9 ms·

> so that if another person is shown your sequence of digits from 1 to 6, he/she should not be able to tell whether these numbers were produced by a real die or

> so that if another person is shown your sequence of digits from 1 to 6, he/she should not be able to tell whether these numbers were produced by a real die or just “made up” by somebody. That instruction is a flaw in the experiment. It's always impossible to tell, for any given sequence, whether it was produced by a fair die. There's nothing an experimental subject can do to make the impossible more impossible. > the kind of sequence you’d get if you really rolled a die Well, there is no such sequence. The instruction is incoherent. You could take it as meaning "Construct a sequence that you think will convince others that it came from a random source". That would be coherent. And then it would be legitimate to eliminate responses that were all-heads ("clearly didn't even try"). But then what are you measuring? The comparative understanding of older and youger people concerning random sources, or the Gambler's Paradox? Their comparative expertise in human psychology? Their comparative willingness to move the mouse-pointer over the screen?

simias 5y ago
>It's always impossible to tell, for any given sequence, whether it was produced by a fair die. If the sequence is long enough you can model how likely it is to have been produced by a fair die. Are all numbers equally distributed? Are some numbers more likely to follow or not follow other numbers? Are some patterns repeating? Of course any sequence can be produced by a fair die, but you can still create some objective metric that will tell you how truly random a sequence is, and the longer it is the more accurate it will be. It's what tests like Diehard are about after all. Can you roll a fair die a thousand times and only get 6s? Well yes of course. It'll never happen though.
- deleted 5y ago
  [deleted]
- marcosdumay 5y ago
  Any single sequence of numbers has the exact same probability of being produced by a fair die (that's how the definition of "fair" goes). The probability of getting all 6 is the same of any other one you get.
  - sp332 5y ago
    That's true given that we have a fair die. But the question is, given some results, what is the probability that the die is fair? And there are statistical tools for that.
  - mxkopy 5y ago
    The probability of a specific sequence occurring is different from the probability of a specific sequence being random.
  - simias 5y ago
    For sure, but I think it's more helpful to this about "classes" of sequences. Sequences which have a uniform distribution of digits (within some margin of error), sequences which do not have repeating patterns, sequences that do not contain the same digit twice in a row etc... Any single one of these sequence is as likely as any other, but some "classes" are vastly bigger (and therefore, more probable) than others. By deciding which classes of sequence any result belongs to, you can decide if it's likely to have been produced by a fair die or not. This intersects with the concept of entropy: assuming that you have a box containing a gas whose particles move randomly about the volume of the box, then at some point you take a snapshot of the position of every single particle in the box and you discover that they're all in the right half of the box, the left side being in a vacuum. Would you assume that it's just random chance? It could be. It certainly isn't. Meanwhile any of the trillions and trillions of snapshots showing particles more or less uniformly distributed within the box are all more "random looking" and are what is expected from such an experiment. These configurations as a group occupy the vast majority of the phase space for the contents of the box.
  - julian37 5y ago
    You work with crypto according to your profile. I hope that when you see a random generator return a series of a hundred 6s, you go and check what's wrong with it instead of assuming you just got lucky this time ;-)
    - marcosdumay 5y ago
      This is much more relevant in crypography than on statistiscs. If your PRNG always returns the same 4, it's buggy, but the really problematic outputs look exactly like random numbers. (Looks like my profile is out of date, by the way.)
- deleted 5y ago
  [deleted]
- gus_massa 5y ago
  > Can you roll a fair die a thousand times and only get 6s? Anecdote time: A few years ago with my friends we were discussing how easy is to roll 5 dices simultaneously and get the same result in all of them. This is a possible way to win a popular game here https://en.wikipedia.org/wiki/Generala https://en.wikipedia.org/wiki/Generala We estimated how often you can roll the dices and the probability, and we estimated that you must try during 2 or 3 hours to get that result. We were young, had a lot of free time, so one of my friend started rolling 5 dices while he was talking with us and eating. After about 2 hour he got the 5 equal dices in a roll. (IIRC he tried again, with a similar result.) (Note that 2 or 3 hours is consistent with how this outcome is used in the game to win automatically. It's possible to get this in a normal game, but it's not super common.) Also, each time you add a dice, the time increase exponentially. With 6 dice it's 14-21 hours, like a day. With 7 dice it's like a week. With 9 dice is like a month. With 10 dice is half a year. 1000 dice will take longer (and x6 if you want to get only 6s instead of any repeated number). (Note that parallelizing this to all humans only adds 14 dices. If everyone on Earth start rolling 24 dices, it will take like half a year until one of us get 24 equal dices.)
  - alisonkisk 5y ago
    Saving a click: Generala is an ancestor of Yahtzee.
    - nonameking2026 5y ago
      i clicked to upvote you and now I'm writing this text. thanks for nothing.
  - codegladiator 5y ago
    They are studying this https://www.quantamagazine.org/elegant-six-page-proof-reveals-the-emergence-of-random-structure-20220425/ https://www.quantamagazine.org/elegant-six-page-proof-reveal...
  - avgcorrection 5y ago
    Saving a click: Generala is a relative of Yatzy.
- RuggedPineapple 5y ago
