# What is the probability of occurrence of natural numbers? [closed]

What is the probability of a number $n ∈ ℕ$ to occur when someone uses a natural number in daily life?

(You can assume that humankind had a number system in which no psychologically distinguished numbers existed like 10000, 9999, 250, or 333. E.g. if humankind would use an ∞-ary number system, then this assumption would be fulfilled.)

• distribute a probability function on N. But then it must be biased since you are dealing with an infinite space. Jul 11, 2015 at 7:29
• I thought it would be similar (but different) to the probability of getting a certain number n ∈ ℕ of tails before the first head when flipping a coin repeatedly. For the numbers 0, 1, 2, ... that's 1/2, 1/4, 1/8, ... . Yet, for any n ∈ ℕ, the probability is 1/2^n ≠ 0, so n may occur with a non-zero probability. Jul 11, 2015 at 11:02
• I don't think this is a mathematical question. You might be interested in this essay: bit-player.org/2014/600613 Aug 9, 2015 at 3:03
• This is a silly question. You try to magically eliminate the bias in favor of $10^n$, which you can't. How can anybody make a sensible answer? Intuitively, small numbers are more likely than large numbers. Once some agency reports the US population as a particular number around $3E11$ (and I have seen it specified down to the $1$!?), that number becomes much more used than any number nearby. Aug 11, 2015 at 5:30
• @GerryMyerson thanks for the link. This is indeed helpful for me. Aug 14, 2015 at 15:13

A distribution like $P(n) = 2^{-n}$ isn't going to give very good predictions. According to that I'm $8$ times more likely to say "$997$" than "$1000$". You mention that base $10$ is sort of a cultural thing that should maybe be ignored, but plenty of numbers distinguish themselves for other reasons and should be more common than their neighbors.

One way of improving this situation is to fix some description language and then define $P(n) = c \cdot 2^{-2^{K(n)}}$ where $K(n)$ is the Kolmogorov complexity of some number $n$ and $c$ is a constant chosen to make all the probabilities add up to $1$. This will give higher probabilities to numbers with shorter descriptions, so "googolplex" will have a much higher probability than some random number with a googol digits. Like base $10$, whatever description language you choose will come with some cultural baggage but some are definitely more natural than others (for example English vs. combinator calculus).

• +1 for proposing a working theory for such a distribution which even gives room for that for any culture (which uses one designated descrition language), there will be cultural specifics for the probability of a number. Aug 14, 2015 at 15:45

I wondered about thinking about it in the following way. Suppose we consider a subset of $\mathbb{N}$ which only goes up to $N$. Then suppose, in daily life there arises the situation which will use a number no higher than $X\leq N$ where $X$ is a random variable uniformly taking a value in $\{1,2,...,N\}$. Then, assuming the probability of drawing each number from this set is equal, the probability of a number $k\leq N$ being used in daily life is \begin{align} P(k) &= \sum_{i=k}^N \frac{1}{i}P(X=i)\\ &= \frac{1}{N} \sum_{i=k}^N \frac{1}{i}. \end{align}

I was then planning on taking the limit of this as $N\rightarrow\infty$, however I think this goes to zero... I wonder if some variant of this would be the right approach?

• I like your answer, because it is very close to what I think. I was also unable to find a final result with this approach, though it looks promising. Aug 14, 2015 at 15:37
• We don't know the probability that X is a number from the Nth set {1,2,...,N}. E.g. the number 3 is presumably more often taken from the 5th set than from the 100th set. Aug 16, 2015 at 15:57

I once did an analysis on the "popularity of numbers" by just googling them and then recording how many "google search results" they obtained.

The result was a pretty consistent relationship where $$\text{num_results}(n) \sim \frac{1}{n}$$.

I know this is only one case of your questions in general, but the internet happens in my daily life at least.

• I appreciate that you've explained your interpretation and research, which I'm willing to stipulate is within the scope of "reasoned mathematical argument", however briefly stated. The Question itself though asks for some hypothetical circumstance in which numbers are not represented by decimal forms. For that reason I"m voting to close the Question. Jul 15, 2016 at 0:33

0, If the distribution is uniform (if you allow any Natural number in the list from which you choose), which I assume you mean when you say there is no distinguished numbers. Otherwise, for an infinite collection, there is no uniform distribution and you need to specify the distribution used.

EDIT: If you want more of a real-life distribution, you could consider, e.g., Benford's law : https://en.wikipedia.org/wiki/Benford%27s_law

• Imagine someone makes a shopping list. He has 3 children (it was more likely I would pick a number below 20 than above, because most people have less than 20 children). Each child wants 2 different kinds of fruit. One child wants apples and dislikes oranges. Another one wants oranges but dislikes apples. So the person needs to make a list with at least 3 different kinds of fruits and the numbers he will buy will total to 3*2=6, so they will all be below 6. Lower numbers are more likey, though I do not know the exact distribution. I'm not specifying the distribution. I'm asking for it. Jul 11, 2015 at 10:56
• @DanielS: Maybe you could consider Benford's law?en.wikipedia.org/wiki/Benford%27s_law Jul 11, 2015 at 14:52
• This is very similar to Benford's law, but not the same. Benford's law only says something about the first digit of a number while I am interested in the complete and exact number. Jul 11, 2015 at 23:04

I'm not sure this question has any kind of assessable answer, but one common assertion that seems to hold some water is the notion that numbers in discourse are distributed roughly logarithmically; this is the basis behind such patterns as Benford's Law. Granted, that usually applies to real numbers (ignoring sign), but it doesn't take too much work to apply it to the natural numbers.

Of course, the sum of the logs (or the integral of the log function) is unbounded, so using that to directly arrive at a probability distribution won't work. Ultimately, we will have to set an upper limit to the numbers we consider. We might use any of a number of very large numbers used in various proofs, but since you mention "daily life" and not "abstruse mathematics," I think it's reasonable to set an upper limit at about a trillion, the largest number that most people are likely to come across in everyday discourse (in connection with government budgets, let's say).

I would also aggregate all numbers with the same first three or so significant digits; that is to say, the number $345217469$ would be assigned a probability of essentially zero, as would most nine-digit numbers beginning $345$, but $345000000$ would be given the sum of all of their probabilities.

That's not quite ideal, but given that the logarithmic distribution is already an idealization (!), that's probably close enough to get in the neighborhood.

• I was looking for an exact solution. But maybe you're right that this probability can only be estimated rather than calculated. Aug 14, 2015 at 15:42
• I also do not believe that we need an upper bound, because e.g. the number of tails before head example is a nice, real world distribution where any number has a non-zero probability. Aug 14, 2015 at 15:49
• As Dan Brumleve's answer mentions, a geometric distribution—even one with a slower fall-off than $1/2$—is not really a likely candidate. I think a logarithmic fall-off is more representative of how often we use numbers of a given size. But within a given size range, there are a lot of inhomogeneities. And real-world considerations (such as education and the desire to conceive of large numbers) impose genuine limitations that are difficult to articulate precisely. And I think there is no point in trying to do so too exactly; it is interesting enough merely to approximate it. Aug 14, 2015 at 15:58