Statistics

Random Means Unknown Cause — And Nothing More

I’ve used this gimmick many times, so regular readers, please, shhh.

I’m thinking of a number between 1 and 4. What is it?

To you the number is random, because you don’t know the cause, or causes, of my picking the number. To me, it is fully determined, because I picked it. There is nothing random to it—to me. So randomness depends on context.

Now regular readers have seen this stunt before, so they have a bit more information than newcomers. It’s up to them how they use this information. But, if they use it right, they can make a better guess. To these clever characters, the answer is less random, because it is more predictable. And it is more predictable because they have more information on the cause. So randomness has a size. (The answer will be revealed at the bottom.)

Given these two characteristics, we deduce randomness is a measure of ignorance: it is epistemological, a thing in your mind, and not in things.

All this is introduction to a small article about the number pi (π): “Pi might look random but it’s full of hidden patterns”, by Humble.

He correctly notes “we will never be able to calculate all the digits of pi because it is an irrational number, one that continues forever without any repeating pattern.” And the reason we cannot is time: we’d need infinite amounts of it in actuality, and not just potentially.

Now there has to be a reason, a cause or causes, why pi equals (or is in base 10) “3.14159…”, where that “…” stands for an infinite number of fixed digits. But not entirely known digits.

But then Humble makes a mistake. He says, “The reason we can’t call pi random is because the digits it comprises are precisely determined and fixed. For example, the second decimal place in pi is always 4. So you can’t ask what the probability would be of a different number taking this position. It isn’t randomly positioned.”

I say pi is random because we don’t know all the causes why it is “3.14159…”. But he’s quite right in insisting that the proposition “The second decimal place in pi is 4” is, given the relevant premises, certain. However, this proposition is random: “The googol-th decimal place in pi is 4”, recalling a googol is 10^100. We only know the values up to a few trillions of decimal places.

Suppose, somehow and per impossible, we knew all the digits of pi. God wrote them on a rock, say, while also providing an infinitesimal microscope to read them all. Given this observation, the proposition “pi is 3.14159…”, where the ellipsis is filled in, is certain.

But unless we knew why it took that infinite precise value and not another, it is still random in part. That is, any proposition containing proposals or the cause, the reason, for this value, are still unknown, and thus random.

Humble then says “‘Is pi a normal number?’ A decimal number is said to be normal when every sequence of possible digits is equally likely to appear in it, making the numbers look random even if they technically aren’t.” Technically they are, if we don’t know them. But what about this “normal” business? (Not to be mistaken for “normality” or normal distributions.)

Well, he gives a small table given by Yasumasa Kanada of the first trillion decimal digits of pi:

    Digit       Occurrences
    0           99,999,485,134
    1           99,999,945,664 
    2           100,000,480,057 
    3           99,999,787,805
    4           100,000,357,857 
    5           99,999,671,008  
    6           99,999,807,503
    7           99,999,818,723  
    8           100,000,791,469 
    9           99,999,854,780 
    Total       1,000,000,000,000

So the first trillion digits aren’t normal, since these entries don’t all equal 100 billion. But perhaps, once we reached infinity, uniformity appears. Somebody might even be able to prove it (if they have, I haven’t heard about it).

What’s the probability of the proposition “pi is normal”? It doesn’t have one until we specify premises. Like “First trillion are normalish, so why not the whole thing?”, then the probability of “pi is normal” is high. Probability of “Pi in first trillion is normal” given Kanada’s table is 0.

But it we had some other irrational number that gave a more obvious non-uniformity (and there are some), like having twice as many 1s as 2s, and twice as many 2s as 3s, and so on roughly in the first trillion, then you not only more information about the non-normalness of the number itself, but also its remaining digits. You could use what you knew learned from the first trillion as a premise for the likelihood of the trillionth and one number, which won’t be 1/10.

As above, there is the sense these numbers are less random, because more is known about them.

But we’d still need to know the entire why of the number—why it was this value and none other—before it loses all its randomness.

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Categories: Statistics

50 replies »

  1. I like this topic, too, Moustaches should be outlawed

    Recall this being an ‘issue’. when I didn’t know why it was, if you get my eaning.
    Had to learn why it wa a problem only to find out that it wasn’t after all

    Argued with m Dad about this, the dictionary was used and everything
    The conclusion was that
    “This Briggs is just being… “

  2. Random Means Unknown Cause

    I disagree. Random means unpredictable. How the digits of pi are what they are is irrelevant. If I can predict what they will be then they are not random. I may not know why the sun comes up every morning but when it does its appearance is not random.