  >Can you roll a fair die a thousand times and only get 6s? Well yes of course. It'll never happen though. Somewhat related, but humans are also REALLY bad at generating randomness and one of our big tells is an aversion to repeats. If you ask someone to pick 0-9 randomly, repeatedly, they will rarely repeat numbers. But in a truly random sample a repeat is likely 10% of the time, and a three-peat will happen roughly once every 100 numbers. An average person will rarely if ever repeat a number and sure as heck won't ever come up with it three times in a row.
  - easymodex 5y ago
    Then again i had exactly this in mind and in OP tasks i did repeat a number here and there.
    - Forgeties79 5y ago
      Yeah but even then most people cap the repeats if they’re faking. If 100 people have to roll 10 random numbers, it is very likely some people will have 3-4 in a row of the same number. It’s less likely none will. My brother in law teaches stats and he runs this scenario with a seminar he teaches. He then runs the results through a simple formula he built in excel and can ascertain who faked vs. did it for real with about 95% certainty IIRC. I love little games like that haha
  - macksd 5y ago
    And funnily enough, you'll often hear this trait as being desirable in a pseudo-random number generator. People often want something that will jump around fairly unpredictably but that will come close to outputting all possible numbers once before getting into re-runs.
    - travisjungroth 5y ago
      I’ve coded up this exact algorithm, it’s really fun. It’s useful for “shuffling” in the music sense (not the cards sense). I think it’s actually the prototypical real-world software engineering problem. User says they want X (random music). X is a term in software, so you give them that (you get a random song). They’re not happy. You dig and find out they really want A, B, and C (next song is unknown, songs don’t repeat too soon or too infrequently). This new problem is harder to verify (how soon is too soon?). Editing in tips on solving this sort of problem. You can turn vague requirements into precise requirements. Rather than make the precise requirements exactly equivalent to the vague ones, it's easier to make them more restrictive. Is playing a song again within 50% of the length of the playlist "too soon"? Maybe. How about within 80% of the length of the playlist? Definitely not. We can give ourselves the requirement "Songs must always play again between 80% and 125% of the length of the playlist." Much easier to solve, much easier to test. Sometimes the extra restriction make the problem harder (not usually I've found). Still, this is a great trade because understanding requirements is harder than solving well defined problems. [To the point of this whole post] Requirements can be turned into testable properties even if it's not programmatic. "When I look at a list of chosen songs, there must be no obvious patterns." Who says what's obvious? You do! Then, have someone else do the same. Consider extreme cases. Extreme cases tend to be the most or least important. If they're least important, create a new set of easier requirements or drop it all together. "If 3 - 10 songs, always play within double the playlist, no obvious patterns, never twice in a row. If 2 alternate, if 1 repeat."
- K0balt 5y ago
  Assuming the die is fair, that outcome would be exactly as likely as any other completely specified outcome.
baby 5y ago
I would have thought that “looking random” would have been calculated by just checking if there is a bias in people’s answers. If someone chooses something that has never been seen before vs something that has been picked a hundred times, then it might be “more random” as the bias clearly comes from humans
graderjs 5y ago
Too deep topic and maybe I don't understand it compared to the megaminds here on hacker News but as far as I get it when it comes to randomness every sequence has the same probability which is simply the inverse of the number of sequences for any given length. In that case it seems to me impossible to tell from only the information of a single sequence. But if you have multiple sequences where one is the binary coding of a Mozart symphony and another is the binary encoding of Shakespeare sonnet and another is the binary encoding of days of the week or a DNA sequence if you have enough data points like that I think you can start to say Well they're probably not random because we see coherent patterns. The second aspect is coherent pattern depends on perspective. So what seems random to us maybe a very common physical constant to an alien civilization. so if we see a binary sequence that represents pi, maybe we say oh well that's not random because it's accurately pi to like a thousand places... but if we see another sequence that looks random maybe it's a physical constant to an alien civilization that's accurate to a thousand places as well. But we don't know that so it looks random...to us. So randomness doesn't just depend on entropy, I think in a realistic point of view, probability is not enough, it also depends on something which Maybe it's a little bit harder to measure perhaps which is perspective. Random depends on the context.
- sacrosancty 5y ago
  I think your way of looking at it requires an explosion of alien civilisations to find one in which our random 1000 digits is their special number. If you generate 1 extra digit, you'd need 10x as many civilizations to expect a match. Getting back to the experiment, obviously it's measuring the human mind so the context is the person's mind. They might enter their bank account number which looks random to the researchers but is actually cheating because they're not using their brain as the source of randomness.
  - graderjs 5y ago
    That's an interesting point about the random sequence selected what's the chance it matches an alien civilization constant. I didn't think about that way. I guess I thought about it more from your other example point of view, where say an alien civilization pretends to be a human and gives you a number that they say is random (just like the guy giving you his bank account number) but actually the alien civilization gave you a physical constant. It's interesting that there's a difference but it's sort of like the difference between you picking a random sequence from a random source and the probability of that being actually random versus you being given a supposedly random sequence by somebody and the probability of that being actually random. I think.
