If you throw a dice a large number of times a random array of the numbers 1-6 will be the result. Why shouldn't we be able to produce a similar array of these numbers ourselves? Why can't our fingers move in such a way to randomly pick one out of 1-6, say written on a piece of paper in front of us? Is it because we are not a random process ourselves? If at each pick we forget about the last, will there always be a preference for a special number, and if why? Have there been experiments that showed this non-randomness? What if I took part in this experiment and I memorized a random array of the numbers? How can you know that I'm not picking randomly? Or am I? How can you know that my picked numbers are not picked by random choice? How do you compare to a real random array?
Is this related to free will? I.e., is my free will maximal if my choices are truly random?
If I have to pick a person blindly, I can just point my finger at one of them, when a group is standing before me. Why can't the same be done with numbers?

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    "my free will maximal if my choices are truly random?" If they were trully random, you wouldn't have influence over them, and thus you wouldn't have free will.
    – armand
    Jun 17, 2021 at 11:25
  • @armand I don't want to influence them. If a choice presents itself, I can always say no or yes to the choice. And wait for another choice to appear.
    – user52804
    Jun 17, 2021 at 14:31
  • "free will" (roughly) means you can choose "yes" or "no". If one of them is selected at random, you are not choosing, so you can't have free will. That's why I don't understand your sentence I quoted.
    – armand
    Jun 18, 2021 at 4:39

4 Answers 4


Having typed a substantial wordcount into the comments, I'm moved to do the honorable thing and post an actual response. It's particularly necessary in this case since two posters gave incorrect or misleading answers, backed up respectively by an equally incorrect and misleading academic study and a Numberphile video.

It is true that the average person on the street is typically very bad at generating a random sequence. That clearly is what @CriglCragl's academic study and @J Kusin's Numberphile video were trying to get across. But we already know that the average person on the street is hopelessly innumerate. That's not the question. The question is, can we verbally express a sequence of bits, say, ones and zeros, that passes every known statistical test for randomness?

Of course we can. The OP suggested some ways. We can flip a coin and memorize the sequence. We can use a pseudorandom number generator (PRNG). We can even use a quantum spin flipper. But suppose we outlaw mechanical devices, preparation, and memorization. Suppose we are required to spontaneously create a random bitstring on the fly. Can we do that?

Yes. There's a simple piece of knowledge required. I'm going to reveal it.

As a conceptual framework, we're going to play a hypothetical game. This was suggested in the Numberphile video, and J Kusin suggested a small variation that I find easier to work with.

The game involves two individuals, the Player and the House. The Player proposes a bitstring of length 20; that is, a sequence of twenty 1's and 0's. The House applies standard statistical tests to the bitstring. The Player wins if the bitstring passes the statistical tests for randomness; and the House wins if the bitstring turns out not to be so random after all.

The claim is that nobody can do this without getting ridiculously lucky. This is false. The Player can win every time, or at least a good percentage of the time, especially against the weak strategy employed by the House in the Numberphile video. I can teach anyone how to do this.

Consider. If you are the House, you would gladly play this game for money with the first person you run into at the local grocery store, or frankly with a member of the US Congress, or the member of the New York Times editorial board who agreed that when Mike Bloomberg spent 500 million dollars on his presidential campaign, he could instead have given a million dollars each to every person in the country. The actual amount is a buck fifty. The New York Times editorial board.

So yes it's true as J. Kusin's Numberphile video and CriglCragl's academic study say, that the average, typical person on the street , in the US Congress, and on the NYT editorial board, is an innumerate moron. Is that too strong? You tell me. But surely we already knew this. Innumeracy is a plague upon the land.

But that is not the question. The question is, can a person spontaneously create a statistically random sequence? We agree that you'd gladly be the House playing against the average member of the NYT editorial board. But would you play this game with the first person you see at the annual meeting of the American Statistical Association? No you would not, because they know what statistically random bitstrings look like. You wouldn't play this game with a person on the street in Cambridge, Massachusetts, because you might pick an MIT student.

How much knowledge is actually required? Very little. Here it is.

Random bitstrings, even short ones, typically have constant runs that look nonrandom.

That's it. Let me show you.