    In a roundabout way, you are right. If I can predict X then the cause can be attributed to my prediction model.

  3. I used to know the “secret sauce”, but apparently not anymore

    I don’t know the number you’re thinking (unless it’s 3.14159

  4. “…randomness is a measure of ignorance: it is epistemological, a thing in your mind, and not in things.”

    Except where it is not all epistemological, like it is aleatory like casinos and lotteries, balls cascading down a Galton board (quincunx), throwing tacks, and so on.

    As far as randomness being other words for ignorance, this may be just semantics. I’d say assuming a probability distribution is just imposing some type of structure as a model. Is using a model and looking at data ‘ignorance’? Maybe. But if that stuff is ignorance, why does it work pretty well in situations? Maybe only partial ignorance. But isn’t partial ignorance also partial knowledge?

    The Strong Law of Large Numbers (SLLN) says that it is almost certain that between the mth and nth observations in a group (ie reference class) of length n, the relative frequency of Heads will remain near the fixed value p, whatever p may be (ie. doesn’t have to be 1/2), and be within the interval [p-e, p+e], for any small e > 0, provided that m and n are sufficiently large numbers. That is, P(Heads) in [p-e, p+e] > 1 – 1/(m*e^2). This happens despite your knowledge of any “cause” – that is simply irrelevant here.

    Another example, is regarding evolution being the non-random selection of random variation, Evolution = NS(RM, OS), where NS is a non-trivial function and is not random, RM is random mutations and is random, and OS is other stuff, some random, and some that is not random: In probability theory, it is known that if X is a random variable, then for a non-trivial function f, f(X) is a random variable. Therefore, because evolution is a non-trivial function of variables, some of which are random, evolution is a random variable, and therefore it is correct to call it overall, random.

    Justin

  5. The Strong Law of Large Numbers (SLLN) says that it is almost certain that between the mth and nth observations in a group (ie reference class) of length n, the relative frequency of Heads will remain near the fixed value p,

    Yet the observations are of random events. How does knowing the distribution have anything to do with whether or not they are random?

  6. The frequency of occurrence of numbers in “data sets” (purported to be “random”) is due to properties that are dependent upon the distribution of the numbers. Benford (a GE electrical engineer) published a paper in the 1930’s that showed that the leading smaller digits (1-5) in many tabulated sets of numbers had a greater probability of occurring than the leading larger digits (6-9). Wikipedia (https://en.wikipedia.org/wiki/Benford%27s_law ) states this was first observed by the astronomer Newcomb when he noticed the wear in pages in a book of logarithm tables (although other accounts attribute the logarithm observation to Benford).
    An interesting real world example of Benford’s Law in action was the draft lottery in the early seventies during the Vietnam War which resulted in picking more birthdays from the first six months of the year with a much greater frequency than those from later months (using the American custom of month, day, year). Ironically, the resulting lawsuits stated that the lottery was not random. Benford’s Law has been applied to detect fraud (and election irregularities) for many years.

  7. FURTHER TO: “But unless we knew why it took that infinite precise value and not another, it is still random in part. That is, any proposition containing proposals or the cause, the reason, for this value, are still unknown, and thus random.”

    THUS: Randomness might also be characterized as a placeholder presumption. One presumes “randomness” in asserting that no explanatory KEY would (or could) enhance (or enable) predictability. An unrevealed KEY may exist ~ but simply be elusive. FOR EXAMPLE: If the PRIME KEY exists then PRIMES would exist as a discernible pattern. Perhaps an infinite series ~ but nevertheless ~ predictable, calculable, discernible; and therefore not random. Ditto for the decimals of ? (3.14>). If this be in error and upon me proved ~ so be it.

  8. Randomness is only a perception not a reality. Nothing is ever truly random. Everything has a cause & effect tied to it even if we do not know the cause or the effect. This is the simple law of nature and reality (I don’t believe in reality either just like some don’t believe in gravity but can’t fly). Wasting time on grasping randomness is futile. Either accept it exist or don’t debating it is not going to change any outcome. It will not win you the lottery or get you that dream job you always wanted.

    And the answer is 3 because the answer is always 3…

  9. Except Benford’s Law doesn’t apply to the 2020 election or so “learned” papers have told me!

  10. “The Strong Law of Large Numbers (SLLN) says that it is almost certain that between the mth and nth observations in a group (ie reference class) of length n, the relative frequency of Heads will remain near the fixed value p,

    Yet the observations are of random events. How does knowing the distribution have anything to do with whether or not they are random?”