HWR_14 5y ago
> It's always impossible to tell, for any given sequence, whether it was produced by a fair die. There's nothing an experimental subject can do to make the impossible more impossible. That's just not true. Or feel free to play a game with me. We'll roll a 20 sided die. If it comes up 20, you give me a ten. If it comes up any other number, I'll get you a dollar. Nice EV on that! Oh, the die has come up 20, 20, 20, 20, 20, 20, 20 the last seven times. Do you play?
- mmoll 5y ago
  You’re comparing apples to oranges. The correct analogy is: you pick any other sequence of numbers between 1 and 20 and then tell me you’re more likely to win because your sequence is more random.
- Ygg2 5y ago
  It can still be a fair die. The context implies it's not, but it's only unlikely, not impossible. See https://youtu.be/8Ko3TdPy0TU https://youtu.be/8Ko3TdPy0TU It introduces a 10 billion human second century i.e. 3.1 x 10^19. If your chance times it is greater than 1 then it's definitely plausible to be done by someone, somewhere.
- denton-scratch 5y ago
  I would; would you? The Gambler's Paradox says you shouldn't (I'm assuming your 20-sided die isn't crooked). Incidentally, you haven't made your case that you can ever tell whether a given sequence was produced by a fair die. You've just asserted it, and then suggested a game that doesn't illuminate anything.
  - dtech 5y ago
    It would be incredibly stupid to take the bet, as it's way more likely that the sequence was not produced by a fair die than that it was (i.e. the dice is rigged in the example) Just because it's impossible to know for certain, doesn't mean you can't make a prediction with very high chance of being correct.
posix86 5y ago
You're wrong. Mathematically. Here's why. When you throw a coin 100 times, each sequence you get is equally likely. However. You can look at properties of the sequence which are more likely to be one way than the other. For instance, it's more likely that the number of heads and tails are about equal than not. The reason is that there are more sequences, in general, where that is true, than those where heads or tails strongly prevail. With the right property, you can make statements such as: This sequence is statistically likelier to be human made than random. One such property is for instance the number of changes from heads to tails, or vice versa. In expectation, random sequences change heads to tails about 50% of all flips. For humans, the expectation is much higher. Hence, if you compare two sequences where one has 51% changes and one 63%, it is mathematically (/statistically) accurate to say that the latter one is likelier human made.
- tidenly 5y ago
  Your point doesnt refute OPs argument. Your final statement "you can say the latter is more likely random" is not the same as "you can say this sequence is not random". I think lots of people (especially programmers) who know about true RNG vs expectations of RNG might intentionally put in strings of same numbers, or not include the full set, because we know its what often happens during plain RNG. It isnt clear what the goal of the sequence is, hence the confusion in the comments.
  - posix86 5y ago
    What does that mean, "you can tell this sequence is not random"? If you show me a blue hat, and ask me if it's blue, I'll say yes, I can say it's blue. But there is always a chance it's not actually blue. It's very conceivable that I'm in a situation where I say with confidence that something is blue, but it isn't. You always only ever speak in probability. Of course you can't say the sequence isn't random, because every sequence can be the result of a random process. "can you tell which is random" to me is equivalent as asking "is one of the sequences such that it is rational to choose it over the other as being random". It's about rational decisions. Consider the frequentist view, where a probability p for an event A means that out of k trials, pk will show A (in expectation), and fruthermore if k -> infty, then the portion of events that show A will converge to p. If you want to choose the right sequence as being random as often as possible, it is rational to choose the one I described above, because it will, overall, be the one that is MORE OFTEN the random one compared to the other.
kevinpet 5y ago
Here are some numbers: 123456123456 Did I get them by rolling a die?
- travisjungroth 5y ago
  Almost certainly not. Given a circle with diameter AB and a point on the perimeter C, is ABC an equilateral triangle?
  - rocho 5y ago
    It's impossible for ABC to be equilateral. Maybe you meant right triangle.
    - travisjungroth 5y ago
      > It's impossible for ABC to be equilateral. Exactly! This is why you can answer "no" to my question, not "almost certainly not".
- denton-scratch 5y ago
  There's no way of telling. I'll guess you didn't, but that's a guess about people, not about sequences of digits. See, it's not about how likely it is that OP rolled 123456123456; it could have been any "unlikely" number (where I suppose "unlikely" is a psychological quality). So OP keeps rolling until an "unlikely" number comes up, and exclaims "Wow, how unlikely was that?". Well, it's impossible to know, without knowing how many "unlikely" numbers there are; but it's much more likely than 123456123456 is. Aren't all sequences really equally unlikely? So whatever you roll, it's as improbable as all-1s?
- m3047 5y ago
  Is there a null hypothesis that some other mechanism has a higher probability of producing that sequence?
scraptor 5y ago
All sequences are equally likely to be produced by a fair die but humans are very biased in the kinds of sequences they produce. It might be impossible to ever be certain but you can certainly look at a sequence of all sixes (or that contains more complicated patterns) and estimate that it was much more likely to be produced by a human than a die.
jancsika 5y ago
