The Volokh Conspiracy

Mostly law professors | Sometimes contrarian | Often libertarian | Always independent


Statistics, and How the World Works

A surprising number of people seem to believe that the "improbability" that Biden could have won (in a fair election) is evidence that he didn't win (in a fair election). It isn't.


Imagine we are back in February 2019, at the start of Spring Training for the 2019 baseball season. I offer you the following wager:

I have $20 that says that the winner of the 2019 World Series will (a) lose 31 of its first 50 games; (b) find itself behind in in the 7th inning of each deciding game of each postseason series; (c) not win a single game in the Series at home; (d) have a starting left fielder from the Dominican Republic and a starting pitcher whose eyes are two different colors; and that (e) game 6 would be played before a crowd of precisely 43,384, and (f) game 4 would be played in precisely 4 hours 3 minutes and 41 seconds; and that (g) the first base umpire's brother-in-law would be hospitalized right before the start of the bottom half of the 6th inning in game 2; AND that (h) the Dow Jones Industrial Average would decline precisely 140.46 points to close at 27,046.23 on the date of the final Series game.

What odds would you have given me, at the start of the 2019 baseball season, that all of those conditions would be met?! Not a single one of them had ever come true in the prior 100+ years of baseball history; how likely was it that all of them would come to pass in a single postseason?!! Gotta be a quadrillion to 1, no?

So the Nationals must have cheated! Wake up America! How much more proof do you need?!


Now that the dust is starting to settle from the Supreme Court's decision in Texas v. Pennsylvania, I want to take a moment to discuss—again—the "One in a Quadrillion" (OIAQ) Affair.

The OIAQ Affair refers, of course, to the preposterous and easily debunk-able claim, made in Texas Attorney General Paxton's brief before the Supreme Court, that Joe Biden had a "one in a quadrillion chance" of having overtaken Donald Trump in any one of the swing states, let alone all four.

I think there's more there than just laughably bad statistical mumbo-jumbo here (though there is that). The OIAQ Affair illustrates a rather important kind of reasoning fallacy, or cognitive bias, to which large numbers of people seem to be susceptible, a misunderstanding about the way that probability and causation operate. I don't know whether it has a name; perhaps it should have.

OIAQ is one part of the a larger "argument"—to the extent one can characterize his position as an "argument"—put forward by the President and his supporters: that the improbability that Biden could have prevailed, given other events that took place on Election Day, constitutes evidence that he didn't, actually, prevail, that there were—there must have been—some "irregularities" or fraud at work to bring about such an unlikely outcome.

As the President's lawyer put it in his brief in the case:

"The fact that nearly half of the country believes the election was stolen should come as no surprise. President Trump prevailed on nearly every historical indicia of success in presidential elections." For example, he won both Florida and Ohio; no candidate in history—Republican or Democrat—has ever lost the election after winning both States. [Ed. note: This is not correct, by the way. Nixon won both Florida and Ohio in 1960 but lost the election.] He won 18 of the country's 19 so-called "bellwether" counties—counties whose vote, historically, almost always goes for the candidate who wins the election.  Republican candidates for the U.S. Senate and U.S. House, down to Republican candidates and the state and local level, all out-performed expectations and won in much larger numbers than predicted, yet the candidate for President at the top of the ticket who provided those coattails did not himself get over his finish line in first place….

These things just don't normally happen, and a large percentage of the American people know that something is deeply amiss.

What are the odds?  What are the odds that Biden could have won given that the early returns in all four swing states gave Trump a sizeable lead? What are the odds that Biden could have won given that Trump ended up carrying 18 of 19 "bellweather" counties across the country? What are the odds that Biden could have won given that Trump carried both Florida and Ohio? And given that Biden under-performed Obama everywhere except in the four swing states; etc.?

This kind of argument shows up repeatedly. The President himself—who, like all talented stand-up comics, has a finely-tuned sense of the lines that resonate with his audience—referred to it over and over in his remarks at the recent rally in Georgia. [I've put some excerpts from his remarks at the end of this post]. Fox News, too, has been all over this claim [see, e.g., here and here, purportedly showing how a Biden victory was "statistically implausible" and "defied a dozen or more metrics that have 100% accuracy" in predicting the results of presidential elections]

It's total nonsense.  It makes as much sense as arguing that the Nationals cheated their way to the 2019 title because of the "statistical implausibility" of their victory.

Here are a couple of basic principles that explain just how nonsensical it is.

Principle 1: Improbable events—highly improbable events—happen.  All the time.

Getting two consecutive four-of-a-kind hands in five-card poker dealt from a fair deck is a highly improbable event; the odds against it are about 16 million to 1. If your uncle Joe is playing poker and turns to you and says: "I'm on a hot streak—I'll bet you $5 I get two consecutive four-of-a-kinds," take the bet.

But if he tells you that he was dealt two such hands in a row last night, don't call him a liar (or accuse him of cheating).  As improbable as it is, it happens all the time; I can pretty much guarantee you that it happened, somewhere, just this past week, probably several times over.** So perhaps uncle Joe was just one of the lucky recipients.

