Statistics

Probability Doesn’t Exist: Nothing Has A Probability

I hope you’ll forgive this minor repetition, my friends. But since the world has not yet converted to the truth, shouted by de Finetti and handful of others, that probability does not exist, and therefore nothing has a probability, it is my duty to press on.

At the least, this truth is badly needed in physics, in which quantum mechanics is mired in an instrumentalist set of beliefs that many have mistaken for Reality. As briefly as possible, a model may make good, even excellent predictions, but this does not prove that the model’s premises accurately describe Reality. Ptolemy’s epicycles are all the argument we need here.

Some of you might recall my dad and I made a coin-flip machine last fall, consisting of a clothespin, some nails, and a chunk of two-by-four. We made a video demonstrating the beast. At last count, it was up to almost a dozen views, or whatever. Is this viral?

Point of the video was to prove that coin flips do not have probabilities. The video met that burden because with every flip the coin came up heads. And continues to. (I tweaked it a bit so that it was more stable. How? By hitting my thumb into the nails.)

In other words, given the operations of the machine and coin and flip, the probability of a head is 1. This probability, like all probabilities, is conditional on the information given or assumed. There is no probability in the coin, or in the flip, or in anything.

That’s it. That’s all of probability. Simple.

Alas, you still hear talk of “data generating mechanisms” and “true distributions” and “parameter estimation” and on and on. So our work is not done.

There are bright spots, though, which form the excuse for today’s post. Take this article from Popular Mechanics: Scientists Figured Out How to Design Dice to Roll Any Way You Want.

Dice throws don’t have probabilities either. Dice outcomes have causes. And these guys, like the coin guys we’ll meet below, have figured out some of the causes. Here’s the barest details we need to get the flavor of this. Their paper can be found in the link.

Scientist Yaroslav Sobolev at the Institute for Basic Science in Ulsan, South Korea—along with his colleagues—have designed an algorithm that creates wonky-shaped objects called “trajectoids” that mathematically travel along any set path. The results of the study were recently published in the journal Nature…

“For any path, you can always find such a sphere, of some radius, that when it completes two periods of the path, it will restore its 3D orientation perfectly,” Sobolev tells New Scientist. “This allows you to make a particle that will roll forever downhill, always tracing the path again and again.”

The algorithm works by tracing a moldable sphere’s contact points with the ground as it travels a predetermined path. The team then created a 3D printed shell to cover the hard metal interior and tested the results against the mathematically designed path. The trajectoids followed the designed path and even successfully repeated the path twice in most cases. (If you’re a 3D printing aficionado, you can download the trajectoid algorithm and try it out for yourself.)

So far this is all abstract. It works in the math, but they haven’t shown it in real life. The reason is simple: the oddly shaped dice have to be tossed on real surfaces. Just as our coin had to land on a real surface, which we jury-rigged as a soft pad to absorb the forces of a fall, those dice guys would have to figure what the surface does to the roll.

Even so, they have nailed the key fact, which is that dice tosses have causes, not probabilities.

Finally we come to this piece, from Smithsonian Magazine: “Gamblers Take Note: The Odds in a Coin Flip Aren’t Quite 50/50: And the odds of spinning a penny are even more skewed in one direction, but which way?”

Their paper is also at the link.

Diaconis is a professor of mathematics and statistics at Stanford University and, formerly, a professional magician. While his claim to fame is determining how many times a deck of cards must be shuffled in order to give a mathematically random result (it’s either five or seven, depending on your criteria), he’s also dabbled in the world of coin games. What he and his fellow researchers discovered (here’s a PDF of their paper) is that most games of chance involving coins aren’t as even as you’d think. For example, even the 50/50 coin toss really isn’t 50/50 — it’s closer to 51/49, biased toward whatever side was up when the coin was thrown into the air.

But more incredibly, as reported by Science News, spinning a penny, in this case one with the Lincoln Memorial on the back, gives even more pronounced odds — the penny will land tails side up roughly 80 percent of the time. The reason: the side with Lincoln’s head on it is a bit heavier than the flip side, causing the coin’s center of mass to lie slightly toward heads. The spinning coin tends to fall toward the heavier side more often, leading to a pronounced number of extra “tails” results when it finally comes to rest.

Persi was for a time one of my PhD advisors when he was at Cornell. We shared an interest in amateur magic.

He also built a coin-tossing machine (way before I did, which is where I got the idea). Here is a picture:

A tad better than mine.

But we share the same conclusion: “We conclude that coin-tossing is ‘physics’ not ‘random’.”

We’ll save quantum mechanics for another day.

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Categories: Statistics

19 replies »

  1. “Alas, you still hear talk of “data generating mechanisms” and “true distributions” and “parameter estimation” and on and on. ”

    Every semester I’d tell my statistics students that all those things were simply MODELS, a short-hand mathematical approximation of some phenomenon. But n-o-o-o! Everyone from the most boneheaded freshman to widely published Ph.D.s wants to reify the damn things and pimp ’em up with the most turgid advanced mathematics. And “random?” Better to say “uninformed as to causes,” just as the term “error” is (almost honestly) described as “stuff we didn’t put in the model.”

