When Does A Small Probability Mean Impossibility?

Question from reader Richard Stevens:

A professional scientist, holding an MS in one of the hard sciences, addressed the argument that the probability of creating a protein or an enzyme by neo-Darwinian means is vanishingly small, e.g. 1E-150 or perhaps very much less. He did not question the mathematics. He argued simply that a chance of 1E-150 means it is “possible, not impossible,” and thus life “could have” arisen from non-life, or animal species could have arisen starting with a replicator molecule. Dr. Briggs, have you written to address the question of whether a sufficiently small probability is equivalent to zero probability?

Impossible is, it is thought, a probability of 0. Any probability greater than 0 would therefore seem to imply possible, just as 0 seems to imply impossible. But this is not so.

First and always remember that all probability is conditional. Every single one. Therefore, though as shorthand we can say “This has a probability of 1E-150”, we always mean “This has a probability of 1E-150 given these assumptions.”

Perhaps non-intuitively, the size of the probability has nothing to do with its possibility.

Start with an example of an impossibility with large probability. Here are the assumptions, or premises or evidence, of the probability. E = “In the room are a unicorn and Pegasus. Exactly one, and only one, will walk out.” Given E, what is the probability of Y = “Pegasus walks out”? Easy (via the statistical syllogism): 50%.

This example can be tweaked in the obvious way to make the probability higher. But 50% is surely much higher than 1E-150. Yet the event will never happen.

I conclude the event will “never happen”. By that I mean the probability of Y is 0. But since all probability is conditional, I must have had some conditioning evidence in mind. Call it E_2 (making the original E_1). My E_2 = “Pegasus and unicorns don’t exist, so no matter what E_1 says, neither can walk out of any room.”

My E_2 is directly related to the cause, or lack of cause, of Y. Y cannot be caused, no matter what, given E_2. Thus the event is impossible, even though E_1 gave it high probability. E_1 said nothing directly about cause. Though even if we put in an explicit cause, nothing would change. The event will still not happen.

I will leave as homework to find an example with small probability for an observable Y that is bound to happen.

It should be clear by now that what’s important are the premises of the probability, and not the probability itself per se. We judge the premises of the probability like we judge any other proposition. Some may be certain, some uncertain, some false. And all these judgments are with respect to other premises which are themselves more certain, and so on. Think of how mathematical theorems are built: chains of argument.

Since the premises are what is important, it does nothing for us, or should do nothing for us, to hear, for any Y, “The chance of Y is 1E-150” or even 1E-15,000,000, or however small you like but still greater than 0. This is because we hear nothing about the premises; therefore, we can say nothing about what the probability means.

It is worth pausing for a moment to emphasize this. Whenever you hear someday say “The chance of that is X%”, and you find yourself nodding along, it means that you have supplied the necessarily premises from which the probability is deduced. Whether your assumed premises match those of the speaker is an entirely different question. This allows the possibility of cheating. Always ask for details!

Thus is all we do is here some low probability and don’t know the defining premises, there is nothing we can say. We can’t even say the low probability is a “practical impossibility”. Impossibility is a causal word, not a probability word. Impossible speaks to the known utter lack of cause, in the given circumstance.

Thus it also does us no good to use the phrase “chances to occur”, which some use in connection with low probabilities. “Chance to occur” makes sense where the causal mechanism is known, or rather the outlines of it known, as in, say, dice throws. We don’t know the precise causes, for if we did, the probability would be extreme (0 or 1) for any throw outcome. But we understand what is means to make the throw, and that when the throw is made, we know what the possible outcomes are.

This is not the case with the 1E-150 probability. We do not know who is throwing the dice, so to speak. For example, suppose we can with reference to accurate biological causes, of the same limited sort as dice throws, deduce the 1E-150. We still do not know how and when the “throw” of amino acids occur. At a casino, dice throws are every, say, 30 seconds. An amino acid throw? Who knows? Every fraction of a second? Every year? What?

This is why I say most (I can’t say all, since I haven’t seen all) probability calculations with regard to Darwinian evolution are a bluff or a mask. We simply have no idea what’s happening. Pointing to small probabilities and then waving hands at very long times is how evolutionists “solve” these probability problems, but these “solutions” rarely hold up.

Here’s a better example, and then a puzzler, also for homework. E = “In a bad we have a slew of black balls and one white one. We reach in, with no peeking, and grab one out.” What is the probability Y = “Ball is white”?

Pr(Y|E) = small.

If you replace “slew” with a number, you replace “small” with one over that number. Any finite “slew” makes it easy for us to understand the causal mechanism, including the how and when. How often you reach into the bag (assuming replacement of the balls or not), makes the causal process clear. All probability calculations are easily deduced, and make physical (biological) sense.

Now let “slew” go to infinity, indeed the size of the infinity of the reals, for now our balls are real numbers, only one of which we’ll call “y”. We reach in and grab “y”. What is Pr(Y = y | E updated)?

Be explicit. What does cause mean here?

This is answered in Uncertainty.

