Good Math/Bad Math

Wednesday, May 31, 2006

Repeat after me: Improbable IS NOT Impossible

I got a request to respond to look at this. It's yet another one of those innumerate jackasses arguing that god must exist, because life is too unlikely otherwise.

It's the same old crap, but with one interesting twist - this guy tries to provide an argument for why anything less probable than a specific level of improbability is absolutely physically impossible. In his own words:
In a previous post, I presented a logical, factual argument for the existence of God. But one of my readers, Seanny McShawn, took issue with my syllogisms, saying that I was simply making an argument “from incredulity”.

When someone tries to win an argument based on simple probabilities, this is called an “argument from incredulity.” This is a logical fallacy. In other words, the sheer unlikeliness of a scenario does not preclude its possibility and cannot be proof against it.

But was I arguing from “incredulity”?
The answer is, pretty much, "yes".
Physicists estimate that in our universe there are 10^80 particles. Mathematicians say that the mathematical level of absolute impossibility is 1 chance in 10^50.
Not any competent mathematician.
However, the physical level of absolute impossibility is 1 chance in 10^80, and here’s why:

On the basic level, probability is defined by the ‘particle’ example: finding a specially marked particle among 500,000 particles is beating odds of 1 in 500,000. In a universe that has 10^80 individual particles, the most improbable scenario is finding a specially marked particle in the entire universe. Due to the size of our universe, it is impossible to have a more improbable set of odds than 1 chance in 10^80. Anything that is more improbable than the most improbable is by all standards absolutely impossible.
I have never seen this particle example nonsense. But nonsense is what it is. It is, in fact, astoundingly easy to demonstrate.

Ready? Are you sure? Ok. Here goes:
That's 286 random characters, produced by pounding my hands on my keyboard, and then deleting any punctuation characters (because I'm too lazy to figure out how many unshifted characters there are on my keyboard, so I stuck to numbers and letters.) The odds of generating that particular string are 36^286. That's quote a bit less likely than 10^80. But guess what? It's possible!

Or, just to annoy the fundies, go get yourself a deck of Tarot cards. (Hey, they're good to have. There are some damned neat poker variants that you play with a Tarot deck, not to mention some funky solitaires.) A tarot deck has 78 cards. Shuffle the deck. Lay the cards out in order. The odds of that ordering? Less than 1 in 10^115. But entirely possible.

What is the probability of a solar flare taking a specific shape? Think of the trillions of trillions of particles in a flare, and the unique shape of each one. The probability of those shapes? Quite a bit worse than 1 in 10^80.

This one in 10^80 stuff is pure nonsense. It's just another attempt to create an arbitrary threshold beyond which we can say something is impossible.

He then goes on to repeat yet another variant of the "odds of life based on the probability of a particular protein" gibberish. I've covered that plenty of times on this blog; for example, my analysis of Berlinski's version of this argument here.

So this guy concludes with:
This is not an argument from incredulity. This is an argument from facts: cold, hard facts. Since any set of odds above 1 in 10^80 is absolutely impossible, random chance could not and did not produce life.
Except that it is an argument from incredulity, utterly lacking in cold, hard facts. In fact, it's trivially refuted by cold hard facts; and it's based on one of the most pathetic attempts I've seen so far to define "impossible" in terms of probability.


  • A very cogent and helpful post! I know several people I'm sending here to read this. Thanks!

    By Blogger The Ridger, FCD, at 8:02 PM  

  • The argument isn't just bad math, it's bad physics.

    You can't mark particles. Particles are completely indistinguishable. Physics, including many important macroscopic properties of matter, depends on this issue of particle indistinguishability, including such obvious things as the apparent solidity of matter.

    Another way to look at the "state of the universe" is imagine all possible states a particle can be in. Each particle can be in one of those state (or some combination of states where the probabilities add up to one). You can then characterize the state of the universe as a set of occupation numbers; that is, a list of all the states and how many particles are in that state. For some kinds of particle, the number can't be more than 1 (electrons, protons) and for other types, you can have as many particles as you want in a given state.

    The reason you can't mark particles is that the mark you'd put on one adds additional state; you've actually changed the system to one that doesn't match nature.

    Of course, the problem gets even more interesting when you factor in field theory; the number of particles in the field becomes a matter of probability as well.

    In short, the whole argument is incoherent.

    By Anonymous Anonymous, at 8:19 PM  

  • Wonderfully succinct. I'd only suggest next time use the generator at and save your fingers.

    By Anonymous usagi, at 8:49 PM  

  • I thought it was silly before... and now anonymous shows me an extra level of nonsense I completely missed.

    ...This guy didn't happen to be named Interesting Ian? I've heard him make similar arguments as if very, very tiny probabilities "might as well be zero." Of course, he performed a lot of Texas Sharpshooter, presuming that this world was the one and only possibility.

    Reminded me of Calvin of Calvin and Hobbes claiming that all of history unfolded the way it did specifically to produce him.