Now I'm curious: in your opinion what is funny about the line "Nine Nine Nine Nine Nine Nine" in the following cartoon strip, vs. something like "Two Nine Eight Three Seven Eight": https://dilbert.com/search_results?terms=Random%20Number%20Generator https://dilbert.com/search_results?terms=Random%20Number%20G...
- drdeca 5y ago
  Not the person you asked, but, P(9,9,9,9,9,9 | Not random ) > average over all values of (sequence) of P((sequence) | Not random ) (under some reasonable assumptions about how likely you consider different processes to be the process producing the output) While P(9,9,9,9,9,9 | random ) = average over all values of (sequence) of P((sequence) | random ) Therefore, P( Not random | 9,9,9,9,9,9 ) > P( Not random )
justinpombrio 5y ago
> That instruction is a flaw in the experiment. It's always impossible to tell, for any given sequence, whether it was produced by a fair die. There's nothing an experimental subject can do to make the impossible more impossible. Baloney! Say you measure the traffic to your website in the morning and the evening, every day for a week. And this is what you see: day: 1 1 2 2 3 3 4 4 5 5 6 6 7 7 time: M E M E M E M E M E M E M E visitors: 51 73 58 72 50 78 55 74 55 77 52 73 55 76 It could be that the traffic you get is uniformly random, and each day the number of visitors you get is a uniformly random number between 50 and 80. Sure, all the morning numbers are less and the evening numbers are more and there are suspiciously no numbers in the 60s. All that could be a coincidence. You know perfectly well, though, that you're getting more traffic in the evenings.
- denton-scratch 5y ago
  You've answered a different question. The question is: can you construct a sequence of dice-casts that an adversary can't distinguish from a real dice-cast? Answer: you can't.
  - a1369209993 5y ago
    > The question is: can you construct a sequence of dice-casts that an adversary can't distinguish from a real dice-cast? Yes, obviously: A: 2 4 4 3 5 2 3 6 4 2 B: 5 5 5 6 4 1 1 2 4 5 One is a real dice-cast with a d6 I had lying around. The other is the (100% deterministic) output of: echo 1651434259 $X | md5sum | grep -o '[1-6]' | paste -sd' ' for some X. Feel free to explain[0] how you are able to distinguish which is which. The real issue is that most people don't bother to produce random numbers in a way that's actually secure (which, to be fair, is rather tedious if you don't have a computer handy, and downright prohibitively impractical if you want to do it all in your head, so why would you bother?), either in the study or in general. 0: If you'd like a more black-box distinguishment, I can provide a longer list; obviously a adversary can get the right answer 50% of the time just by chance.
    - denton-scratch 5y ago
asdfasgasdgasdg 5y ago
> That instruction is a flaw in the experiment. It's always impossible to tell, for any given sequence, whether it was produced by a fair die. There's nothing an experimental subject can do to make the impossible more impossible. I don't think it's a flawed instruction -- i.e., I believe most participants will read it as intended, and it will produce interesting results. I read it as follows: "Suppose we show a person your sequence and a sequence generated via an actual random number generator. Minimize the average probability that that person correctly guesses which sequence was generated by a human, across all possible real random sequences." I guess most other people would read it that way, even it is not formally correct from a statistical perspective. And I believe it gets at what they're trying to measure.
- denton-scratch 5y ago
  > I guess most other people would read it that way Not me. But as I said upthread, that's the only way I can construe the question as being coherent. And I agree that the question thus construed invites the experimental subject to make predictions about the behaviour of others; so the question tests the subject's accuracy in psychological judgements. If you construe it as being a coherent question. I think it's silly to construe it that way; I just think it was a crap question, and you can't draw any conclusions at all from the results.
Strilanc 5y ago
Let's not confuse P(observed_rolls|used_dice) with P(used_dice|observed_rolls). P(observed_rolls|used_dice) is always the same, independent of observed rolls, assuming the dice are fair. But P(used_dice|observed_rolls) can vary, because other ways of generating rolls which are under consideration may be biased towards certain answers, and this allows you to perform inference. For example, suppose the rolls you will be shown were either generated by fair dice or by the program "always return 4" [4]. The rolls you are given are "4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4". Are you really thinking you'd make the same prediction for this sequence of rolls as you would for the sequence "1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1", or is there perhaps some SMALL INKLING OF A HINT as to which answer is correct? The simple fact of the matter is that, in the real world, if you see a sequence like "4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4" you can be surprisingly confident that it was not generated by die rolls. This is because there are plenty of other ways to get sequences and those other hypotheses didn't just pay a Bayes factor penalty of a trillion. Seeing the instructions as incoherent is the mistake of trying to over-isolate the study to the abstract mathematical realm, instead of the world people actually operate in. If someone tells you their luggage combination is 1234, do you really think it's meaningless to guess that it was the default combination as opposed to being generated by secure die rolls? Do you not form opinions about whether or not someone is using secure randomly generated passwords when you find out their password is "password2"? 4: https://xkcd.com/221/ https://xkcd.com/221/
zem 5y ago
Martin Gardner had a wonderful demonstration in one of his columns, where he had people write out 36 "random" numbers, then fold them into a 6x6 square. there were inevitably doubles or triples of the same digit going vertically, and almost never horizontally, since repeating a digit didn't seem random enough to the person generating the sequence.

