Good Math/Bad Math

Wednesday, May 31, 2006

Improbable is not impossible, redux

I actually decided to say a bit more about the concept of improbable vs. impossible after my last post.

As I've said before, the fundamental idea behind these arguments is what I call the "big numbers" principle: We (meaning human beings) are very bad at really grasping the meaning of very large numbers. It's hard to really grasp what a million is: if you counted one number a second, with no breaks of any kind, it would take you more than 11 days to count to one million. When we start getting to numbers that dwarf a million, our ability to really understand what they mean - to really grasp on a deep, intuitive level what they mean - just completely goes out the window.

What does 10^20 mean? If it takes 11 days to count to one million, then it would take roughly 11 times 10^14 days - or 3 times 10^12 (aka 3 trillion) years to count to 10^20. That just doesn't mean anything on an intuitive level. So what does 10^100 mean?

What happens when we start to look at probability? Well, it's really easy to get huge numbers in probability calculations - the kinds of numbers that we can't really grasp. And because those numbers are so meaningless to us, so utterly beyond the reach of what we really comprehend, the probabilities of astonishingly common events can seem like they're entirely beyond the realm of possibility: they must be impossible, because how could anything so easily comprehensible create numbers so incomprehensible?

Who hasn't shuffled cards? Who hasn't played a game of klondike solitaire? And yet, every time we do that, we're witnessing an event with a probability of roughly 1 in 8 times 10^67!

If something as common as shuffling a deck of cards can produce a result with odds of 1 in 10^67, then think of the probability of something uncommon: what's the probability of the particular arrangement of molecules in a particle of dust? Or the shape of a comets tail? Those are all natural, possible events - and the probability of them happening the way that they do is pretty much beyond our capacity to imagine.

When you grasp that, then you start to realize just how meaningless the big numbers argument is. No matter how improbable something is, that does not mean that it is impossible. Impossible is a probability of 0. And any probability, no matter how small, is almost certainly the probability of something that's occurring in the universe, somewhere, right now.

21 Comments:

  • Ah, the wonders of taking a brain adapted for the plains of Africa and making it try to understand the universe.

    By Blogger Thomas Winwood, at 11:07 PM  

  • I once belonged to a large computer users group that could draw major speakers to its monthly meetings. There were lots of giveaways, too. Members were given raffle tickets at the door. Product reps would show their wares and then draw winning numbers from a bucket. One fun-loving guy drew a couple of tickets in the usual away, bestowing prizes on the folks with the right numbers, then changed the script. He reached into the bucket, grabbed a handful of tickets and dumped them on the floor. "Okay, those guys lost," he said. Then he picked some more winners, giving away several more software packages.

    People went nuts. I heard absolutely furious people demanding that no one ever be allowed to do such a horrible thing again. "What if he threw away my ticket?" they cried. "He robbed me of my chance to win!" There was no way to persuade them that their chances of winning a prize during the evening had in no way been affected by his method of drawing (or discarding) raffle tickets. Everyone had the same overall chance. But lots of people fixated on the possibility that their winning numbers had been trampled under foot.

    By Blogger Zeno, at 11:24 PM  

  • "And any probability, no matter how small, is almost certainly the probability of something that's occurring in the universe, somewhere, right now."

    What a neat comment! Very subtle and true. I misread it at first, then looked again. The problem for me happens when the number of individual outcomes is immense, but the number of classes of outcomes I care about is small. Then one has a vanishingly small probablility of an individual outcome, but perhaps even odds of some outcome in a class we are looking at.

    Say for example cards and blackjack, There are those zillions of possible shuffles, but really only 20 classes of them that matter for me playing blackjack. That is contrived but I do stumble on that point sometimes.

    So when you turn it around you say look, only a one in 10^68 or so chance of this particular thing, not noticing that its not 1 but 10^67 possiblities out of 10^68 that I would say the same thing about...

    Anyway hope that made some sense...

