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What is the rarest play in baseball?

MLB "One In A Billion" Plays (Lucky/Rare Plays) ᴴᴰ - YouTube

The rarest of rare plays, that I’m sure will never actually happen but would be amazing to see, would be one that can only happen under National League rules. It would be where, in the bottom of the ninth, the home team pitcher hits a walk off home run to preserve his own perfect game, and in the process breaks up a bid for a perfect game by the opposing pitcher.

EDIT: With the National League now adopting the designated hitter as of the 2022 season, this scenario becomes even more potentially rare than when I wrote the answer a few years back. Going forward, for this scenario to happen, a team would have to make the unusual decision to allow their pitcher to bat, rather than using a DH. This even in and of itself is a fairly rare occurrence. Madison Bumgarner did so on June 30, 2016, followed by Shohei Ohtani a number of times in 2021 such as Aug 18 when he homered and pitched eight innings in an AL game, but it is rare.

The “quadruple play” which someone else mentioned is pretty rare too, although probably not quite as unlikely. As the other response mentioned, this would be a situation where a team turns what is already a triple play, but must record a “fourth out” to negate a run that would otherwise count. BTW officially this would “only” be a triple play. Only three outs can officially be recorded. If a team records four outs, though, then only three outs will count, but the official “third out” will be whichever one is most advantageous to the team recording it. Still a very, very rare play especially if in combination with a triple play.

As others have noted, the unassisted triple play which has occurred about 15 times in MLB is probably the rarest play for which there is a specific statistic and which is truly about a single play, and not the context in which the play occurred. However, it turns out that the antithesis of an unassisted triple play—three errors by one player on play—is even rarer, having occurred only once as far as I can tell in MLB history. Tommy John, pitching for the NY Yankees against the Milwaukee Brewers, achieved this “accomplishment” on July 27, 1988. Ironically, apart from this one very bad play, John appears to have pitched a good game. He went eight strong innings for the win in a 16–3 blowout. Two of the three Brewers’ runs occurred on the three-error play, although one of them became an earned run due to a subsequent single allowed by John.

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