Since Joe Biden won the 2020 presidential election, there’s been a lot of analysis aimed at somehow proving that his victory was a result of fraud or illegal voting. None of that analysis has offered credible proof of fraud, as dozens of judges in various courts and any number of independent observers have determined. But the goal is often less to prove the case than to suggest the case, to continue to present the well-settled issue as unsettled and thereby to present President Trump as having not yet lost his reelection bid instead of having clearly lost it a month ago.

In service of this objective, Trump’s supporters and the president himself have taken to declaring that Biden’s win was not just unlikely but “statistically impossible,” a term they generally use to mean something like “not possible — to the extreme.” But Biden’s win was possible, as made clear both through a detailed consideration of the claims of statistical impossibility and, more directly, by Biden’s having won the 2020 presidential contest.

Before we parse the claims of impossibility that have been floating out there, it’s worth pointing out that the term “statistically impossible” doesn’t really mean much. If something’s impossible, it’s impossible, and Biden winning the 2020 election was never impossible in any legitimate sense of the word. What people generally mean is that something is very, very unlikely, implying a sort of finality by using “statistically impossible” even when things have nothing to do with statistics.

The case at hand offers a lot of examples of how this bit of exaggeration works. Consider Trump’s presentation of the “statistical impossibility” of Biden’s win at his rally in Georgia on Saturday. Trump walked through a number of the more common claims about how various things were “impossible” according to “statistics,” by which he meant that they were “things he would like people to think didn’t happen” even though they “did.”

Since Nov. 4, President Trump has repeatedly claimed his election loss as a result of massive fraud. The following is a roundup of his claims. (The Washington Post)

For example, he reiterated his consternation that somehow Biden got more votes than he did.

“We got 74 million plus,” Trump said, “and they’re trying to convince us that we lost. We didn’t lose. They found a lot of ballots, let’s be nice about it, they got rid of some, too.”

He cited “the great pollster, John McLaughlin,” who said that “there’s no way this could’ve happened other than the obvious cheating or a rigged election.”

McLaughlin is the pollster who said that then-House Majority Leader Eric Cantor had a 34-point lead in his reelection bid in 2014. Cantor lost by 10 points.

How did Biden get more than 80 million votes? Two ways, as we’ve explained before. The first is that the population has grown a lot over the past 12 years. The number of voting-eligible adults increased by 25 million relative to when Biden was first elected vice president. That Trump won more votes than any prior incumbent means less when you consider that there are more Americans than at any prior point in history.

The other factor was Trump himself. Trump was the most polarizing president in modern political history, inspiring fierce loyalty from Republicans and deep hostility from Democrats. In 2016, a lot of voters were indifferent about both Trump and his opponent, Hillary Clinton. After the past four years, few Americans don’t have strongly held opinions about Trump, helping power record turnout for his 2020 general-election opponent.

This pattern will come up a lot as we consider Trump’s “statistical” case: He was very popular with his own party and unusually unpopular with the opposition. And that by itself explains a lot.

For example, Trump made this argument for the unlikelihood of Biden’s win:

“No incumbent who receives 75 percent of the total primary vote lost reelection in the history of our country,” he read from a sheet of paper. “President Trump received 94 percent — not 70, 94. Which is one of the highest in history.”

Sure, fine. But the sample size here is really small. The era of political primaries replacing the smoke-filled-room selection of presidential nominees is fairly young, really beginning in the 1960s. The number of incumbents since that time is smaller still.

Combine that with Trump’s popularity among Republicans, and you see how this wasn’t particularly complicated. Particularly given that the GOP worked to freeze opponents out of the primary process and that a number of states declined to hold primaries at all.

Trump continued on this train of thought, claiming that he “set a record for the most primary votes ever received by an incumbent ever,” adding that “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.”

Again: population growth and boxing out the competition. Not complicated.

Next, Trump talked about the composition of his support.

“Nationally, initial numbers show that 26 percent of President Trump’s voting share came from non-White voters,” he said, again reading from the paper. “The highest percentage for a GOP presidential candidate since 1960.”

Accepting exit poll data at face value — a fraught endeavor — about 18 percent of Trump’s support came from non-White voters compared with 46 percent of Biden’s. That is an improvement for Trump over 2016. But Biden did better with White voters than did Clinton, earning 41 percent of the White vote. Given that White voters made up two-thirds of the electorate, that helped boost Biden past Trump: Forty-nine percent of Clinton’s support in 2016 was from non-Whites, a higher percentage than Biden received this year.

Again, that’s according to exit polling and should be taken with a grain of salt. But it gets at another reason that the Trump loss was not impossible. White voters were a bit less overwhelmingly supportive of Trump than they had been four years ago. Part of that erosion for the president was a function of White suburban voters swinging against him — and him specifically.

“Think of this one,” Trump continued: “President Trump won 18 of 19 bellwether counties. You know what a bellwether county is? A big deal. So I won 18 of 19, a record.”

Bellwether counties are bellwether counties because they predict winners, not the other way around. In other words, it is not the case that someone wins because they win a bellwether county; a bellwether county is a bellwether county because it aligns with who won.

The point is this: If bellwether counties got the winner wrong, that means they weren’t good bellwethers this time, and nothing more. Trump also delineated the time frame for those bellwethers as 1980 to 2016 — meaning 10 elections, hardly a massive sample size.

He didn’t identify the bellwether counties, but, given that most of this was cribbed from an article published by the hard-right Federalist website, we can assume they are the ones cited in that article, identified by the Wall Street Journal.

How did the Journal explain what happened?

“America has become so polarized that presidential bellwether counties — those that consistently back the national winner, switching between parties in the process — are nearing extinction,” the Journal’s John McCormick wrote.

The gulf between people who did and didn’t like Trump (groups that often don’t overlap) was also apparent in Trump’s next presentation of the “impossibility” of Biden’s win: that the president-elect “didn’t demonstrate coattails [in] down-ballot races.”

Trump’s made this claim before, arguing on multiple occasions that statistics prevented him from having lost when Republican House and Senate candidates won. Except, of course, that there are a lot of reasons that might have happened.

One is that not every House and Senate Republican candidate won. Those who did win in swing districts may have benefited from the increased turnout of voters coming out to support Trump. But those candidates may also have enjoyed the support of more of those independent and Republican voters who are skeptical of Trump.

Imagine a swing House district in the suburbs with a relatively moderate pool of White voters. Trump’s presence on the ballot brings out a chunk of people who didn’t vote in 2018, hurting Republican House candidates. But some of those more-moderate voters might have decided not to vote for Trump even while they supported the Republican running for Congress. Just like that, you have a split ticket — precisely as appears to have happened in a number of jurisdictions.

That entirely conceivable scenario explains a lot of the “impossibilities” Trump raises. While this wasn’t extremely common, according to an analysis conducted by FiveThirtyEight, it did happen in a number of swing states.

Speaking of swing states, Trump also marveled that Biden “beat [Clinton] in the swing states, but she beat him everywhere else.”

Biden outperformed Clinton in every state but six: Arkansas (where Clinton was once first lady), California, Florida (a swing state), Hawaii, Nevada (barely) and Utah (which voted heavily for a third-party candidate in 2016).

In other words, this point is simply wrong. Or, to put it in terms Trump would appreciate: It is statistically impossible that Trump’s claim is correct.