    P.S.
    And strongly typed languages never led to any discernable drops in software bug reports in my 15 years when I still did software... (heh heh, had to put that in :-) Function point analysis was still the best predictor when I quit dealing with that...

    By Anonymous Markk, at 11:25 PM  

  • "Everyone had the same overall chance."

    Really? I'm currently too tired to model the question properly, but I believe I can see that drawing procedure affects chances.

    Say you have 10 prizes and 100 tickets. If the first ticket takes a prize, the chances for the remaining participants drops from 1/10 to 1/11.

    Would removing tickets to a nondrawing pot during drawing make a difference in chances?

    By Anonymous Torbjörn Larsson, at 2:47 AM  

  • Er, that's not quite right: there's an important distinction between "probability zero" and "impossible". For example, if you pick a real number uniformly at random between 0 and 1, you obtain a rational number with probability zero — but there are rational numbers in that range!

    This distinction matters because sometimes you need "there is no number in the desired range such that [some condition]" — if a rational number fulfils [some condition], you don't have it, no matter that they occur with probability zero.

    By Anonymous Anne, at 3:34 AM  

  • Yes, I was too tired. It is the drawn tickets that wins here. So removing tickets doesn't change anything. (Unless you remove so many that you end up with fewer tickets than prizes, of course.)

    By Anonymous Torbjörn Larsson, at 4:14 AM  

  • Really, I always like the explanation of calculus, mostly because that "ding!" moment of getting it is still one of my fondest memories from high school.

    You can add an infinite number of infinitely small things and end up with a finite number. Make them as small as you want and it still works just fine (it only gets more accurate).

    It's the same with probability.

    By Blogger sailorman, at 12:20 PM  

  • Anne:

    You wrote: "if you pick a real number uniformly at random between 0 and 1, you obtain a rational number with probability zero — but there are rational numbers in that range!"

    Two errors:
    1. It is not possible to have a uniform probability distribution over the reals between 0 and 1. So you've set up a false example.
    2. There's a slight difference between limits, and real zero. The probability of choosing a rational might head to zero as the set of reals that you choose from grows, but that's not quite the same as saying there is zero probability of choosing a rational.

    By Anonymous Don Geddis, at 1:31 PM  

  • Some of what you argue boarders on a more fundamental question that we as people have been asking ourselves for eons; is there a god? I agree with you on zero probability and the impossible vs. improbable argument. It’s interesting to think about but no matter how small the probability of something is, if given enough time which for all we currently know is infinite then it WILL happen. Its this question of time that becomes vitally important in this particular argument. If the common acknowledgement of big bang theory are correct then the answer is no. How does time “stop”? How do we know when it stops? Does time need an observer for it to exist? However more importantly in the big bang model regardless if the theory that the universe collapses or expands infinitily are correct they both are on an infinite time table. Some might say that the big bang theory is theory after all so effectivly bringing us back to the question; does God exist. Its this question that arises because we simply don't know for sure. Yet oddly enough its also this same benefiet of the doubt that allows the fundamentialist with any crediable foundation (the existence of god). Its intresting to think about but proving that life was created out of pure "chance" does not prove that god does not exist. I understand that fundamentalist believe the Bible word for word and feel threatened from the prospect of life being created outside this “model” but they've put the question how did God create the universe? Does God exist? as mutally inclusive.

    By Anonymous Anonymous, at 2:16 PM  

  • A.C. Doyle's great creation put it best:
    "How often have I said to you that when you have eliminated the impossible whatever remains, howevere improbable, must be the truth?" The Sign of the Four

    And the wonderful corollary:
    "It IS impossible as I state it, and therefore I must in some respect have stated it wrong." The Adventure of the Priory School

    By Anonymous usagi, at 2:22 PM  

  • Sure it's possible to have a uniform distribution on [0,1] - the Lebesgue measure is just that. If that bothers you, look up the definition of a probability distribution - it's a positive measure with total measure one.

