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.

gdc said...

If you throw 6 inn no hit it means you probably have good command of at least 2 pitches to get through the order twice and the batters have more to think about (as opposed to having excellent FB location and missing the breaking ball in the dirt, then the next time they are not chasing the curve and getting into the hitters count the next time through). If what people say about hitting is true, the more you have to think the harder it is to hit.

A tiny help to the defense might be the “unwritten rule” of not bunting for a base hit that might allow the corner infielders to stay back a bit on the fast slap hitters.

Dave P said...

I love this post – mainly because I’d been thinking of doing the exact same thing but never got around to parsing the data out to do it. Very interesting stuff!

Zach said...

I wonder if the last out is skewed by that baseball “rule” that you don’t hit on the last out of a no-hitter. I find the “rule” silly but all it takes is some mindful batters to not try their hardest. Considering everyone is trying their best all other 26 outs, this could increase the likelihood of a 27th.

Dave said...

The odds of the first few batters of the game getting hits are higher than the model because the model assumes batters are all average – the top of the lineup tends to be better than average.

Likewise, we should always see more no-hitters than the model predicts because pitchers are not all average either – the above average pitchers are more likely to pitch a no-hitter than this (and vice versa) raising the overall odds of one occuring.

I believe graph #3 is actually a graph of how often a team DOES have a hit by that point, not how often they are hitless.

I sound negative – this is a cool little article about an interesting topic. And like you said, a topic limited by small sample size. I am surprised the late-inning data matches up this well.

sal baxamusa said...

Dave:

- Yes, this is definitely true; I intentionally kept things simple here.

- I disagree with your second statement. Your statement implies that the body of pitchers that complete no hitters is different than the body of pitchers that takes a no-hitter 8 2/3 into the game.

- Yes, correct. My mistake.

Northern Rebel said...

This is not statistical or sabermetric analysis, but the vast majority of no-no’s happen early or late in the season, because of wheather, and september call ups from the minors.

For me, the best time to root for a no-hitter, is when it’s the pitcher on yor favorite team! ;o)

Northern Rebel said...

Wheather? Oops1

Kahuna Tuna said...

the vast majority of no-no’s happen early or late in the season, because of weather, and september call ups from the minors.I’ll go along with the September part of that. Regular-season no-hitters since 1901, by month:

April: 33

May: 37

June: 36

July: 30

August: 30

September: 53

October: 2

Oddly enough, the two dates on which the most no-hitters have been pitched since 1901 (six) are April 27 and May 15. There has never been a no-hitter pitched in the six-day period from May 24 to May 29.

Michael Van Kleeck said...

I’d love to see some further analysis on that last at-bat, and the pitchers who’ve thrown 8 2/3 innings of no-hit ball. What are the stats for the last at-bat (or the one that breaks the no-no…)?

Thanks for an excellent read!!!

Northern Rebel said...

Thanks for the research, Tuna!