By the time Giants rookie right-hander Chris Heston took the mound for what he hoped would be a ninth hitless inning at Citi Field last week, even the umpires appeared to be pulling for him. Up 0-2 on Daniel Muno, who was pinch-hitting in the pitcher’s spot, Heston threw a curve off the outside corner for a questionable called strike three.

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In slow motion, it’s easy to see that the pitch never nicked the plate. Instead, it arced around the edge, a glove’s width from the actual corner. Fortunately for Heston, he was throwing to a guy with a very good glove: Buster Posey led all catchers in framing runs saved last season and leads them again this year. On 0-2, the strike zone typically contracts, making it more difficult to get calls in this location. But some combination of Posey’s pitch presentation and the excitement of the moment convinced umpire Rob Drake to send Muno back to the bench.

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Two batters later, Heston threw a 2-1 sinker to Ruben Tejada roughly the same distance off the opposite corner, and again he got the call.

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With his next pitch, a better but still borderline strike three to Tejada, Heston completed a no-hitter that was more unlikely than most. Both Muno and Tejada are below-average hitters, and Heston probably would have made history even without the assists from Drake. But the pained looks on Muno’s and Tejada’s faces remind us that Heston did have some help.

We can quantify that pain. Using PITCHf/x data, the fine minds at Pitch Info assign a called-strike probability to every pitch thrown in the majors, based on the results of millions of previous pitches and adjusted for a number of factors that can affect the odds of a strike: location, pitch type, count, year, and batter/pitcher handedness. The farther away from the center of the batter’s strike zone a given pitch is, the lower its called-strike probability, regardless of its result.

The average called-strike probability of all MLB pitches this season is 45.9 percent, which means that the typical pitch has slightly less than a 50-50 shot to be a called strike if the batter doesn’t swing. If we limit the sample to actual called strikes, the average probability rises to 79.0 percent. It might seem confusing that the called-strike probability of a sample of actual called strikes is less than 100 percent, but as the Heston examples above illustrate, many pitches that become called strikes probably shouldn’t be — and wouldn’t be, with a different pitcher/umpire/catcher combination.

On the whole, the 28 called strikes in Heston’s no-hitter had a 77.8 percent called-strike probability, slightly lower than the 79.0 percent MLB average but slightly higher than the 76.3 percent average with Posey behind the plate this season. In other words, Heston wasn’t getting a giant zone throughout the game: It was only in the ninth that the borders opened up. The pitches to Muno and Tejada that went Heston’s way were only 5.3 and 7.1 percent likely, respectively, to be called strikes. Essentially, one could throw the same pitches to the same batters in the same counts and expect them to be strikes no better than one in 14 times.

Heston’s no-hitter wasn’t Drake’s first: He also umped Felix Hernandez’s perfect game on August 15, 2012. And sure enough, there’s evidence of a ninth-inning strike-zone enlargement in that game, too. The lowest-probability strike Hernandez received in that start came on a 1-1 pitch to Desmond Jennings leading off the last inning. This one wasn’t as egregious — its called-strike probability was 34.2 percent, several times higher than those of the gifts Heston got — although the off-center camera angle makes it look more reasonable than it was.

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This isn’t all the work of one rogue, no-hitter-happy umpire, though. In the ninth inning of no-hitters, things get strange. For one thing, pitchers start nibbling: Eight innings in, no one wants to catch too much of the plate. The table below shows the average called-strike probability by inning of all pitches thrown in the 27 no-hitters of the PITCHf/x era. It’s not that these pitchers throw more off-speed stuff at the end of the game. Although they throw fewer fastballs after the first time through the order (as do almost all starters), they don’t throw fewer in the ninth than they do in the fourth, fifth, or sixth. But the average called-strike probability of their pitches — a proxy for proximity to the center of the zone — declines in each of the last three innings, reaching its nadir in the ninth. As they try to record those final few outs, pitchers treat their opponents like dangerous sluggers and try to stay away, hoping that they’ll chase.

