In this Numberphile video, the question: "What is a number nobody has thought of?" is addressed. The method is as follows:
- Estimate a number $N$ as the number of times humans have thought of numbers
- 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
- 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?