Explaining perfect games and other baseball rarities through poker odds (and vice versa)
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Grant Brisbee
May 2, 2024
On June 13, 2012, my father-in-law attended his first baseball game in over 20 years. A friend of his had good seats, and he hadn’t had a chance to see the new ballpark in San Francisco yet, so why not? There had been about 235,000 modern-era Major League Baseball games before that date, and there had been 21 perfect games thrown. My father-in-law sat down with a beer and a hot dog, and he watched Matt Cain throw a perfect game.
He loved to needle me about this. Not only have I never seen a perfect game in the hundreds and hundreds of games that I’ve attended, but I’ve never seen a no-hitter. I’m not sure if I’ve seen a no-hitter get into the eighth inning. Yet my father-in-law went to a game for the first time in decades, and he got the best regular-season baseball experience possible. We would joke about it, saying things like, “That’s like going into a casino, sitting down for your first poker hand and immediately getting a royal flush.”
Sounds about right.
Is it?
Let’s learn a little about baseball and poker at the same time.
First, a note about cards. They’re a great way to break brains.
Pick up a standard deck of cards. Shuffle it. Flip all 52 cards over in a row from left to right. Congratulations. You’ve created a sequence of cards that has almost certainly never existed before and will never exist again. Even if every person who ever lived had shuffled a deck of cards every second since the Big Bang, there would be only a 1 in duodecillion chance that an exact sequence of 52 cards would be repeated. This is nearly impossible to grasp, and it’s just one reason why I think math should be illegal.
Second, when you’re talking about events like a perfect game or the odds of a moth getting stuck in Matt Holliday’s ear, you can’t just say, “Here’s how many times x has happened. Here’s how many games there have been. Therefore, here are the odds.” The odds of a moth getting stuck in Holliday’s ear aren’t a matter of how many baseball games have been played in history. You have to consider the odds of Holliday being in that exact spot, at that exact time. The ballpark lights had to be just so. You have to consider the odds of the moth avoiding a predator the night before. The moth’s parents had to avoid predators long enough to create a new moth. Matt Holliday’s great-great-great-great-great-great-great grandparents had to avoid being eaten by a bear that was startled out of its den by a moth. There are an infinite number of variables stretching back to the creation of the universe.
If it’s statistically impossible to repeat the exact sequence of a 52-card deck, the moth in Holliday’s ear is more like the exact sequence of a duodecillion-card deck. Good luck repeating that. The odds of a moth getting stuck in another player’s ear isn’t 1 in 239,799 (the number of games played since 1871 as of this writing). It’s roughly 1 in ∞.
The odds of a perfect game are much easier to grapple with, but that doesn’t mean that you can just divide the number of perfect games by the number of total games played. There are plenty of factors to consider, such as the rising strikeout rate and the fact fielders are much better now. Also, players in 1910 were all 4-foot-10 with rickets, and they played with oblong balls and raccoon-hide gloves, et cetera.
Doesn’t mean we can’t have fun with some of the stranger events in baseball history, even if the math isn’t perfect.
(And that also doesn’t mean we can’t hope another moth gets stuck in a player’s ear, just for yuks.)
The royal flush, explained in baseball terms
A royal flush — ace, king, queen, jack and 10, of all the same suit — is the highest hand in poker. There are four suits in a standard deck of cards, which means there are four types of royal flushes. The odds, however, depend on which variation of poker you’re playing.
Five-card stud is a variation where the player is dealt five cards, and that’s it. There are no discards or community cards to use. Because there are 2,598,960 different five-card combinations possible in any five-card deal, that means the odds of getting a royal flush are 1 out of every 649,740 deals.
Think of all the innings that have been played since the first unassisted triple play, which happened in 1909. (Back then, ballplayers were paid so little that the Cleveland News took up a collection to raise money for Neal Ball, the shortstop who made the play.) There were just 15 unassisted triple plays, including the postseason, in that entire span.
While play-by-play data gets sketchier the closer you get to the start of the 20th century, we can estimate just how many opportunities for an unassisted triple play there have been. An unassisted triple play needs two or three runners at the same time, with zero outs. According to Baseball-Reference’s Stathead, there have been 387,590 such opportunities, including the postseason, since 1912. They estimate that about 3 percent of the play-by-play data was lost in the very early days of recordkeeping, so go ahead and round that up to an even 400,000.
So an unassisted triple play (1 in 26,667 opportunities) is much more common than a royal flush in five-card stud (1 in 649,740). You’re 24 times more likely to see an unassisted triple play than get a royal flush. You’re even more likely to get an unassisted triple play (.0000375 probability) than a four-of-a-kind hand (.00002401).
Which means video poker machines don’t offer very good odds at all, and they might actually favor the casino. Be careful out there.
When people think of poker and all its variations, though, they usually aren’t thinking of five-card stud, but Texas Hold ’Em, where a player makes the best possible hand out of seven cards — two in their hand and five community cards. The handy website Wizard of Odds includes odds for different hands that depend on how many players are at the table. To get the odds of a royal flush even close to that of an unassisted triple play, you’ll need a 10-player table. That’s 20 dealt cards and five community cards, which means that nearly half the deck is in play, with a pool of three out of five shared cards available to everyone.
