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In this Numberphile video, the question: "What is a number nobody has thought of?" is addressed. The method is as follows:

  1. Estimate a number $N$ as the number of times humans have thought of numbers
  2. Estimate a probability distribution $\mathbb{P}$ for what number you think of when you have a thought and suppose that each though is an independent draw from this
  3. Calculate the distribution of the maximum of $N$ independent draws from $\mathbb{P}$

More precisely, this is the answering the question: "What is the largest number someone has thought of?" (and if you add 1 you will surely get a number nobody has thought of).

A more interesting question in my view is "What is the smallest number nobody has thought of?". If we agree with step 1 and step 2 from the Numberphile video, then this ammounts to:

What is the distribution of the smallest integer $I$ so that $N$ iid draws from a distribution $\mathbb{P}$ on the integers does not contain $I$. (In other words: the event that $\{I > x\}$ is the event that the numbers $\{1,\ldots,x\}$ all appear in our sample).

Surely someone has thought of this before....is there an elegant way to calculate this?

Mihai Nica
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  • I would have thought this depends (a) on the distribution and (b) the number of people and (c) the particular case is that specific sample – Henry May 17 '22 at 17:48
  • Right: the problem is how do you comibine the number N and the distribution P to get the answer( for example, for the maximum seen number as they did the video, the answer is to do something like (1-P(X > x))^N). What is the right operation to do to get the minimum unseen number? – Mihai Nica May 17 '22 at 17:53
  • I just realized that the problem of finding the probability that all numbers from {1,...,x} are thought of in N thoughts is the coupon collector problem with unequal probabilities! So https://math.stackexchange.com/questions/600012/coupon-collectors-problem-with-unequal-probabilities or https://mat.uab.cat/matmat_antiga/PDFv2014/v2014n02.pdf may be helpful – Mihai Nica May 17 '22 at 17:58

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