simias 5y ago

>It's always impossible to tell, for any given sequence, whether it was produced by a fair die. If the sequence is long enough you can model how likely it is to have been produced by a fair die. Are all numbers equally distributed? Are some numbers more likely to follow or not follow other numbers? Are some patterns repeating? Of course any sequence can be produced by a fair die, but you can still create some objective metric that will tell you how truly random a sequence is, and the longer it is the more accurate it will be. It's what tests like Diehard are about after all. Can you roll a fair die a thousand times and only get 6s? Well yes of course. It'll never happen though.

deleted 5y ago
[deleted]
marcosdumay 5y ago
Any single sequence of numbers has the exact same probability of being produced by a fair die (that's how the definition of "fair" goes). The probability of getting all 6 is the same of any other one you get.
- sp332 5y ago
  That's true given that we have a fair die. But the question is, given some results, what is the probability that the die is fair? And there are statistical tools for that.
- mxkopy 5y ago
  The probability of a specific sequence occurring is different from the probability of a specific sequence being random.
- simias 5y ago
  For sure, but I think it's more helpful to this about "classes" of sequences. Sequences which have a uniform distribution of digits (within some margin of error), sequences which do not have repeating patterns, sequences that do not contain the same digit twice in a row etc... Any single one of these sequence is as likely as any other, but some "classes" are vastly bigger (and therefore, more probable) than others. By deciding which classes of sequence any result belongs to, you can decide if it's likely to have been produced by a fair die or not. This intersects with the concept of entropy: assuming that you have a box containing a gas whose particles move randomly about the volume of the box, then at some point you take a snapshot of the position of every single particle in the box and you discover that they're all in the right half of the box, the left side being in a vacuum. Would you assume that it's just random chance? It could be. It certainly isn't. Meanwhile any of the trillions and trillions of snapshots showing particles more or less uniformly distributed within the box are all more "random looking" and are what is expected from such an experiment. These configurations as a group occupy the vast majority of the phase space for the contents of the box.
- julian37 5y ago
  You work with crypto according to your profile. I hope that when you see a random generator return a series of a hundred 6s, you go and check what's wrong with it instead of assuming you just got lucky this time ;-)
  - marcosdumay 5y ago
    This is much more relevant in crypography than on statistiscs. If your PRNG always returns the same 4, it's buggy, but the really problematic outputs look exactly like random numbers. (Looks like my profile is out of date, by the way.)
deleted 5y ago
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gus_massa 5y ago
> Can you roll a fair die a thousand times and only get 6s? Anecdote time: A few years ago with my friends we were discussing how easy is to roll 5 dices simultaneously and get the same result in all of them. This is a possible way to win a popular game here https://en.wikipedia.org/wiki/Generala https://en.wikipedia.org/wiki/Generala We estimated how often you can roll the dices and the probability, and we estimated that you must try during 2 or 3 hours to get that result. We were young, had a lot of free time, so one of my friend started rolling 5 dices while he was talking with us and eating. After about 2 hour he got the 5 equal dices in a roll. (IIRC he tried again, with a similar result.) (Note that 2 or 3 hours is consistent with how this outcome is used in the game to win automatically. It's possible to get this in a normal game, but it's not super common.) Also, each time you add a dice, the time increase exponentially. With 6 dice it's 14-21 hours, like a day. With 7 dice it's like a week. With 9 dice is like a month. With 10 dice is half a year. 1000 dice will take longer (and x6 if you want to get only 6s instead of any repeated number). (Note that parallelizing this to all humans only adds 14 dices. If everyone on Earth start rolling 24 dices, it will take like half a year until one of us get 24 equal dices.)
- alisonkisk 5y ago
  Saving a click: Generala is an ancestor of Yahtzee.
  - nonameking2026 5y ago
    i clicked to upvote you and now I'm writing this text. thanks for nothing.
- codegladiator 5y ago
  They are studying this https://www.quantamagazine.org/elegant-six-page-proof-reveals-the-emergence-of-random-structure-20220425/ https://www.quantamagazine.org/elegant-six-page-proof-reveal...
- avgcorrection 5y ago
  Saving a click: Generala is a relative of Yatzy.
RuggedPineapple 5y ago
>Can you roll a fair die a thousand times and only get 6s? Well yes of course. It'll never happen though. Somewhat related, but humans are also REALLY bad at generating randomness and one of our big tells is an aversion to repeats. If you ask someone to pick 0-9 randomly, repeatedly, they will rarely repeat numbers. But in a truly random sample a repeat is likely 10% of the time, and a three-peat will happen roughly once every 100 numbers. An average person will rarely if ever repeat a number and sure as heck won't ever come up with it three times in a row.
- easymodex 5y ago
  Then again i had exactly this in mind and in OP tasks i did repeat a number here and there.
  - Forgeties79 5y ago
    Yeah but even then most people cap the repeats if they’re faking. If 100 people have to roll 10 random numbers, it is very likely some people will have 3-4 in a row of the same number. It’s less likely none will. My brother in law teaches stats and he runs this scenario with a seminar he teaches. He then runs the results through a simple formula he built in excel and can ascertain who faked vs. did it for real with about 95% certainty IIRC. I love little games like that haha
- macksd 5y ago
  And funnily enough, you'll often hear this trait as being desirable in a pseudo-random number generator. People often want something that will jump around fairly unpredictably but that will come close to outputting all possible numbers once before getting into re-runs.
  - travisjungroth 5y ago
    I’ve coded up this exact algorithm, it’s really fun. It’s useful for “shuffling” in the music sense (not the cards sense). I think it’s actually the prototypical real-world software engineering problem. User says they want X (random music). X is a term in software, so you give them that (you get a random song). They’re not happy. You dig and find out they really want A, B, and C (next song is unknown, songs don’t repeat too soon or too infrequently). This new problem is harder to verify (how soon is too soon?). Editing in tips on solving this sort of problem. You can turn vague requirements into precise requirements. Rather than make the precise requirements exactly equivalent to the vague ones, it's easier to make them more restrictive. Is playing a song again within 50% of the length of the playlist "too soon"? Maybe. How about within 80% of the length of the playlist? Definitely not. We can give ourselves the requirement "Songs must always play again between 80% and 125% of the length of the playlist." Much easier to solve, much easier to test. Sometimes the extra restriction make the problem harder (not usually I've found). Still, this is a great trade because understanding requirements is harder than solving well defined problems. [To the point of this whole post] Requirements can be turned into testable properties even if it's not programmatic. "When I look at a list of chosen songs, there must be no obvious patterns." Who says what's obvious? You do! Then, have someone else do the same. Consider extreme cases. Extreme cases tend to be the most or least important. If they're least important, create a new set of easier requirements or drop it all together. "If 3 - 10 songs, always play within double the playlist, no obvious patterns, never twice in a row. If 2 alternate, if 1 repeat."
K0balt 5y ago
Assuming the die is fair, that outcome would be exactly as likely as any other completely specified outcome.