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27 Thoughts

  1. It is not possible to attach probabilities to everything. For example, attaching probabilities to a scientific theory is meaningless. The notion of attaching a probability to a product of evolution seems dubious, partly at least in that there is no stochastic element in the theory, despite the propaganda.

    In an attempt to remain is the spirit of the post, I’m tempted to ask, what is the probability of an enzyme critical to the existence of a species to have evolved, given that the species exists?

  2. Typo: “Thus is all we do is here some low probability”
    Typo: “In a bad we have a slew of black balls”

  3. Because all probability is conditional on the assumptions, how do we evaluate the “truthfulness” of the assumptions? By truthfulness I mean having confidence that the assumptions are accurate/reasonable/relevant. Without some assurance the assumptions have validity, it seems the probability of the model conclusion being useful is unhelpful and pointless. A practical answer is desired.

  4. @Gary

    A practical answer from 1878:

    “Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.”

    —C.S.Peirce

    What has the most interesting practical consequences., e.g. predictions? Confusing natural philosophy with theology seems to be common in this area of little detailed data.

  5. @Gary
    When there’s lots of data and experimenting is practical, people seem to say “micro evolution” and roll with it. When there isn’t, it’s “macro evolution” and people act as if their hair is on fire.

    I am unaware of any experimental fossil formation experiments. Some folks are experimenting with practical abiogenesis or artificial life in the lab.

  6. Getting injured or property damage by a meteorite is very rare (tiny probability) — it has happened — but is “equivalent” to zero for most of us.

    The simple example may inform more than numbers.

    The question posed is also curious: kind of looks like the questioner was striving to get a “yes” to “equivalent” to then say that natural processes cannot generate like … therefore god … The probability that that is what the questioner was really aiming for is, probably, high…100 percent perhaps?

    Briggs didn’t take the bait.

    But that sort of manipulation is common – people pull such intellectual brinksmanship to get answers wanted vs real truth. Much more than other genuine errors.

  7. This regular reader poses a question: what happened to Good News Week? Not that I mind especially the presented content; it’s just that something was promised and not fulfilled. Could it be that Good News is highly improbable or even an impossibility? I don’t think so …

    Here’s a suggestion: what is “grace”? Where does it come from? What good is it? What is the probability of grace? Answer that one, Doctor.

  8. “I will leave as homework to find an example with small probability for an observable Y that is bound to happen.”

    As Prince sang: “Sometimes in snows in April.”

  9. Consider the Monty Hall problem. You guess that the car is behind door #1. The announcer opens door #2, revealing a goat, and asks if you want to change your guess to door #3. Based on what you know, there’s now a 66.6% chance that the car is behind door #3. But the announcer knows the car is behind door #3 with 100% probability. Someone just returning from the bathroom doesn’t know which door the contestant chose first; he only sees one goat and two closed doors, so to him it’s 50/50 between doors #1 and #3. In macro-scale classical physics, there’s no such thing as probability, only imperfect knowledge and casino games with high sensitivity to input conditions.

    Nuclear physicists can say, “Uranium-235 decays to Actinium-231 with 75.3% probability”, based on the observation of a large number of decay events. When a climate scientist says, “the north polar icecap will be gone by 2030 with 78% probability”, he’s just pulling numbers out of his ass.

    We know that the probability P of intelligent life arising on any randomly-selected planet in the universe is not zero, because we exist. We’ve discovered over 4,000 planets around other stars and can infer the existence of trillions more, yet see no signs of intelligence. No radio signals, no rocket plumes from interstellar ships, no giant solar panels capturing the energy that stars radiate into space, no E.T.s landing on the White House lawn, so P must be very, very close to zero. Civilizations might be so widely spaced as to be causally disconnected, meaning that not even a photon could ever travel from one to another.

  10. Dave,

    Your non zero probability of life “arising” is itself contingent on an assumption that life can arise. What if it can only be Created and the Creator only did that here? Then zero probability.

  11. Embedded within this cloud of tedious confusion there is exactly one specific assertion:

    ‘If you replace “slew” with a number, you replace “small” with one over that number. ‘

    and it’s incorrect. The student taking a first course in probability or statistics usually stops making this error, if he was ever befuddled enough to make it, after the first couple weeks of class.

  12. from the conclusion of “Nexus”
    On 7 February 2016, a bus driver named Kamraj was struck and killed by a meteorite on the campus of Bharathidasan Engineering College in Vellore, Tamil Nadu State. Scientists and media marveled at the unlikeliness of the event and said he was killed by mere chance. But Kamraj was not killed by chance, he was killed by a meteorite. Chance is not a cause, even if she strikes like a hammer.

    Causation is vertical, not horizontal. That is, there is a cause for your flat tire and a cause for the moon being in quarter phase; but for getting a flat tire while the moon is in quarter phase, seek no cause. That way lies madness. Or astrology.

  13. “in a bad we have…”

    i’m not putting my hand into a bad. a bin, perhaps. a bad? no way. there might be a unicorn, or pegasus. that bites. that would be bad.
    with p=1 i might add, with appropriate wee pee value. or not. it’s still bad, way bad.