    The relative frequency converges to a flat line (the probability) irrespective of where we enter the stream of observations (or your beliefs/knowledge about the process). This was von Mises idea of randomness.

    Justin

  11. “Everything has a cause & effect tied to it even if we do not know the cause or the effect.”

    Darren, except those that believe in their preferred version of god say that while “everything obviously” has a cause, their preferred version of god (which would have to be more complex than all the effects) does not have a cause.

    Justin

  12. I know I’m rather off-topic, but I am reminded of a useful trick for computing pi back in the early days of pocket calculators. As you may recall, 1970’s era calculators were often limited to simple arithmetic functions. If you needed a value for pi, you could punch in a memorized value like 3.14159265. Or you could use 22/7, but that’s off starting at the 3rd decimal place. Here is a memory trick I found: Think 113355, and on your calculator compute 355/113. That simple trick gives pi to 6 decimal places. Coincidence? Random chance? Fun with figures? 🙂

  13. Pi is the ratio between the circumference of a circle to its diameter. That’s why it is 3.14159 …. , for that’s what God made it be ;p, if I am remembering my plane (Euclidean) geometry from more than 50 and now approaching 60 years ago.

  14. I have no idea of how things like Casinos are supposed to be counterexamples to randomness being an epistemological quality.

    Take a game of blackjack. In statistical analysis of this game the card given on a hit will be treated as random (with restrictions from known information about which cards have been dealt, but random nonetheless.) But surely this is only a question of knowledge: the card is to be given on a hit is already on the top of the deck and if we knew what it was we would be able to predict the next hit with perfect accuracy.

    Now the objection might be raised that at this point the card selection is not random, but the shuffling process was random. But even there, the problem is one of knowledge. If we knew the precise order of the cards before they were shuffled, the precise motions of the dealer’s hands which caused the cards to move into their positions, etc. we would be able to determine where the cards ended up and therefore predict with perfect accuracy the cards to be dealt and given on hits. The same thing if the cards were shuffled by a machine. It’s only “random” because we don’t know these things, but that makes “randomness” a question of epistemology.

    You can say the same things about the dice rolled at the craps table, the place that the ball ends up in roulette, and so on. In fact gamblers have been known to try to determine if a certain table has imperfections that make it easier to stop at a certain point than others, and then adjust their bets accordingly. Since they don’t know everything about how the table is made, spun, and where the ball will be released they still have some “randomness” in their observations, but it’s less than what the people who know nothing about the table have to deal with.

  15. Justin,

    No one who argues for the existence of God by a first way type argument says that “everything obviously” has a cause. In fact the whole point of the argument is that there must be something uncaused, or nothing could exist. Framing it that way is like saying that mathematicians say that “all numbers obviously are rational, except things like the square root of two are arbitrarily not rational.” It only shows that you don’t understand the argument.

    But the discussion is irrelevant to the question of “randomness” in probability anyway. Probability only studies the contingent, since necessary things obviously can’t fail to happen even in principle. Since every contingent thing has a cause probability acts as though all things are caused. It’s the same as how there are obviously non-material things (impressions, mathematical truths, etc.) but since a discipline like Chemistry only studies matter, there is nothing wrong with saying that Chemistry acts as though there is nothing non-material.

  16. Briggs, you always bring up such devilishly difficult mental conundrums. Oh, I know, it’s clear enough to you, but I’m afraid I won’t know what to think until Shecky Green and Lee Phillips have weighed in.

  17. 2.3614827452681006378255638215
    Just a guess

    There isn’t really a question here at all It’s about fractions of matter, ultimately
    A circle is in the imagination where a perfect circle exists. Actual things that look like circles are jagged. The human eye can’t see the edge of things properly to the degree of accuracy that pi suggests it should
    The mind is full of circular thinking if you do too much of it because you have no way of knowing where to start without clear boundaries
    The answer is three because everybody says it and you started speaking of Pi which is suggestive of 3
    Does anybody know the trick where you ask someone at the end to think of a cloud an animal and a vegetable?
    They all say an elephant, the number three and a carrot? Or something
    If you don’t know something, it’s a good substitute for randomness!
    If randomness is what you’re looking for

  18. NEVER typed “cloud”. that’s an insert
    I typed a “number’ ‘cloud’…..?????

  19. Someone who works for GCHQ told me that it is possible to read live what people are writing on the internet

  20. When I was a freshman at the Big U, the first new words I learned were “stochastic” and “heuristic”. They basically mean “we don’t know” and “but we could guess”. After careful study I discovered that the frequency of usage of these words is inversely proportional to the size of the minds that use them.