baby 5y ago

I would have thought that “looking random” would have been calculated by just checking if there is a bias in people’s answers. If someone chooses something that has never been seen before vs something that has been picked a hundred times, then it might be “more random” as the bias clearly comes from humans

graderjs 5y ago

Too deep topic and maybe I don't understand it compared to the megaminds here on hacker News but as far as I get it when it comes to randomness every sequence has the same probability which is simply the inverse of the number of sequences for any given length. In that case it seems to me impossible to tell from only the information of a single sequence. But if you have multiple sequences where one is the binary coding of a Mozart symphony and another is the binary encoding of Shakespeare sonnet and another is the binary encoding of days of the week or a DNA sequence if you have enough data points like that I think you can start to say Well they're probably not random because we see coherent patterns. The second aspect is coherent pattern depends on perspective. So what seems random to us maybe a very common physical constant to an alien civilization. so if we see a binary sequence that represents pi, maybe we say oh well that's not random because it's accurately pi to like a thousand places... but if we see another sequence that looks random maybe it's a physical constant to an alien civilization that's accurate to a thousand places as well. But we don't know that so it looks random...to us. So randomness doesn't just depend on entropy, I think in a realistic point of view, probability is not enough, it also depends on something which Maybe it's a little bit harder to measure perhaps which is perspective. Random depends on the context.

sacrosancty 5y ago
I think your way of looking at it requires an explosion of alien civilisations to find one in which our random 1000 digits is their special number. If you generate 1 extra digit, you'd need 10x as many civilizations to expect a match. Getting back to the experiment, obviously it's measuring the human mind so the context is the person's mind. They might enter their bank account number which looks random to the researchers but is actually cheating because they're not using their brain as the source of randomness.
- graderjs 5y ago
  That's an interesting point about the random sequence selected what's the chance it matches an alien civilization constant. I didn't think about that way. I guess I thought about it more from your other example point of view, where say an alien civilization pretends to be a human and gives you a number that they say is random (just like the guy giving you his bank account number) but actually the alien civilization gave you a physical constant. It's interesting that there's a difference but it's sort of like the difference between you picking a random sequence from a random source and the probability of that being actually random versus you being given a supposedly random sequence by somebody and the probability of that being actually random. I think.