  14. Per, I assume that it’s possible for life to come into existence because it did, somehow. How do you know that the Creator did not create life elsewhere as well? I doubt that anyone but Richard Dawkins would see the discovery of extra-terrestrial life as a disproof of Christianity, though I would be very curious to know what the E.T.s believe about life, death, and resurrection.

    BTW there’s a trope in sci-fi where a mad scientist creates sentient life in a test tube, this life reads Origin of Species and concludes that it evolved from primordial slime, fanatically clinging to this belief against the scientist’s insistence that no, I created you guys.

  15. “An amino acid throw? Who knows? Every fraction of a second? Every year? What?”

    From Evolution FAQ:

    “For example, the simplest theorized self-replicating peptide is only 32 amino acids long. The probability of it forming randomly, in sequential trials, is approximately 1 in 10^40, which is much more likely than the 1 in 10^390 claim creationists often cite.

    Though, to be fair, 10^40 is still a very large number. It would still take an incredibly large number of sequential trials before the peptide would form. But remember that in the prebiotic oceans of the early Earth, there would be billions of trials taking place simultaneously as the oceans, rich in amino acids, were continuously churned by the tidal forces of the moon and the harsh weather conditions of the Earth.

    In fact, if we assume the volume of the oceans were 10^24 liters, and the amino acid concentration was 10-6M (which is actually very dilute), then almost 10^31 self-replicating peptides would form in under a year, let alone millions of years. So, even given the difficult chances of 1 in 10^40, the first stages of abiogenesis could have started very quickly indeed.”

  16. All,

    swordfishtrombone’s comment is a perfect example of why you can’t trust probabilities in evolution arguments. Notice how every rule was violated in his quotation, just as is explained in the post. “The probability of it forming randomly, in sequential trials, is approximately 1 in 10^40, which is much more likely than the 1 in 10^390 claim creationists often cite.”

    1 in 10^40 versus 1 in 10^390 forsooth! Neither means anything. And you should now be able to say why.

  17. “1 in 10^40 versus 1 in 10^390 forsooth! Neither means anything. And you should now be able to say why.”

    But ignorance is bliss, and the fish-horn is incredibly happy!

    Passive aggressive ignorance is the worst kind. What sort of sick sado-masochist is that dude to continue spewing on your posts?

  18. @ Kent Clizbe,

    “What sort of sick sado-masochist is that dude to continue spewing on your posts?”

    LOL – Sounds like something Robin might have said to Batman in the 1960’s Adam West TV version.

  19. Spot on, fishy!

    And to carry that analogy further–everything you say sounds like it was lifted from the lines of a Batman villain!

    The Joker! The Penguin! Louie the Lilac!

  20. @ Briggs,

    “1 in 10^40 versus 1 in 10^390 forsooth! Neither means anything. And you should now be able to say why.”

    Even though you’re an expert statistician, I can disagree with you because you’re not making a statistical argument, only one about assumptions which are physical in nature.

    “We simply have no idea what’s happening. Pointing to small probabilities and then waving hands at very long times is how evolutionists “solve” these probability problems, but these “solutions” rarely hold up.”

    We don’t have “no idea” what’s happening. You should look into the research that’s been done into abiogenesis before dismissing it as “waving hands”.

  21. ‘We don’t have “no idea” what’s happening.’

    But if by “we” the author meant himself and a few equally benighted pals, he’s correct. They have no idea, about biology, physics, or, apparently, how to solve elementary probability problems. At least, that’s how I interpreted the admission.

  22. @ Briggs,

    “Good grief”

    As far as I can see, the question asked by your reader, “have you written to address the question of whether a sufficiently small probability is equivalent to zero probability?” Seems to be purely mathematical in nature, and the answer should have just been “no”.

    “On abiogenesis, this:”

    The informal opinion of one scientist who isn’t actually studying the field he’s criticising is ultimately irrelevant.

    @ Lee Phillips,

    “But if by “we” the author meant himself and a few equally benighted pals, he’s correct.”

    We _don’t_ have “_no_ idea” means we DO have an idea – have you not heard of a double negative? “We” refers to mankind. Mankind’s current state of knowledge isn’t zero.

  23. Swrodfish,

    The informal opinion of one blog commenter who isn’t actually studying the field he’s criticising is ultimately irrelevant.

  24. Swordfish: we’re both commenting on the author’s remark that

    “We simply have no idea what’s happening.”

    and I was pointing out that his statement is correct if “we” refers to himself and others who share his lack of knowledge, rather than to mankind. I hope this is clear now.

    “The informal opinion of one blog commenter who isn’t actually studying the field he’s criticising is ultimately irrelevant.”

    Or blog author, right?

  25. “1 in 10^40 versus 1 in 10^390 forsooth! Neither means anything. And you should now be able to say why.”

    One of the assumptions seems to be that each combination is equally likely, whereas some combinations are more likely and some less likely, maybe even impossible because of the way the electric charges present themselves on the surfaces of molecules.

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