    By Blogger Bronze Dog, at 9:58 PM  

  • Sorry to pick nits but:

    "That's quite a bit less likely than 10^80" and I think you meant that a tarrot deck has 78 cards.

    By Anonymous Anonymous, at 7:53 AM  

  • Tarot is the spelling I see most places. I've also seen Taroc. I've never seen Tarrot.

    Let's consult Google:

    "Tarot deck" 1,120,000 hits
    "Tarrot deck" 519 hits
    "Taroc deck" 2 hits

    *shrugs* Maybe "tarrot" is a regional variant some place, but it doesn't seem to be the preferred spelling.

    By Blogger Qalmlea, at 11:43 AM  

  • If what this nut is saying is true, I wonder what would happen if I rolled a fair d(10^80+1). (A die with that many sides.)

    By Blogger Bronze Dog, at 12:24 PM  

  • bronze:

    His reponse to that is perfectly predictable: since there are only 10^80 particles in the universe, you couldn't create something with 10^80 + 1 sides. :-)

    By Blogger MarkCC, at 12:41 PM  

  • If what this nut is saying is true, I wonder what would happen if I rolled a fair d(10^80+1). (A die with that many sides.)

    Being more perfectly spherical than any existing sphere, said die would never stop rolling and hence prove the existence of God.

    By Anonymous BMurray, at 12:55 PM  

  • I've encountered the "too improbable to be possible" argument before. My argument in the past has been something along the lines of: you not only have to know the odds of something happening, but also the number of attempts you are allowed. Let's say that your odds of winning the lottery are 1 in a million. Are those good odds? Are you going to win? Well, you have to know how many lottery tickets you have to figure out what your actual odds are. The same thing is true for very unlikely outcomes. Let's say that the odds of something happening are 1 in 10^100 and that you get 10^100 attempts (chosen randomly, not sequentially). Your overall odds of making that 1 in 10^100 is around 67%. I've run some tests on this and, based on my calculations, for any situation where your odds are 1 in n, and you get n attempts, your overall odds approach 67% as n increases. (1 in 2 with 2 attempts = 75%, 1 in 3 with 3 attepts = 70%, 1 in 4 with 4 attempts = 68%, 1 in 5 with 5 attempts = 67.2%, ...)

    His use of "1 chance in 1.0 x 10^120,412" isn't relevant because (1) this isn't the only sequnce that would work (based on comparisons of genes, every gene has a large variety of workable versions), (2) there are almost certainly other classes of genes that can perform the same function, (3) the universe gets more than one attempt (the universe gets billions of years + trillions of interactions) to make the desired outcome, and (4) his number of "200,000" base pairs is almost certainly an overestimate (assuming a primitive organism must have x genes because the world's *current* simplist organism has x genes is flawed because a primitive organism lived in a simpler environment - free of predators among other things).

    By Anonymous BC, at 5:19 PM  

  • And while we are at it nobody knows how many particles are in the universe. When physicists speak of the size of the "universe" or the number of particles in the "universe," they are normally talking about the "visible" universe--that part of the universe that is potentially observable, given the speed of light and the amount of time since the big bang. That doesn't mean that's all that there is--only that there hasn't been enough time for light from beyond that point to reach us.

    By Blogger tgibbs, at 6:07 PM  

  • "I've run some tests on this and, based on my calculations, for any situation where your odds are 1 in n, and you get n attempts, your overall odds approach 67% as n increases."

    Not to nitpick, but it approaches 1-e^-1 which is about 63.2%; that's the odds against having no successes. It's a good ol' fashioned Poisson distribution.

    By Anonymous Anonymous, at 7:29 PM  

  • When I said a tarrot deck had 78 cards, I meant to correct the original post which said a solitaire deck has 78 cards (it still says that). I don't care how you spell "tar(r)ot".

    By Anonymous Anonymous, at 12:25 PM  

  • anon:

    Sorry, I misinterpreted the original correction; like the other commenter, I thought you were saying I misspelled Tarot. At the moment, Blogger won't led me edit the post; once things get working correctly again, I'll fix that. Thanks for the correction!

    By Blogger MarkCC, at 12:52 PM  

  • Mark, you're probably already aware of this, but I got to wondering where the creationist got the magic number 10^50. Why is an event with a probability of 1 in 10^50 impossible while an event with a probability of 1 in (10^50)-1 only very unlikely?

    If anyone else is interested, this number comes from some comments made by Emile Borel and (surprise!) ripped completely out of context. Some have even gone so far as to claim that the rule, "Events with a propbability of 1 in 10^50 are impossible," is something called "Borel's Law," known to every mathematician and scientist (but strangely absent from every standard mathematical dictionary and encyclopedia).

    If you do a Google search for "Borel's Law," you'll get a nice combination of pages that apply the law in creationist arguments and pages that patiently explain that there is no such thing. John Stockwell's piece at is especially nice, and was at the top of the list when I searched.

    By Blogger bcarson, at 3:50 PM  

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