Even in that situation, the unassisted triple play is more likely than a royal flush. Ten players, each with two cards, all sharing the same five cards to make the best five-card hand, are less likely to get a royal flush than a baseball team is to hit into an unassisted triple play with two (or three) runners on and nobody out. The odds of baseball have always bothered me, but playing cards scare me.
The straight flush, explained in baseball terms
Forget the royal flush, then. Let’s talk about the straight flush — any numerical sequence with five cards of the same suit. There are eight numerical sequences possible with a straight flush instead of just one with the royal flush. The odds are boosted eightfold.
Except if you’re talking about five-card stud, those odds are still less likely than the unassisted triple play. It’s been more than twice as likely for a team to get an unassisted triple play throughout baseball history (1 in 26,667) than it is for anyone to get a straight flush of any variety in five-card stud (1 in 72,202).
So move on to Texas Hold ’Em. In a head-to-head matchup (just two players), a straight flush is a little more likely than an unassisted triple play. Heck, yeah.
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When it’s you and four, five or six friends playing Texas Hold ’Em, your odds of getting a straight flush actually go up. More cards are removed from the pool, which means there are more cards to match with the suited sequence in your hand.
Comparing poker odds to a perfect game or an unassisted triple play isn’t the right move, though. A better comparison for this exercise would be the humble no-hitter, of which there have been 323 in major-league history. That’s about a 1-in-739 probability (.00133) against the nearly 240,000 MLB games played. So when you’re talking about a straight flush, those are the same odds as getting one in a Texas Hold ’Em game with four other players (1-in-762).
When you get that straight flush, one of those players will be Will. That’s right, Will, you’re not so cocky now, are you?
Seriously, what’s the deal with Will?
Two asides when it comes to the straight flush. The first is that I had one in video poker on a cruise ship last July.
The next day, I tested positive for COVID-19. I’m not sure if that’s less likely or more likely than the straight flush, but I’m pretty sure when you multiply them together, you start getting into Matt Holliday-and-moth territory. Life is filled with peaks and valleys.
The other aside is that a straight flush is what James Bond had in the 2006 version of “Casino Royale”:
Dude just threw a no-hitter.
Also, Daniel Craig sounds like a pitcher who went 7-9 with a 4.76 ERA for the 1998 Pirates.
Other poker hands, explained in baseball terms
If you’re playing five-card stud, here are the comparisons:
Four of a kind: 1 in 4,164
This one gets complicated. According to this SABR article, these odds are close to what you’d give a player like Joe DiMaggio in his prime having a 56-game hitting streak or longer (1 in 4,505). But those odds are disputed by another statistician who specializes in these sorts of baseball-probability hunts. While I’m not smart enough to give you my opinion, this did lead me to discover that the Pirates have never had a hitting streak longer than the 27-game hitting streak from Jimmy Williams in 1899.
Hitting streaks are hard. It’s like getting four of the same cards out of a 52-card deck.
Full house: 1 in 693
Half as likely as a no-hitter or a cycle in any given game. Maybe put down the cards. They’re not in your favor.
Flush: 1 in 508
Roughly the same as a game that features a player hitting a grand slam in his first major-league at-bat. There have been four of those in history. Not just four grand slams in the history of first at-bats, but four games in history, including all of the ones that didn’t feature a first at-bat from a rookie at all.
Straight: 1 in 254
Half as likely as an inside-the-park home run, relative to standard home runs. Although, if you adjust for sprint speed and funky outfield configurations, you’ll get plenty close. (Not that close: Remember that Rickey Henderson never had an inside-the-park home run.)
Three of a kind: 1 in 46
Two pair: 1 in 20
According to Baseball-Reference, there have been 22,859 major-league players since 1876 (excluding the National Association), and there have been 820 different managers. Of those, 675 played in the majors. So there’s a 1-in-34 chance of any given major leaguer becoming a manager. That’s somewhere between getting three of a kind and two pair in five-card stud.
If you get a two pair with an ace kicker, congratulations. You’re an MLB manager now.
The perfect game, explained in poker terms
Before my father-in-law went to that game in 2012, the odds of seeing a perfect game were roughly 1 in 11,190. That’s using the simplistic odds of perfect games divided by total MLB games played. Except those odds were tainted because Philip Humber had thrown a perfect game less than two months before. They were tainted even further when Félix Hernández threw another perfect game only two months later. Three perfect games in the same season signaled the rise of the perfect game.
If the odds of seeing a perfect game were 1 roughly in 11,190 in 2012, then the odds of there being three perfect games that year were roughly 1 in 1.5 billion:
The exact probability is impossible to pinpoint, but it will almost certainly never happen again, even if pitchers start striking out 18 batters for every nine innings they pitch. Three perfect games in the same season is more than three times less likely than a royal flush in five-card stud. So congratulations, you’ve already won. The prize is nothing, but it’s still neat.
Back to the original question, then. What is the best way to explain my father-in-law’s luck in poker terms?
It’s as if he sat down at a Texas Hold ’Em table with six other players. Everyone gets their cards, and nobody folds. They stick with the flop, the turn and the river, and when everyone flips over their cards, the winner has the best high card out of seven different high-card hands.
A perfect game is roughly the same as someone winning a hand of Texas Hold ’Em with nothing but a high card if there are six other players in at the end of the hand.
That’s not nearly as romantic as a perfect game, but it’s almost better somehow. Baseball is weird. Cards are weird. Neither one of them is to be trusted, but our lives are so much more interesting because of them.
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