The return of the probability argument

We should have known. We’ve heard it for so long. Creationists love the Argument from Big Numbers — if we chain together a whole series of improbabilities and multiply them, we can get a really big exponent, therefore God. This approach is so familiar there’s a FAQ by Ian Musgrave on the errors in the calculations of the evolution of proteins.

Problems with the creationists’ “it’s so improbable” calculations

1) They calculate the probability of the formation of a “modern” protein, or even a complete bacterium with all “modern” proteins, by random events. This is not the abiogenesis theory at all.

2) They assume that there is a fixed number of proteins, with fixed sequences for each protein, that are required for life.

3) They calculate the probability of sequential trials, rather than simultaneous trials.

4) They misunderstand what is meant by a probability calculation.

5) They seriously underestimate the number of functional enzymes/ribozymes present in a group of random sequences.

We’ve seen it all. People seem to be fundamentally statistically innumerate and, without training, incapable of grasping the basic principles. There are whole books about innumeracy and its consequences.

So I wasn’t surprised at all when I saw that the Texas Attorney General had filed a lawsuit claiming there was a less than one in a quadrillion chance that Biden could have honestly won the Texas election, and that it’s based on a familiarly stupid argument. Also unsurprising: that an old compatriot, Wesley Elsberry, would jump on the faulty reasoning. We’ve all been here before.

Texas filed a lawsuit in the Supreme Court against four other states (Michigan, Wisconsin, Pennsylvania, and Georgia). Others have already weighed in on how unserious a lawsuit this apparently is.

But I want to have a look at something that is a bit more approachable, which is the statistics opinion that Texas Attroney General Ken Paxton relied upon in crafting the lawsuit. It makes some remarkable claims:

The probability of former Vice President Biden winning the popular vote in the
four Defendant States—Georgia, Michigan, Pennsylvania, and Wisconsin—
independently given President Trump’s early lead in those States as of 3 a.m.
on November 4, 2020, is less than one in a quadrillion, or 1 in
1,000,000,000,000,000. For former Vice President Biden to win these four
States collectively, the odds of that event happening decrease to less than one
in a quadrillion to the fourth power (i.e., 1 in 1,000,000,000,000,0004). See Decl.
of Charles J. Cicchetti, Ph.D. (“Cicchetti Decl.”) at ¶¶ 14-21, 30-31 (App. 4a-7a,

Read Wesley’s post for a thorough deconstruction (or this thread for a similar take), so I’ll keep it simple. What kills Paxton’s claims are the assumptions: he assumes that voters should have voted exactly as they did in 2016, that people who voted early on election day would have the same statistical preferences as those who voted later, that people who voted in person would vote the same way as those who voted by mail, and that different precincts would show no change in their preferences over time. He doesn’t seem to realize that what he has shown is not that Biden couldn’t have won, but that his assumptions were all wrong.

Now this has gotten me thinking about genetics, and it’s too early — this is my break, people — and the very first cross we’re going to do. It’s a boring cross to get heterozygotes out of two true-breeding strains, just a preliminary to the real experiment, but I have the students do observations to test their assumption that they’ll get half males and half females. They never do, and the statistics all say it’s a significant difference, with more females than males. Further, when you sample the population at different times after eclosion, it changes, with more females eclosing early. You don’t get to say, “it’s supposed to be 50:50!” and pretend your results are wrong — you’re supposed to question your assumption that sex is a random binary choice. There are a lot of factors that bias the outcome!


  1. Reginald Selkirk says

    claiming there was a “less than one in a quadrillion” chance that Biden could have honestly won the Texas election

    Well he didn’t. Biden lost Texas, but won the federal election.
    Something to keep in your back pocket: that the attorney general of Texas should be telling other states who they could have elected does not square well with “state’s rights.”

  2. says

    This is what is called “Standing” in court cases. If you ain’t got it you don’t get heard and are thrown out right away. We will see what the SC decides, is the law the same as it has been since 1789 or did it suddenly change.

  3. JoeBuddha says

    My favorite answer to the probability question? I bet I can flip a fair coin in a fair manner and get 5 heads in a row almost every time.

    Methodology: Start with 64 coins (to stack the deck, so to speak). Every time a coin comes up tails, remove it. Odds are you’ll end up with at least one flipping heads 5 times in a row. Or tails. You just can’t pick the coin ahead of time.

  4. PaulBC says

    There were already widely publicized statistical models based on polling data. Trump did much better than the median on FiveThirtyEight, but well within the range of realistic outcomes. Down-ballot Republicans also did better. Susan Collins’s win and its extent were, again, towards the tail in her favor. Republicans also did very well in House elections relative to polling. I’d have to look to find a number, but the probability that Republicans would do as well as they did is on the low side, but also well within the range of an ordinary chance outcome.

    This was not 2016 and turnout was much higher on both sides. I think that even the courts, which are not great with numbers typically, will get that point.