  2. What does this imply? That the physical world is purely deterministic? If we know all “ins” then we know with 100% certainty the “outs”?
    What about us? Are we purely deterministic machines or is the spiritual different? And if it is different, how much effect does it have on physical world, causing the “ins” not always resulting into predetermined “outs”?

  3. The implications of this article is that the world is determined which leads to the next question of free will by Hun.

    Does it exist? A very popular Stanford professor says free will does not exist. His name is Sapolsky and has written a book titled “Determined.” He claims there is no creator who one would think could establish free will in his creations.

    But he has no basis for his claim that there is no creator. He just assumes it.

    Evidence overwhelming points to a creator.

  4. While there exists a math model (yes, a math model, a generating mechanism) to describe the deterministic coin-toss process, I think coin flip can be argued to be random in two ways. A small change in the initial parameter (e.g., strength and angle of tossing, landing surface) can produce a different outcome, which means the process behavior is chaotic. And the outcome of a coin flip is random, i.e., unknown, regardless of whether it is 50/50 or not.

    Anyhow, models (math) only say what they’re told to say. There can therefore be no discovery by models. Ha.

  5. Well, no – every discrete event has a probability. 1 if it happens, 0 if it doesn’t.

    if P(E)=1 or 0 than E has happened (exists, whatever) and we have perfect knowledge. Otherwise
    0<P(E)<1 means that P(E) is an estimate describing our knowledge (and our confidence in that knowledge) about E.

    Thus, before reading this article I'd have said P(twirling coin lands tails up) = 0.5 because I have no knowledge about the outcome – but you would have said p(twirling..) = 0.8 because you know something about it – same event, different estimates reflecting different levels of knowledge.

    Bottom line: P(E) is about our knowledge of E, not about E.

    re: JerryR and JH: Yes, the world is 100% deterministic and free will is an illusion reflecting a lack of knowledge – but it is indistinguishable from some imaginary "real" free will because there's no way for us to even list the predecessor events for most E -(remember it's really always P(E|given a whole bunch of stuff) never mind know anything about their outcomes. See telearb.net/node/14 for more.

  6. Paul Murphy, if everything is deterministic and free will is just an illusion, then there is no morality, no responsibility, nothing matters and anything goes.

    Of course, the world being 100% deterministic is just your belief. It’s not possible to know this with 100% certainty. If you think you do know, then you’re lying to yourself. But then that doesn’t really matter, because according to your own belief, we are just automatons and everything is predetermined and thus there is no point in arguing or doing anything at all.

  7. Arguing, that free will is an illusion is committing a logical fallacy.

    It is based on assumptions that are not true. So it is based on something that has no place in reality.

  8. The purpose of a trajectoid is not to be rolled. Its purpose is to be pushed over, one face at a time, so that it follows a set path along the ground. A normal six sided die, when pushed thus, will travel a straight line.

    There is a relatively fun video about this.

  9. Models are maps, reality the territory. Never confuse the two.

    Physicists have spent generations in the futile pursuit of the models while ignoring reality. The current “crisis in cosmology” is caused by reality not adhering to their most beloved model. Astrophysicists blame reality, and demand new and more expensive observations.

  10. Probability Doesn’t Exist: Nothing Has A Probability

    The second part I’m OK with, in that probability is not a property of any thing, but is the output of a model of the world and things.
    It’s the first part. Probability being the output of a model, it clearly exists in the same way that an integer exists,or pi exists (or even the Love of God exists?). So the “Doesn’t Exist” refers to a different sort of existence; of what sort I am unclear…
    Put another way: were I to say to some bloke down the pub “Probability Doesn’t Exist”, how would I justify that statement?

  11. gareth,

    I mean by it only that it doesn’t have physical existence; it is not a substance; it has no powers.

    But that makes for a mighty long headline.

  12. So how about “Probability doesn’t exist, except as an imagined number. It doesn’t “cause” anything, such as dice throw outcomes; they just happen exactly as they happen. Nothing “has” a probability – Probability isn’t a thing or a property of something. It’s just a number we assign to whatever we suppose might happen in the future, based on what we think we know about what might happen (which we never fully do).”
    ???

  13. Great article and super comments. Regarding the one about random numbers, there was a Scientific American article about 30 years ago that plotted “random” numbersbin a 3D point cloud chart, and they did seem random, until the cloud was rotated to reveal a clear pattern in the points’ arrangement. This reality is pregnant with meaning. It implies meaning in the universe, not the meaninglessness that’s so worshipped by secularists. I canardly wait for Briggs to scold the Quantum Mechanickers—and that’s no lie!

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