I found a lovely random bitstring generator online in less than 30 seconds of searching. We live in wonderful times. Here are the first three 20-bit bitstrings I generated. I did not cherry-pick these. This is what came out.




The first number has a run of five 1's, a run of five 0's, and another of six 0's. The second has a run of four 1's, a run of four 0's, and ends with a run of five 0's. The third contains three runs of three 0's. The amateurs would never pick strings like these; but the professionals know, and now you know, that this is exactly what random bitstrings look like.

That's it. That's the trick. The average person will go 101011010. They'd be afraid to write 00000 in a 20-bit string because they think it wouldn't look random. But that IS exactly what you should do.

Would J Kusin or the Numberphile guy play this game with anybody who's read this far?

No. They'd lose. I rest my case and I have made my point. Humans can easily spontaneously verbalize 20-bit random bitstrings with no preparation, memorization, or mechanical aids; using only the tiniest bit of knowledge about the surprising frequency of constant runs in even short random bitstrings.

There are other ways to win as well. The OP suggested them himself. You can flip a fair coin 20 times and convert that to a bitstring. You can use a pseudorandom number generator (PRNG).

You could even use a quantum electron spin detector, but that would not generate any better results than a fair coin or a PRNG. This point bears examination.

When it comes to statistical randomness, electron spins aren't any more random than coin flips or PRNGs. Electron spins may be "truly" random -- or they may not. Depends on if you're a Copenhagen or Many Worlds fan. The truth is that nobody knows whether electron spins are "truly" random or not, or if the concept is even meaningful.

But when it comes to statistical randomness, PRNGs, coin flips, and quantum spin detectors are equivalent. There's no difference among them, even though their degree of "true" randomness is different. The PRNG is absolutely deterministic, it produces the exact same sequence every time. A coin flip is the deterministic outcome of physical variables such as flip force, air pressure, and so on; but is random in practice because we can never know all those variables to sufficient accuracy. And the quantum spin? That's a matter of philosophy. Nobody knows if it's random or not. But in terms of statistical tests, none of that matters.

The Numberphile game reminds me of gambling casinos that offer Blackjack, but throw you out of the casino if you happen to be good at it. They only allow bad players to play. What kind of sportsmanship is that? The same kind showed by Numberphile. You can only beat bad players. In the Numberphile version, the player isn't even told the rules of the game before sitting down to play! If the House DID tell the player the nature of the game, the Player would read up on random sequences and win. The only way the House wins is to ban players who know how to play. And that proves nothing. It's like saying, "Nobody can beat Blackjack," because everyone who CAN beat Blackjack gets thrown out of the casino. That does not constitute proof of the thesis.

A rare miss by Numberphile, though I only watched the first five minutes and perhaps they redeemed themselves later. It turns out that it's easy to verbalize a random bitstring with only a tiny bit of knowledge of what random bitstrings look like. The fact that the great mass of humanity isn't good at it doesn't prove anything at all. The great mass of humanity can't dunk a basketball but the NBA is full of people who can. Numberphile's logic is bad.

Likewise CriglCragl linked a study titled, "Humans cannot consciously generate random numbers sequences." I just showed how they can. All that's true is that the average person on the street can't. But the average person on the street can't do high school algebra. That's no proof of anything, other than that our educational system is an unmitigated disaster. But we already knew that. Humans with a very small amount of training can easily be taught to generate statistically random sequences.

I might add that if the digit-run trick isn't good enough, statisticians no doubt know some other common characteristics of random bitstrings. And even with the constant run trick, you won't win all the time. But you will definitely beat the House over the long run. The Numberphile guy was just using the fact that people instinctively shy away from unlikely-looking runs. He'll lose to a Player who includes them. The larger point is that you would not play this game with a professional statistician. MOST humans aren't very good at making random sequences, but SOME are very good.

But most people aren't good at the things that some people are very good at. If you saw a study that said, "People aren't good at playing the piano," you'd immediately spot the bad logic. Most people can't play but some can play brilliantly. Likewise creating statistically random sequences. It's not worth an academic paper or a Numberphile video, because the fact that most people are bad at activities that some people are brilliant at is a very trite observation. Somehow the bit about randomness made some otherwise smart people not realize that.