    PS — if it’s not Euler’s e, then I don’t care.

  21. Dean Ericson seems to want less Joy in his life.
    Perhaps he hasn’t considered that a joyless (Joy-less?) life is a miserable one.

  22. Pi has several known, mathematically equivalent derivations (causes), all of which are infinite series. Here are three of the most commonly known.
    PI = 3 + 4*(1/(2*3*4) – 1/(4*5*6) + 1/(6*7*8) – 1/(8*9*10) + …)
    PI = 4*(1/1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + …)
    PI = 4*(1/1 * 1/3 * 2/3 * 2/5 * 3/5 * 3/7 * 4/7 * 4/9 * 5/9* 5/11 * …)

    Any given digit is knowable, given sufficient time and computing resources. Of course, an infinite series, by definition, takes infinite time to compute a definitive answer.

    Here is the important part: The digits of pi are discovered, not created by man. They are fixed and unchanging, even if unknown.

  23. Ah, Dav! A Davless life is a whole lot of running around in circles. Everybody should have a Dav

    Deany, sorry, I was a bit unkind? in your direction a few weeks back and if it’s any help I still feel guilty about it
    Will try to shut up, it’s a compulsion at the moment. WAS going to say sorry eventually

  24. McChuck’s formulas (above) brought to mind this amazing exhibition by Englishman with SYNESTHESIA ~
    who visualizes numerals as colors:

    (Daniel Tammet): Count pi to 20 thousands digit
    https://youtu.be/X3AlKU7dfOM

  25. john b(), He previously asked this and the answer was e.

    The number, incidentally, was e, the base of the so-called natural logarithm. https://wmbriggs.com/post/13075/

    Of course it may be different with this post but Briggs is not all that random so the number is unlikely an integer.

    Incidentally, the same post gives the formula for calculating any hexadecimal digit of pi, the Bailey–Borwein–Plouffe formula (BBP formula).
    https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula

  26. DAV

    Thanks

    He used to “hide” the number in “View Page Source” for mentalist, Uri Geller like posts which is what confused me. (It wasn’t an integer then either – Sheri had to explain it back then.) I kept looking in “Page Source” and “Page Info” and even in “Expect Element”

    I probably saw that post but that was 5 years ago!

  27. Jan Van Betsuni,
    The title of that youtube is: Count pi to 20thousands digit. see how by Daniel Tammet but it seems see how was clickbait and watch him name some would have been more apt. Perhaps I missed the explanation?

    In my opinion, remembering digits using 22,000 colors is just as hard as remembering the digits itself. But then, I remember colors with only their labels (not the color itself) and that’s limited to a dozen or so. Most of my mental images are gray scale.

  28. DAV

    It’s to laugh
    The eight or so pack of first grade “Crayola’s” is as much as I can go

    When we played “Duck Duck Grey Duck”, after Red Duck, Blue Duck, Orange Duck and Yellow Duck, we’d start making up colors (I don’t have to explain the Minnesota variant of Duck Duck Goose, do I?).

  29. Here is the important part: The digits of pi are discovered, not created by man. They are fixed and unchanging, even if unknown.

    No. Mathematical concepts are created by man. They may appear to be related to nature but that’s only because man is doing the observation and impressing ideal models on nature. There are no ideal circles in nature. Those that seem close depart from the ideal of pi * diameter.

  30. John B(S), Duck Duck Goose

    Must be a local thing. We played tag yelling “IT!” Not as imaginative.

  31. Duck Duck Goose is Universal

    Perhaps. If it is, I’m guessing we are from different universes.
    We didn’t play it when I was growing up — instead we played “IT!”

  32. Briggs even used the term “GIMMICK”

    “I’ve used this gimmick many times, so regular readers, please, shhh”

    Going to an old post is a gimmick? That’s not a gimmick [spoken in Australian accent]

  33. DAV

    It might be an age thing? I’m older than dirt, almost as old as the SSGT

  34. DAV,

    In your account on the theory of mathematics, it is mere coincidence that mathematics can be useful. If there is no relation of mathematics to nature then mathematical theorems should not even give approximate answers in real world situations. That is, if an ideal circle is nothing more than a manmade idea with no relation to nature outside of our perceptions, then “real world circles” shouldn’t even have an area close to pi times r squared. In order for a shape to approximate an ideal, the ideal must exist in some way (not necessarily as a platonic form; there are other explanations for the existence of mathematical objects. But they must exist.)