    I'm not taking any kind of limit here; I really am just drawing from the [0,1] interval according to the Lebesgue measure. So, at the start, all the rationals are possible.

    By Anonymous Anne, at 2:31 PM  

  • How does time “stop”? How do we know when it stops?

    It's probably self-contained, like space. No stop necessary.

    Does time need an observer for it to exist?

    If you're referring to the way quantum mechanics actually refers to "observers," probably. If you're talking about conscious observers, why would it need one?

    However more importantly in the big bang model regardless if the theory that the universe collapses or expands infinitily are correct they both are on an infinite time table.

    One hypothesis I've heard: After the universe chills down to a sea of weak particles (I forget which kind), the laws of probability will inevitably cause all of those particles to jump back into one spot, creating the seed for another big bang.

    Some might say that the big bang theory is theory after all so effectivly bringing us back to the question; does God exist. Its this question that arises because we simply don't know for sure.

    It's hard to answer that question without a good definition of "God."

    Its intresting to think about but proving that life was created out of pure "chance" does not prove that god does not exist.

    1. No one says life arose out of pure "chance" ...Except maybe the Last Thursdayists.

    2. True, but only because you can't prove that sort of negative. Especially since god beliefs tend to be constructed, often deliberately, to be unfalsifiable.

    By Blogger Bronze Dog, at 2:33 PM  

  • One hypothesis I've heard: After the universe chills down to a sea of weak particles (I forget which kind), the laws of probability will inevitably cause all of those particles to jump back into one spot, creating the seed for another big bang.

    Exactly, point being that it’s an infinite process. Big bang --> inevitable collapse to a singular point --> Big Bang…



    2. True, but only because you can't prove that sort of negative. Especially since god beliefs tend to be constructed, often deliberately, to be unfalsifiable.


    We could discuss this, point by point but it’ll just lead us to the old philosophy question of what came first the chicken or the egg. If god doesn’t exist what happened to create the “stuff”? Was the “stuff” just there from the start? Not sure if we could answer that question in terms of the classical model of “god” as some higher BEING implying that it takes form, something tangible.


    It's hard to answer that question without a good definition of "God."

    How a seemingly trivial question might actually create the model to answer the chicken and egg question and more importantly the existence of God. God could be anything.

    By Anonymous Anonymous, at 3:19 PM  

  • We could discuss this, point by point but it’ll just lead us to the old philosophy question of what came first the chicken or the egg. If god doesn’t exist what happened to create the “stuff”? Was the “stuff” just there from the start? Not sure if we could answer that question in terms of the classical model of “god” as some higher BEING implying that it takes form, something tangible.

    So far, we seem to live in an uncreated universe: Matter and energy are apparently eternal. No need to invoke a special agent yet.

    By Blogger Bronze Dog, at 4:02 PM  

  • It's not that sort of probability at all because it involves selection. An analogy: two boxes of 100 dice. The object is to get all sixes. One box is shaken until all 100 dice show 6. The other is shaken, but the dice showing 6's are removed and set aside, i.e., selected. Guess which set of dice will reach the end (100 sixes) first?

    By Anonymous Anonymous, at 4:32 PM  

  • It's not that sort of probability at all because it involves selection. An analogy: two boxes of 100 dice. The object is to get all sixes. One box is shaken until all 100 dice show 6. The other is shaken, but the dice showing 6's are removed and set aside, i.e., selected. Guess which set of dice will reach the end (100 sixes) first?

    Selection for what? If your talking about a number (single value) then yes, what you said applies. However what if we said that it required all of the 100 dices to show 6 or if you needed a combination of numbers to show the wanted outcome. Point is that for life to have had appeared it required many different conditions to be met. Just wondering how selection would apply to particle probability for life.

    So far, we seem to live in an uncreated universe: Matter and energy are apparently eternal. No need to invoke a special agent yet.

