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Imagine the following experiment: you wait say at a subway exit, and ask everyone passing "please tell me a number" (positive integer, of course). You do this day after day, until you reach say 1M people.

  • What is the distribution $\mu$ on the positive integers that you get?

This is a serious question, obviously some numbers are "nicer" than some other, say arithmetically speaking, so $\mu$ is probably a very interesting measure!

Of course, if you would do the poll with very small kids, $\mu$ would be more or less uniform on $2,3,4,5$ or so, perhaps with some mass at 1, and probably at $0$ too (coming from scientists's kids, proud of knowing what 0 is :)

My question of course concerns adults, and results obtained via a real poll like the one suggested above: does anyone know, is anything written on this subject?

(question inspired from The human body's random number generator, I mean from the title of that question.)

[Edit Jan 2: thanks very much everyone, as a partial conclusion: (1) the measure $\mu$ certainly depends on the precise location of the poll, interesting would be for instance the results of an experiment - I mean, the picture/precise formula of $\mu$ - in a "random" place, say Times Square, (2) there are lots on interesting links signaled below (by MP, Joel, JSE..), papers by cognitive scientists, plus some interesting math interpretations/speculations (by quid, Alexander, Andreas, Yuichiro..) but I'm still afraid there's no picture of some particular $\mu$ emerging from all this, (3) will keep looking etc., and of course, if I ever get a huge grant, with some freeness in spending it :) think I'll conduct such a poll experiment myself - it's probably worth it.]

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closed as off topic by Qiaochu Yuan, Felipe Voloch, Chris Godsil, GH from MO, Alex Bartel Jan 2 '13 at 14:59

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Interesting question, +1. Not really number theory, in my opinion, and also I doubt the measure is interesting (in a mathematical sense), but still now that you brought it up I'd be interested if this was done and what the result would be. (I'd guess it would also depend on cultural context, some numbers have a certain meaning in some cultures while being just-some-number in others.) I once heard, but could now not find it, of a study which 'random picks' people tend to make on some grid of numbers for a lottery and how it depends on the form of the grid. –  quid Jan 1 '13 at 16:29
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This is not a mathematical question; it's purely empirical. I suppose some statisticians familiar with some such experiment might be able to answer, but in my opinion the question doesn't belong on MO. –  Todd Trimble Jan 1 '13 at 16:53
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Interesting question, but I don't see any mathematical content. Voting to close. –  Goldstern Jan 1 '13 at 17:05
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Meta thread at tea.mathoverflow.net/discussion/1508 [extra sign up is necessary, MO login will not work, but simple and instant, depsite the wording 'apply for membership'] –  quid Jan 1 '13 at 19:17
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psycnet.apa.org/… –  M P Jan 1 '13 at 19:58

2 Answers 2

up vote 12 down vote accepted

Cognitive sychologists study this kind of question, as you might expect. Here's a paper (behind a paywall, sorry) where they asked people to name random digits. You don't get uniform distribution on 0,..,9.

I learned a little about this stuff when I was writing a blog post about detecting election fraud by looking for digits which looked more like "numbers made up by humans" than "numbers arrived at randomly."

Update: I spoke to my colleague Gary Lupyan, a cognitive psychologist here who studies such things. There are lots of interesting results, although he hasn't done the precise experiment suggested in the question. If you ask people to name a number between 1 and 100, the modal responses are between 1 and 10, with maybe a slight continuing dropoff afterwards. People disprefer even numbers and multiples of 5 and 10. He also replicated the folk belief that if you ask people to name a number between 1 and 20, the modal response is 17.

It doesn't look to me like the results he's getting are well-modeled by any particularly natural distribution, though you could certainly fit some kind of decay to it.

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This isn't really an answer, but I couldn't post this in the comment field. So allow me to write this here.

There can't be a single definite answer to this question. But if you restrict your mathematical model and ask it in an applied math way, there might be a very interesting answer.

For example, you may choose a more mathematical side of phonetics or phonology in linguistics and focus on how each phoneme affects your choice of words in this kind of experiment, where the choice seems random on the surface (e.g., "Say a number") but shouldn't be if you think about it (e.g., numbers have cultural connotations that aren't random).

A phoneme is, in a very rough sense, a unite of sound in human language. For example, the word "word" is pronounced with the "w" sound followed by the "r" sound and ends with the "d" sound (at least in most part of US, I think). So "word" has three phonemes in it: "w", "r", and "d".

It is known that some phonemes are more natural to humans and others are more difficult. For instance, the "m" sound is universally found in pretty much every natural language. But, the English "th" sound and "r" sound, for example, are mastered by children at a later stage than other sounds during language acquisition. Adults who are learning a foreign language have more difficult times mastering certain phonemes.

Can this intrinsic difficulty of each phoneme affect your choice of words? If so, how much? What about a combination of phonemes? Are some popular combinations more difficult? Can we quantify the difficulty?

I guess this kind of question is known and answered somewhere in linguistics. But it would be interesting to know, for instance, the distribution of "random numbers humans pick" when asked to provide a number between 1 to 9 in a well-controlled or ideal situation, where all other nonrandom effects such as cultural connotations are eliminated and only phonetic difficulty is important. So, for example, among the nine numbers, "7" is the only two syllable guy (s-e-v-V-n, where V is the phoneme representing the neutral vowel). So most likely we should have prior knowledge on how much the number of syllables of a word affects your choice too. But it would be very intriguing if linguists have already done various different experiments and theoretical work and quantified the difficulty of, say, the two phoneme combination "thr" so you can derive the probability that a native English speaker randomly picks "3" in an ideal no-other-bias situation.

The above model is too simplistic, of course. For example, a phoneme typically has allophonic variations (e.g., "p" in "pin" and the same phoneme in "spin" are actually different sounds, though native English speakers typically can't even hear the difference and think they're exactly the same "p" even though they subconsciously differentiate the two and never use the wrong one). But very cruder quantification is already interesting to me. It's fun to know which number is slightly more favored because of how it is pronounced. It's interesting to know if in principle we can predict if someone is a bilingual by observing abnormality in his choice of numbers in such ideal experiments; a certain combination in one language may be easy if you speak a certain different language.

Of course, this post is my sneaky trick to see if someone knowledgeable in this line of work can answer my question in a more math oriented way (e.g, giving a distribution in an ideal pick-a-number experiment) that would be closed at the speed of light if asked as a stand-alone question on MO.

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