Now in terms of the OP, we might ask, even if we did learn how to verbalize random sequences, can I really pick a random number? What does that even mean? A coin flip isn't random if you knew enough about the variables; and even a quantum spin measurement is random in some interpretations and not in others. Nobody actually knows. So the question's too philosophically deep to sensibly address. When I picked my statistically random sequence, I don't know if my choice was determined at the moment of the big bang.

What we do know is that with a little bit of knowledge of the properties of random sequences, we can easily verbalize sequences that will pass statistical tests of randomness, defeat J Kusin's Numberphile game, and falsify CriglCragl's academic study. I think that's as much as we can say.

ps -- I kept referring to the Numberphile video but the link is actually buried in the comments under J Kusin's response.

Here's the video.

  • What a wonderful answer (also the part about the 500 million dollars)! I can do nothing else than accept! The internet is wonderful indeed!
    – user52804
    Jun 18, 2021 at 10:23
  • I was reading the comments below the answer of J.Kuskin. Great stuff!
    – user52804
    Jun 18, 2021 at 10:42
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    @Methadont Thanks, I owe J Kuskin a thank you for inspiring me to this.
    – user4894
    Jun 18, 2021 at 10:48

Yes, lots of studies, eg 'Humans cannot consciously generate random numbers sequences: Polemic study' Małgorzata Figurska et al.

Our cognitive biases mess us up, like say underestimating how frequent coincidences like the same roll in a row, are.

True randomness is pretty rare. It is basically just found in quantum mechanics. Other cases are pseudo-random, based on various algorithms to simulate randomness but which over very large numbers, or in certain circumstances, they usually have limitations or flaws.

The Many Worlds interpretation of QM, restores determinism, but over the sum of all histories, which all happen.

  • I see you are a fan of the MWI. Don't we have an internal dice? Can't I just let my finger move over different kinds of icecream and with closed eyes pick a taste?
    – user52804
    Jun 17, 2021 at 19:20
  • @Methadont: Absolutely not random. As you do that more times, deviations from chance will become very apparent. See eg recency bias, gambler's fallacy (non-preductive patterns), confirmation bias (preconceptions), outcome bias (over-focus on what has already happened, like 'on a winning streak' ), hindsight bias (playing revisionust historian). All kinds of deviation from randomness can be used for cracking cryptographic systems, based on predicting humans or algorithms.
    – CriglCragl
    Jun 17, 2021 at 22:01
  • But why can't I practice to evade biases? Why can't I practice to employ the pure probability of QM? There are plenty of these random processes going on in my head. I can imagine I measure them and choose accordingly.
    – user52804
    Jun 17, 2021 at 22:05
  • Methadont: It's like asking, why can't you just stop being human? Sure you can, but then you wouldn't be you, or an us. 'If my grandma had three wheels she'd be a tricycle', as Italians say.. Have a read of royalsocietypublishing.org/doi/10.1098/rsta.2015.0100 You might consider using the 'Universe Splitter' app for your decision making.
    – CriglCragl
    Jun 17, 2021 at 22:16
  • Then the question reduces to: Is it possible to act in a non-human way while being human? I can imagine my nonna to have three wheels :) So why don't imagine a rolling dice?
    – user52804
    Jun 17, 2021 at 22:22

A choice is the logical opposite of random by definition. If a sequence of random numbers is chosen, that is a pseudo-random sequence designed to give the appearance of randomness, to pass some statistical tests for randomness.

Statistical randomness is a property of a series of values. It means the lack of any pattern or predictability in the series. Statistical randomness does not detect whether the values are pseudo-random (=fake random) or truly random.

True randomness is a property of a single value. It means that the value is not chosen by anyone or the product of an algorithm (which must be chosen).

People can only do pseudo-random, because we have to choose the values. For truly random results we have to roll the dice or shuffle the cards.


Because there WILL be a pattern no matter how hard you try. In quantum experiments there is ZERO pattern in successive measurements in certain types of experiments, like incoming electrons being deflected up or down. You may challenge the world's leading mathematicians, cryptographers, super computers, and machine learning algorithms, and you will find your array [produced from your mind, not from reading coins or die rolls - blurt out a series of 0's, and 0'1, try to be random and disguise any pattern, for example] has a detectable pattern.

That is an empirical way to tell the difference.


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