  35. In order for a shape to approximate an ideal, the ideal must exist in some way

    It only exists in your mind and your mind imposes it on reality.
    If you think it can exist outside of a mind, show me an ideal one and not an approximation.
    Just because its useful doesn’t make it real.

    then “real world circles” shouldn’t even have an area close to pi times r squared
    So, models are real when they approximate closely?

  36. “To me, it is fully determined, because I picked it.”

    So where’s your free will?

  37. DAV,

    It only exists in your mind and your mind imposes it on reality.
    If you think it can exist outside of a mind, show me an ideal one and not an approximation.

    You are assuming two things: first that mental objects are personal, and second that they are not real. Note that I never claimed that things like ideal circles exist in the physical world, only that they are real.

    If there is no ideal circle in any sense, and when I think of a circle it is merely a concept that I have created myself, there is no reason why “my” circle should be the same as anyone else’s. Each mental object would be in a completely separate brain with no connection. So then why should we be able to agree on the properties of these ideal circles? You need some sort of universal object for this to make any sense. Even saying something like “we agree on the properties because we’ve agreed on a definition of circles that forces those properties to be true” only says that we have reduced things down to the same universal object.

    How they are real is a different question. Platonic forms are one possibility, though one that I find unlikely (and I suspect you do to). For a better possibility see proof three of Edward Feser’s Five Proofs of the Existence of God

  38. For swordfishtrombone a free will apparently requires not having actions determined, even after you have done them. That is, free will requires being free even from our own decisions (i.e. free will must even be free from itself).

  39. Rudolph Harrier,
    If there is no ideal circle in any sense, and when I think of a circle it is merely a concept that I have created myself, …

    It is a real concept that is itself not real. In fact, it’s all in your head.

    The definition of “real” is: actually existing as a thing or occurring in fact; not imagined or supposed. Ideal circles and other concepts are imagined or supposed. They are not real.

    there is no reason why “my” circle should be the same as anyone else’s. Each mental object would be in a completely separate brain with no connection.

    Using that logic, words are real despite being codes for concepts but otherwise possessing no meaning outside of a mind.

    Even saying something like “we agree on the properties … only says that we have reduced things down to the same universal object

    You are confusing objects with concepts. There are no objects (physical things) with the property of an ideal circle. Definition of “object”: a material thing that can be seen and touched.

  40. There are no objects (physical things) with the property of an ideal circle.

    By equating “object” with “physical thing” you are requiring a reductionist materialist framework, even though I’m obviously contending such a framework.

    But even if we grant such a framework you still haven’t explained why mathematics has any consistency or why it can be used to model the real world.

    It’s not as though mathematics is mere “model fitting” either. For example, the conclusions of spherical trigonometry were reasoned out through geometric axioms, before anyone had precise enough measurements to see if they matched paths on the actual globe. Yet the results were a good approximation nonetheless. So far you’ve given no reason for that to be anything more than coincidence.

  41. “Definition of “object”: a material thing that can be seen and touched.”

    I object! 🙂

  42. By equating “object” with “physical thing” you are requiring a reductionist materialist framework

    Tough. It’s the dictionary definition. Look it up.

    It’s not as though mathematics is mere “model fitting” either. For example, the conclusions of spherical trigonometry were reasoned out through geometric axioms,

    So? It’s still a mental construct. It isn’t real. It becomes a model when applied to the real world. Not all models are “fitted” in the sense of “adjusting parameters”. There are no “circles”, “triangles”, etc. in the real world. Those are just what we imagine we are seeing. In fact, we tend to see them even when incomplete. For example, how many dots are needed to make a triangle? Zero, triangles are made of lines. We imagine the lines then imagine the triangle. It’s all in your head. You do realize we can only see dots, yes?

    Yet the results were a good approximation nonetheless

    That’s what makes it useful. It still shouldn’t be confused with reality.

    you’ve given no reason for that to be anything more than coincidence.

    It’s not a coincidence. Mathematics (or geometry at least) started off as a model of reality. It’s a useful description of how we view the physical world but is not itself that world.

    Kneel, I see what you did there 🙂

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