HWR_14 5y ago

> It's always impossible to tell, for any given sequence, whether it was produced by a fair die. There's nothing an experimental subject can do to make the impossible more impossible. That's just not true. Or feel free to play a game with me. We'll roll a 20 sided die. If it comes up 20, you give me a ten. If it comes up any other number, I'll get you a dollar. Nice EV on that! Oh, the die has come up 20, 20, 20, 20, 20, 20, 20 the last seven times. Do you play?

mmoll 5y ago
You’re comparing apples to oranges. The correct analogy is: you pick any other sequence of numbers between 1 and 20 and then tell me you’re more likely to win because your sequence is more random.
Ygg2 5y ago
It can still be a fair die. The context implies it's not, but it's only unlikely, not impossible. See https://youtu.be/8Ko3TdPy0TU https://youtu.be/8Ko3TdPy0TU It introduces a 10 billion human second century i.e. 3.1 x 10^19. If your chance times it is greater than 1 then it's definitely plausible to be done by someone, somewhere.
denton-scratch 5y ago
I would; would you? The Gambler's Paradox says you shouldn't (I'm assuming your 20-sided die isn't crooked). Incidentally, you haven't made your case that you can ever tell whether a given sequence was produced by a fair die. You've just asserted it, and then suggested a game that doesn't illuminate anything.
- dtech 5y ago
  It would be incredibly stupid to take the bet, as it's way more likely that the sequence was not produced by a fair die than that it was (i.e. the dice is rigged in the example) Just because it's impossible to know for certain, doesn't mean you can't make a prediction with very high chance of being correct.

posix86 5y ago

You're wrong. Mathematically. Here's why. When you throw a coin 100 times, each sequence you get is equally likely. However. You can look at properties of the sequence which are more likely to be one way than the other. For instance, it's more likely that the number of heads and tails are about equal than not. The reason is that there are more sequences, in general, where that is true, than those where heads or tails strongly prevail. With the right property, you can make statements such as: This sequence is statistically likelier to be human made than random. One such property is for instance the number of changes from heads to tails, or vice versa. In expectation, random sequences change heads to tails about 50% of all flips. For humans, the expectation is much higher. Hence, if you compare two sequences where one has 51% changes and one 63%, it is mathematically (/statistically) accurate to say that the latter one is likelier human made.

tidenly 5y ago
Your point doesnt refute OPs argument. Your final statement "you can say the latter is more likely random" is not the same as "you can say this sequence is not random". I think lots of people (especially programmers) who know about true RNG vs expectations of RNG might intentionally put in strings of same numbers, or not include the full set, because we know its what often happens during plain RNG. It isnt clear what the goal of the sequence is, hence the confusion in the comments.
- posix86 5y ago
  What does that mean, "you can tell this sequence is not random"? If you show me a blue hat, and ask me if it's blue, I'll say yes, I can say it's blue. But there is always a chance it's not actually blue. It's very conceivable that I'm in a situation where I say with confidence that something is blue, but it isn't. You always only ever speak in probability. Of course you can't say the sequence isn't random, because every sequence can be the result of a random process. "can you tell which is random" to me is equivalent as asking "is one of the sequences such that it is rational to choose it over the other as being random". It's about rational decisions. Consider the frequentist view, where a probability p for an event A means that out of k trials, pk will show A (in expectation), and fruthermore if k -> infty, then the portion of events that show A will converge to p. If you want to choose the right sequence as being random as often as possible, it is rational to choose the one I described above, because it will, overall, be the one that is MORE OFTEN the random one compared to the other.

kevinpet 5y ago

Here are some numbers: 123456123456 Did I get them by rolling a die?

travisjungroth 5y ago
Almost certainly not. Given a circle with diameter AB and a point on the perimeter C, is ABC an equilateral triangle?
- rocho 5y ago
  It's impossible for ABC to be equilateral. Maybe you meant right triangle.
  - travisjungroth 5y ago
    > It's impossible for ABC to be equilateral. Exactly! This is why you can answer "no" to my question, not "almost certainly not".
denton-scratch 5y ago
There's no way of telling. I'll guess you didn't, but that's a guess about people, not about sequences of digits. See, it's not about how likely it is that OP rolled 123456123456; it could have been any "unlikely" number (where I suppose "unlikely" is a psychological quality). So OP keeps rolling until an "unlikely" number comes up, and exclaims "Wow, how unlikely was that?". Well, it's impossible to know, without knowing how many "unlikely" numbers there are; but it's much more likely than 123456123456 is. Aren't all sequences really equally unlikely? So whatever you roll, it's as improbable as all-1s?
m3047 5y ago
Is there a null hypothesis that some other mechanism has a higher probability of producing that sequence?

scraptor 5y ago

All sequences are equally likely to be produced by a fair die but humans are very biased in the kinds of sequences they produce. It might be impossible to ever be certain but you can certainly look at a sequence of all sixes (or that contains more complicated patterns) and estimate that it was much more likely to be produced by a human than a die.

jancsika 5y ago

Now I'm curious: in your opinion what is funny about the line "Nine Nine Nine Nine Nine Nine" in the following cartoon strip, vs. something like "Two Nine Eight Three Seven Eight": https://dilbert.com/search_results?terms=Random%20Number%20Generator https://dilbert.com/search_results?terms=Random%20Number%20G...

