How unlikely are two immaculate innings in the same game against the same batters? (very!)

This week the Astros threw two immaculate innings in the same game against the same batters – something that’s never been done before in MLB! As I’ve noted before, there have been a lot of baseball games, so finding something that has never happened before (at least as far back as we have pitch data) is pretty exciting!

In Jayson Stark’s column I calculated there was a 1 in 67.2 million chance of this happening in a game – I thought it would be fun to break down where that came from! (and correct a mistake I made; oops!)

For reference, an immaculate inning is an inning where the pitcher strikes out all three batters on three pitches a piece. It’s the kind of accomplishment inside a game that you might not even notice if you’re not paying attention, but as we’ll see it’s rarer than a no-hitter! (or a walk-off walk!)

Since strikeouts are much more common than they used to be, let’s just look at data since the year 2000. Thanks to this handy list of immaculate innings, we can count that there have been 57 immaculate innings from 2000-2021, or less than an average of 3 per season! But more precisely:

• In a normal season, there are 30 teams, they each play 162 regular season games, but each game involves 2 teams, so there are a total of 30 * 162 / 2 = 2430 total regular season games. (for our purposes I’m going to ignore the postseason; there are only a tiny number of games compare with the regular season) There are 9 * 2 half-innings per game (again, we’re ignoring any shortened games, and the fact that the bottom of the 9th inning doesn’t always happen, but this also ignores extra innings so my guess is it doesn’t matter much), so that’s 2430 * 9 * 2 = 43,740 half-innings per normal season.
• In the pandemic-shortened 2020 season, each team played only 60 games, so a total of 30 * 60 / 2 = 900 games, or 900 * 9 * 2 = 16,200 half-innings.

So the period from 2000-2021 had 21 normal seasons plus 2020, so that’s a total of 43,740 * 21 + 16,200 = 934,740 half-innings. Since there were 57 immaculate innings in this time, this means an immaculate inning happens about once every 16,400 half-innings.

For a comparison, from 2000-2021 there were 64 no-hitters, so immaculate innings are slightly more rare!

Anyway! Now that we know the probability of any half-inning being an immaculate inning, let’s get to some game-wide probabilities! Note that this assumes the chance of an immaculate inning is independent of the pitcher or the hitters, which is of course absurd, but that’s what we’ve got to work with.

• The probability of two immaculate innings in the same game – First we have to pick which half-innings they happen in; since there are 18 half-innings to choose from and the order doesn’t matter, there are (18 choose 2) = 18 * 17/2 = 153 choices. The two immaculate innings happen with probability 1/16400^2, so this is a total of 153/16400^2, or around one in 1.76 million.
  • Note that technically we should multiply by (16399/16400)^16 for the non-immaculate innings, but this value starts with 0.999 so I’m just going to ignore it from here on.
• The probability of two immaculate innings in the same game by the same team – This is a similar analysis, except there are 9 half-innings to choose from. This is where I made a mistake; I forgot to multiply by two because it could be for either the home team or the visiting team! Whoops. Anyway, that gives (2 * (9 choose 2))/16400^2, or around one in 3.74 million. (not one in 7.47 million)
• The probability of two immaculate innings in the same game by the same team against the same batters – This starts with the previous probability, but the two half-innings have to start at the same position in the batting order. We’re assuming that half-innings have an equal chance of starting at any batter, which is a big simplification but I think a reasonable one. We’re also assuming that there are no pinch hitters or anything. Anyway, this just means we have to divide by 9 for a grand total of one in 33.6 million!

Since there are 2,430 regular season games in a normal season, you would expect this to happen once every 13,835 seasons! Truly a very very unlikely thing 🙂

More articles written with data from the Baseball Win Expectancy Finder:

• How common are walk-off walks (on four pitches!) in baseball?
• How likely are walks and strikeouts by count?
• How often does a bases loaded no out situation end with 0 runs?
• Why are so many runs scored in the bottom of the first inning? (also on FanGraphs Community)
• Did the Astros set a record by challenging a play while up 13 runs? (no)
• Does the length of the top of the first inning affect the number of runs scored in the bottom of the first? (somewhat!)
• How effective are teams at manufacturing runs?