**That's based on the assumption that 10 million poker hands are dealt around the world—a million players, an average of 10 hands each—each day.  [Probably an underestimate, by the way].  In which case we would expect there to be two consecutive four-of-a-kinds every day or two, somewhere.

By the same token, the likelihood that all eight conditions listed in my opening paragraph would come true in 2019 was very, very low.

But if you had bet, in February 2019, on them all happening, you'd be rich today, because happen they all did. Wow!

It is just how the world works.  Any event—my spilling coffee on myself this morning—can be described, once it occurs (i.e., ex post), as having been, ex ante, spectacularly improbable.

Here's a spectacularly improbable event:  that 30 years from now, on December 15, 2050 at 8:48 AM EST, a man

(a) whose great-grandfather owned a fruit stand in Vinetsa, Ukraine; (b) whose father drove a Pontiac for several years; (c) who grew up on a street where two of the other children were named "Solomon"; (d) who married someone whose older brother moved, within the preceding 6 months, to Gainesville, Florida, where he had lived previously; (e) who recently re-read John Steinbeck's The Grapes of Wrath; and (f) who got in trouble, in 4th grade, for tossing art supplies out the window,

would spill coffee on himself.

What are the odds of that!?!? Surely around a gazillion to one.

And if they're a gazillion to one today, regarding what may or may not happen in 30 years, the odds would have been a gazillion to one 30 years ago, in regard to an event happening today.

And yet it happened! Of all the people in the world, I meet all of those conditions, and damned if I didn't spill coffee on myself at precisely 8:48 this morning!!  Unbelievable! What are the odds!?

This leads to Principle 2: The notion of the "probability" or "likelihood" that such-and-such an event will or will not occur ceases to have any meaning once the event occurs.

This is probably just a fancy way of saying: It is nonsensical to talk about whether an event can happen, or the likelihood that it will happen, once it has happened.

Consider the dinosaurs. What are the odds, they surely said to one another, ex ante, back in 94 million BCE, that a comet would rip into Earth and throw up enough debris to cloud over the skies for many thousands of years and destroy 90% of all living things? Relax!

Once it happened, though, it no longer made sense to talk about the probability that it could happen. And however low that probability might have been ex ante, that is not evidence that it didn't happen, once it did.

So I don't care—and you shouldn't care, either—if there are 10 or 20 or 200 "historical indicia of presidential success" that are "defied" by a purported Biden victory. None of that is relevant to the question of whether he did or did not win, or to whether the election was or was not fair.


Here are some excerpts from the President's remarks at the Georgia rally [full transcript here]:

Well, thank you very much. Thank you. No, we won a lot of places. We won Florida. We won Ohio. Big, big. We won them big. Remember we were going to lose Florida they said. We were five down in Florida. We won by a lot. We were way down in Ohio and we won by a lot. I think they say that if you win Florida and if you win Ohio in history, you've never lost an election. This has got to be a first time. But the truth is they were right, we've never lost that election. We're winning this election.

Think of it, with over 74 million votes, over, think of that, more than … I got more votes than any sitting president in history. 11 million more votes than we got in 2016. And we thought that if we could get 68 million, 67 million that would be the end. All of our great, brilliant geniuses said you'd win if you get 67 or 68. It's over. We got 74 million-plus and they're trying to convince us that we lost. We didn't lose.

But listen to this. These are the facts. And each fact is irrefutable and it means win. President Trump received, done by one of the most brilliant political people, President Trump received more votes than any previous incumbent president in history. And we lost. And we lost. So, we received more votes than anybody in history, any incumbent in history, and we lost, supposedly. We didn't lose. You're going to see that. No incumbent who received 75% of the total primary vote lost reelection in the history of our country. President Trump received 94, not 74. 94, which is one of the highest in history.

In fact, President Trump is the only one of five incumbents since 1912 to receive more than 90% of the primary vote. And again, anybody received over 75%, they won. We got 94%. President Trump set a record for the most primary votes ever received by an incumbent, ever. And nobody that's received all of the primary votes, nobody's received at a much lesser level than what we, they always won. But we didn't according to what they say. It's rigged. It's a fixed deal.

Think of this one. President Trump won 18 of 19 bellwether counties. You know what a bellwether county is, it's a big deal. So I won 18 of 19, a record, never happened. That between 1980 and 2016 voted for the eventual president in every single election, and before that, it was almost every election. And we won a record 18 of 19. Never lost. Nobody's ever lost with anything like that.

Think of this. The first time that anybody, I think in history, we didn't lose one seat. Normally, you lose two, three, five, and you replay. We lost nobody sitting in Congress.

President Trump won. President Trump won as we said both in Florida and then Ohio. And by the way, won by a lot. Remember the fake polls where they said he's down by four in Florida, and I won by a lot. He's down by two in Ohio, and we won. I think we got eight or nine or something, up. But nobody's ever done that.

And many of these swing states, it's a very interesting statistic. President Obama beat Biden all over the country, except in some of the swing states where Biden beat him badly. How does that work? And they say it's statistically impossible. He beat crooked Hillary. Think of this. He beat crooked Hillary in the swing states, but she beat him everywhere else. Let me tell you, this election was rigged and we can't let it happen