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# Math says Larsen > Halladay (by a lot)

Today marks the 54th anniversary of Don Larsen's perfect game, which in this Postseason of the Pitcher, takes on even greater signficance with what's happened in the playoffs the last two days.

Larsen and Roy Halladay are the only pitchers to throw no-hitters in postseason play.

Halladay tossed his no-no against a Reds’ offense that led the National League with a .272 batting average in the regular season. Larsen’s perfecto came against the 1956 Brooklyn Dodgers, who led the NL with a .342 on-base percentage that regular season.

So taking account the high level of competition in both cases, which of the two historic performances was most statistically impressive?

We're here to show you that a mathematical approach says Larsen's is better. And if you can be patient with the explanation below, you'll see that it's not close.

One way to go about evaluating this is to look at what each accomplishment entails, exactly, and calculate the likelihood of the series of events necessary for it to occur.

Let’s start with Halladay. In order to get his no-hitter, Halladay had to deny a hit to the batter in each of the 27 at-bats against him (we discount the walk he gave up since no-hitters only deal with a lack of hits).

First, he had to make sure Reds’ leadoff hitter Brandon Phillips did not get a hit. The probability of that happening can be thought of one minus Phillips’ .275 batting average in the regular season, or 0.725. Then, he had to ensure Orlando Cabrera (.263 BA this season) did not get a hit, which has a probability of 0.737.

Assuming those two at-bats are independent, the probability of a pitcher not allowing a hit to BOTH Phillips and Cabrera in the first two AB of the game is the product of those two, (0.725*0.737 = 0.534). Using this method for all the subsequent batters, we can calculate the probability of denying a hit to each one in the series of 27 batters that recorded an AB against Halladay (Phillips, Cabrera, Votto, … - obviously including repetitions for multiple at-bats by the same player).

When you get through multiplying the 27 probabilities, the likelihood of what Halladay did is quite low: 0.0001542. That number’s tininess may be difficult to comprehend, so think of it this way: one in 6,500!

So clearly Halladay’s performance is a rare feat, but how does it compare to Larsen’s?

A similar method can be used to calculate the probability of retiring each of the 27 Dodgers batters that came to the plate against Larsen in Game Five of the 1956 World Series.

The only difference is that instead of using each player’s batting average, the player’s on-base percentage is used because Larsen not only denied them hits, but also kept them from reaching base at all –- essentially, the difference between a no-hitter and a perfect game.

So the probability of retiring Dodgers’ leadoff man Jim Gilliam in his first PA is one minus his .399 OBP, or 0.601. The probability of retiring Pee Wee Reese in his first PA is .678, then there is a .601 probability of retiring Duke Snider, and so on.

Putting it altogether, the likelihood of throwing a perfect game against the lineup Larsen faced is almost nil: 0.000009! You’re reading that correctly: nine times out of a million, or about one in 111,000. By comparison, Halladay’s no-hitter against the Reds was about 17 TIMES more likely than Larsen “perfect gaming” the 1956 Dodgers!

Why is Larsen’s performance so much more unlikely, statistically speaking? As you may have guessed, it comes down to the how much more difficult it is to throw a perfect game than a non-perfect no-hitter.

The Reds’ hitters used in Wednesday’s game had a BA of about .280, so denying them a hit is a 72 percent proposition, on average. The 1956 Dodgers who came to the plate in Game 5 of the World Series had an on-base percentage of .362, so an out happens about 64 percent of the time. The eight percent difference doesn’t seem huge for a single event, but when extended to 27 straight occurrences in each case, the perfect game becomes much, much more unlikely.

Of course, this analysis doesn’t take into account that in addition to Halladay not allowing a hit, he also gave up only one walk (to Jay Bruce in the fifth inning) and didn’t allow anyone else to reach base.

This would probably make Halladay’s accomplishment look even more impressive, since he was “almost perfect”, or at least much closer than a no-hitter with several walks or other runners reaching base.

So the point of this is not to take away from Halladay’s fantastic performance Wednesday -- one in 6,500 is still quite amazing! But this statistical take shows that even after another postseason no-hitter was finally thrown 54 years later, Larsen’s perfect game back in 1956 remains in a class of its own.

Alok Pattani is a researcher for ESPN Stats and Information