    Republicans should be celebrating or breathing a sigh of relief at least. The fact that they’re whining instead just shows their core belief that they should be given office without any opposition at all, particularly from “urban” voters.

  5. Reginald Selkirk says

    Furthermore, the time between the two presidential elections is short enough that significant changes as discussed in this declaration could not be due to underlying changes in demographic factors.

    WRONG. The number of people who have actually experienced a Trump presidency, and therefore know how horrible it is, has gone way up in the last 4 years.

  6. PaulBC says

    Also, if Paxton had any argument (which he does not) shouldn’t Democrats be able to keep Doug Jones as a senator and grab back a bunch of House seats that we lost. (Obviously due to “fraud” right?)

    Fraud is when people of color vote for Democratic candidates. They should just come out and say it.

  7. says

    Okay, let’s talk probability…

    So how powerful and knowledgeable is this God who supposedly created the entire Universe and everything and everyone in it? Why, he’s INFINITELY powerful and knowledgeable, of course — that’s a given.

    So how many different universes can such an infinitely powerful and knowledgeable God consider creating? Why, an INFINITE number and range, of course.

    So what’s the probability that God would create this particular universe, out of all those possible options he knew were available to him? That would be one in infinity. Which is to say, zero. Not “essentially zero,” mind you, not “near zero” or “infinitesimal,” but really, truly, exactly, literally zero point zero zero zip zilch nada buggerall fuggedaboudit.

    So there we have it: the evolution of humans and all other known life-forms from non-living matter is INFINITELY more probable than God creating the Universe we see. QEDone. (And I showed my work!)

  8. says

    Something to keep in your back pocket: that the attorney general of Texas should be telling other states who they could have elected does not square well with “state’s rights.”

    I was thinking this lawsuit might be taken as an excuse for AGs in, say, California, Massachusetts, New York, and Virginia, to countersue Texas, Alabama, Mississippi and maybe others over their long-running voter-suppression tactics. THAT suit would come with actual evidence and a more solid line of reasoning.

  9. PaulBC says

    Raging Bee@8 If this goes far enough that a countersuit makes sense, the judicial system has already failed us. Sen. Cornyn isn’t even buying it. It’s another dumb stunt, doomed to fail.

    However, it will succeed in convincing some Americans that “where there’s smoke there’s fire.” No, as Krugman said in another context, it’s a bunch of guys with a big smoke machine. They are now completely open about it, but it doesn’t make it ineffective.

  10. bionichips says

    What is the probability I exist? If it had not been for WW1 which led to WW2 my parents never would have met. My dad was stationed in NYC – what are the odds? If my mother had not been at the USO at the same time as my dad – what are the odds? So now lets go back to my grandparents. My maternal grandmother escaped Russia hiding in hay stacks? What are the odds he would have escaped the soldiers with pitch forks and made it to NYC where he met my maternal grandmother who escaped Poland to make it to NYC. And so forth and so on. Say 1/10 for each predecessor. For 64 generations. So the probability I exist is 1 one a trillion – yet here I am.

  11. jenorafeuer says

    @Raging Bee, PaulBC:
    Several people have been pointing out that Paxton has his own (multiple) problems with pending cases against him personally, and this whole thing may just be a way of angling for one of the various Pardons that Trump is handing out like party favours.

  12. PaulBC says

    jenorafeuer@12 Indeed, and if the goal is to prove your loyalty to Trump, then maybe the more asinine the argument, the better. If Trump said he’d give you a pardon to pee your pants on television, then you might do it if the alternative is a prison sentence. Dance little monkey, dance!

  13. Reginald Selkirk says

    @12: That would only be possible if any charges against Paxton were federal, not state.

    @13: You would also have to calculate in the probability of Trump keeping a promise.

  14. jenorafeuer says

    @Reginald Selkirk:
    Well, the FBI is investigating him; a lot of the charges appear to be things like securities fraud, which are definitely federal in nature.

    As for ‘probability of Trump keeping a promise’, that’s definitely an issue. (See also the ‘Trump Tax’ New York construction companies used to levy on his projects where they’d double the initial bid because they expected to only ever see the half paid up front.) And it’s not looking good in this case, because the case fell apart so quickly.

  15. KG says

    one in a quadrillion to the fourth power (i.e., 1 in 1,000,000,000,000,0004)

    Er… wot? 1 in 1,000,000,000,000,0004 is 1 in ten quadrillion and 4, with commas in unusual places. The “4” at the end should be superscripted.

  16. chrislawson says

    I love the probability argument. There are 3,116,480 letters in the King James Bible. Given 52 letter choices, that means the probability of the KJV being the word of god is 52^3116480, which is 10^823,555*. Clearly not feasibly possible. Right?

    (To be clear, this is a googol to the power of 8236. Or roughly the numbers of atoms in the known universe to the power of 1000.)

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