or MLB Network, when do you start giving live updates? If you’re flipping through games on Extra Innings, when should you put the clicker down?

The following plot shows the probability of a team recording its first hit as a function of inning. The red dots are data culled from the roughly 14,600 MLB team games from 2007-2009, and the black line is a simple statistical model. The model is grounded in basic probability: The probability that the Nth hitter records a team’s first hit (treating walks, HBP, errors, etc. as non-events) is

PROB = (AVG)x(1-AVG)^(N-1)

where AVG is a composite batting average. I used .265, the MLB batting average from 2007-2009. You can see that the model matches the reality pretty well.

A team recorded its first hit after “zero” innings (eg, the leadoff batter gets a hit) about 27 percent of the time, a tick higher than the league batting average. The fourth batter got a hit after the first three failed about 10.9 percent of the time; the model predicts 10.5 percent.

Something interesting happens around the fifth and sixth innings (zoomed in view below). The data points do not drop as rapidly as the model, suggesting that maintaining a no-hitter the third time through the batting order is a tough hurdle for pitchers looking to make history.

Another way of asking the question is: In what percentage of games is a team held hitless for X innings? That’s what the next plot shows:

A team records its first hit within the first inning 60 percent of the time. After 2.1 innings, 90 percent of teams record their first hit. And 99 percent of the time, a team picks up its first hit within the first five innings. Five no-hit innings is a 1-in-100 kind of feat.

But no-hitters are super rare. Rare like The Last Unicorn rare. The model suggests that the chance of a no-hit inning is about 40 percent. That means that sustaining a no-hitter for nine innings has only a .025 percent chance of happening, roughly once every 4,000 games. Strangely, the model predicts that we should have seen three or four no-hitters over the last three years. We’ve seen eight. And the funny thing is that the model works pretty well until the last out. Is there something magical about that last out? Is there really a difference between having no-hit stuff and near no-hit stuff? Do fielders concentrate extra hard on the last out when the no-hitter is at stake? (Given the sample size, it is probably a statistical fluke. But that is a boring answer.)

Let’s split the difference and say that we should have seen about six no-hitters in the last three years. That’s .04 percent of all the games played over that time. Using the simple model, .08 percent of teams record their first hit after outs 19 and 20, roughly speaking. So, if a pitcher takes a no-hitter a few outs into the seventh inning, he has about a 50 percent of chance of completing the feat. Whether you choose to invest your emotional energy in rooting for a no-hitter before then is up to you.

References & Resources
The data are courtesy Harry Pavlidis.