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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 for 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.

1

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 for someone knowledgeable in this line of work can answer my question that would be closed at the speed of light if asked as a stand-alone question on MO.