drdeca 5y ago
Not the person you asked, but, P(9,9,9,9,9,9 | Not random ) > average over all values of (sequence) of P((sequence) | Not random ) (under some reasonable assumptions about how likely you consider different processes to be the process producing the output) While P(9,9,9,9,9,9 | random ) = average over all values of (sequence) of P((sequence) | random ) Therefore, P( Not random | 9,9,9,9,9,9 ) > P( Not random )

justinpombrio 5y ago

> That instruction is a flaw in the experiment. It's always impossible to tell, for any given sequence, whether it was produced by a fair die. There's nothing an experimental subject can do to make the impossible more impossible. Baloney! Say you measure the traffic to your website in the morning and the evening, every day for a week. And this is what you see: day: 1 1 2 2 3 3 4 4 5 5 6 6 7 7 time: M E M E M E M E M E M E M E visitors: 51 73 58 72 50 78 55 74 55 77 52 73 55 76 It could be that the traffic you get is uniformly random, and each day the number of visitors you get is a uniformly random number between 50 and 80. Sure, all the morning numbers are less and the evening numbers are more and there are suspiciously no numbers in the 60s. All that could be a coincidence. You know perfectly well, though, that you're getting more traffic in the evenings.

denton-scratch 5y ago
You've answered a different question. The question is: can you construct a sequence of dice-casts that an adversary can't distinguish from a real dice-cast? Answer: you can't.
- a1369209993 5y ago
  > The question is: can you construct a sequence of dice-casts that an adversary can't distinguish from a real dice-cast? Yes, obviously: A: 2 4 4 3 5 2 3 6 4 2 B: 5 5 5 6 4 1 1 2 4 5 One is a real dice-cast with a d6 I had lying around. The other is the (100% deterministic) output of: echo 1651434259 $X | md5sum | grep -o '[1-6]' | paste -sd' ' for some X. Feel free to explain[0] how you are able to distinguish which is which. The real issue is that most people don't bother to produce random numbers in a way that's actually secure (which, to be fair, is rather tedious if you don't have a computer handy, and downright prohibitively impractical if you want to do it all in your head, so why would you bother?), either in the study or in general. 0: If you'd like a more black-box distinguishment, I can provide a longer list; obviously a adversary can get the right answer 50% of the time just by chance.
  - denton-scratch 5y ago

asdfasgasdgasdg 5y ago

> That instruction is a flaw in the experiment. It's always impossible to tell, for any given sequence, whether it was produced by a fair die. There's nothing an experimental subject can do to make the impossible more impossible. I don't think it's a flawed instruction -- i.e., I believe most participants will read it as intended, and it will produce interesting results. I read it as follows: "Suppose we show a person your sequence and a sequence generated via an actual random number generator. Minimize the average probability that that person correctly guesses which sequence was generated by a human, across all possible real random sequences." I guess most other people would read it that way, even it is not formally correct from a statistical perspective. And I believe it gets at what they're trying to measure.

denton-scratch 5y ago
> I guess most other people would read it that way Not me. But as I said upthread, that's the only way I can construe the question as being coherent. And I agree that the question thus construed invites the experimental subject to make predictions about the behaviour of others; so the question tests the subject's accuracy in psychological judgements. If you construe it as being a coherent question. I think it's silly to construe it that way; I just think it was a crap question, and you can't draw any conclusions at all from the results.

Strilanc 5y ago

Let's not confuse P(observed_rolls|used_dice) with P(used_dice|observed_rolls). P(observed_rolls|used_dice) is always the same, independent of observed rolls, assuming the dice are fair. But P(used_dice|observed_rolls) can vary, because other ways of generating rolls which are under consideration may be biased towards certain answers, and this allows you to perform inference. For example, suppose the rolls you will be shown were either generated by fair dice or by the program "always return 4" [4]. The rolls you are given are "4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4". Are you really thinking you'd make the same prediction for this sequence of rolls as you would for the sequence "1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1", or is there perhaps some SMALL INKLING OF A HINT as to which answer is correct? The simple fact of the matter is that, in the real world, if you see a sequence like "4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4" you can be surprisingly confident that it was not generated by die rolls. This is because there are plenty of other ways to get sequences and those other hypotheses didn't just pay a Bayes factor penalty of a trillion. Seeing the instructions as incoherent is the mistake of trying to over-isolate the study to the abstract mathematical realm, instead of the world people actually operate in. If someone tells you their luggage combination is 1234, do you really think it's meaningless to guess that it was the default combination as opposed to being generated by secure die rolls? Do you not form opinions about whether or not someone is using secure randomly generated passwords when you find out their password is "password2"? 4: https://xkcd.com/221/ https://xkcd.com/221/

zem 5y ago

Martin Gardner had a wonderful demonstration in one of his columns, where he had people write out 36 "random" numbers, then fold them into a 6x6 square. there were inevitably doubles or triples of the same digit going vertically, and almost never horizontally, since repeating a digit didn't seem random enough to the person generating the sequence.