Inning Pitches Called-Strike Probability Hard% Off-Speed% 1 362 45.0 70.7 29.3 2 351 43.0 63.0 37.0 3 306 49.8 66.3 33.7 4 332 46.8 54.2 45.8 5 313 44.6 55.3 44.7 6 327 47.4 53.8 46.2 7 335 45.3 57.0 43.0 8 335 43.1 53.7 46.3 9 351 40.8 56.1 43.9

In most games, this wouldn’t work. Here are the average called-strike probabilities by inning of all pitches that have resulted in swings and called strikes in non-no-hitters from 2008 on.

Inning CS Prob, Swings CS Prob, Called Strikes 1 63.1 76.8 2 63.0 76.7 3 62.7 76.6 4 62.5 76.1 5 62.4 75.9 6 62.5 76.0 7 62.3 76.1 8 62.3 76.2 9 62.2 76.3

There’s no noticeable difference. In normal games, hitters swing and umpires call strikes more or less the same way in the ninth that they do in the fifth or seventh. In no-hitters, though, opponents often play into the hands of extra-cautious starters. Below are the average called-strike probabilities by inning of only the pitches that resulted in swings in the 27 no-hitters. The numbers jump around a bit — 27 starts is a pretty small sample — but two of the three lowest values come in the last two innings. In the eighth and ninth innings of no-hitters, batters are more likely to chase, either because they’re not getting good pitches to hit or because they’re anxious to get on the board.

Inning Swings Called-Strike Probability 1 152 60.5 2 147 55.6 3 129 63.1 4 146 60.1 5 148 59.6 6 160 59.5 7 149 65.2 8 155 54.8 9 157 56.6

Or maybe there’s a third reason: It could be that batters are afraid that if they don’t swing, they’ll get Draked. Check out the inning-by-inning breakdown of the called-strike probabilities of pitches that resulted in called strikes in no-hitters only:

Inning Called Strikes Called-Strike Probability 1 67 84.2 2 69 80.5 3 68 82.4 4 68 79.6 5 56 73.3 6 62 81.1 7 57 72.4 8 62 73.1 9 58 70.5

At the start of the game, the pitchers paint the zone, with a called-strike probability well above the league average. That makes sense, since starters who throw no-hitters tend to be pretty good. By the end of the game, though, the starters are getting called strikes on pitches that are far less likely to be strikes under normal circumstances. There’s a clear change throughout the game that doesn’t show up in the leaguewide numbers: In the late innings of no-hitters, the strike zone actually expands. And it’s not just Drake.

So what were the most generous calls in the ninth innings of no-hitters in the PITCHf/x era? Heston’s pitches to Nuno and Tejada had the fourth- and fifth-lowest called-strike probabilities, respectively. Here are the four other called strikes in the ninth innings of no-hitters that had called-strike probabilities below 10 percent:

July 2, 2013: Homer Bailey vs. Gregor Blanco, Ryan Hanigan Catching (8.3 Percent)

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May 29, 2010: Roy Halladay vs. Wes Helms, Carlos Ruiz Catching (4.8 Percent)

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June 13, 2012: Matt Cain vs. Jason Castro, Buster Posey Catching (2.9 Percent)

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May 2, 2012: Jered Weaver vs. Denard Span, Chris Iannetta Catching (0.5 Percent)

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PITCHf/x hasn’t taught us that umpires make mistakes; that’s something fans have always known (and loudly made known). But it has taught us that there’s a consistency to their inconsistency. Umpires, as The Hardball Times author John Walsh once put it, are compassionate: When batters are in a hole, they shrink the zone, and when pitchers are down in the count, they make it slightly larger. Although those fluctuations might seem like mistakes, they actually, in a sense, make baseball better: They’re predictable, and they make plate appearances more competitive by giving a small edge to the underdog. Late in no-hitters, the umpire’s compassion appears to be aligned with ours. We want to see something special, and umpires, perhaps subconsciously, make it slightly more likely we’ll get our wish. Except, that is, for the occasional time when they take our wish away.

Thanks to Rob McQuown and Harry Pavlidis of Baseball Prospectus for research assistance.