    “Uncreated universe” implying no “god” in the classical sense. Like I mentioned I don’t find the “matter and energy are eternal” a valid argument. It’s the same logic that god is eternal. What do you mean by “apparently”? As in, it’s absolute that energy and matter are eternal? Let’s leave stipulation for laymen, what is “god” (i.e. some higher being, energy, matter, anything) is nothing compared to why. I could care less if god was a living breathing woman, or even if god didn’t exist but rather just matter and energy. I want to know why its here, always eternal.

    Either way I think it’s peeked my interests on physics again and I want to remind Mark that his topology articles are much anticipated.

    By Anonymous Anonymous, at 5:57 PM  

  • Like I mentioned I don’t find the “matter and energy are eternal” a valid argument. It’s the same logic that god is eternal. What do you mean by “apparently”? As in, it’s absolute that energy and matter are eternal?

    I don't understand what the problem is. So far, no one has been able to find a way to create or destroy energy. That's where their apparent eternal nature comes from. If someone invents a free energy machine, we'll know otherwise. No absolutes are involved.

    I want to know why its here, always eternal.

    To be cheeky with a point, why not?

    By Blogger Bronze Dog, at 8:15 PM  

  • "Like I mentioned I don’t find the “matter and energy are eternal” a valid argument. It’s the same logic that god is eternal. What do you mean by “apparently”? As in, it’s absolute that energy and matter are eternal? Let’s leave stipulation for laymen, what is “god” (i.e. some higher being, energy, matter, anything) is nothing compared to why. I could care less if god was a living breathing woman, or even if god didn’t exist but rather just matter and energy. I want to know why its here, always eternal."

    Interesting. You have turned the usual argument (finite time, therefore god) around (infinite time, therefore god).

    You can't prove the existence or nonexistence of gods - too many has tried too long to think that is feasible. You could possibly verify an ordinary theory beyond reasonable doubt - if you could come up with a definition that is agreeable.

    Time is more basic than emergent spacetime - quantum theory tells us so. There are indeed infinite time cosmologies. Endless inflation embeds our bigbang universe in an infinite time multiverse. (New universes wormholes from old due to vacuum fluctuations.)

    Massenergy conservation follows from infinite time and Noether's theorem. Noethers theorem follow from observations and has been verified beyond reasonable doubt.

    For example, the last bound on the change of the fine structure constant was to combine astronomical and lab measurements to argue that the upper bound for the change during the existence of the Solar system is one part per million.

    This is a bound on energy constance for most interactions. There are others. The most general ones relies on inflation as observed by WMAP CMB going back to within parts of a second from bigbang.

    Unless you believe in supernatural actions intruding in our natural causal continua. Prove it with an observation!

    But remember, supernatural actions can at most explain about one part per million of what has happened on Earth. The rest is nature alone.

    Coincidentally, this is beyond the usual choosen limits for physical theories to be considered verified beyond reasonable doubt. In this case the absence of other than natural actions. Perhaps you will think of this as the evidence for that no gods exists. I do; and there are other similar ideas with similar results.

    You ask "why"? If the universe is infinite and creationless, there is no reason to ask why. If the universe isn't you can't ask the question. If the universe is you can ask the question. That is an
    observer effect.

    The observer effect is usually interpreted such that the question is not only meaningless but false. If you think this case is different, you must prove why!

    By Anonymous Torbjörn Larsson, at 1:17 AM  

  • Math good.
    Physics good.
    Philosophobabble bad.

    By Anonymous Anonymous, at 3:33 AM  

  • Anonymous, you are funny.

    The essence of formalised religion is philosophy disconnected from facts. This framed your questions.

    Both bronze and I made efforts to give physics based answer.

    AFAIK, your question "why" is always a philosobabble question. But in this case it happens to have a physics answer. From science verified methods we can easily conclude asking "why is the universe" is a failed question.

    By Anonymous Torbjörn Larsson, at 11:35 PM  

  • what came first the chicken or the egg. .

    There is fossil evidence for eggs that predates chickens. What's left to think about?

    By Blogger Stephen, at 1:52 PM  

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