We can choose at random between two options, between a hundred, between a million. But when we want all positive integers to have exactly the same chance, randomness breaks.
The Lottery Without Enough Randomness
Riddle statement
You want to design a perfectly fair lottery for choosing a positive integer:
The condition is that every number must have exactly the same probability of being chosen.
Is this possible?
Show solution
Solution
No, it is not possible.
Suppose every number has the same probability.
Call that probability $p$.
Then:
the number 1 has probability $p$; the number 2 has probability $p$; the number 3 has probability $p$; and so on.
The sum of all probabilities should be 1, because some number must be chosen.
But there are two cases.
If:
then adding $p$ infinitely many times makes the sum grow without bound. It cannot equal 1.
If:
then every number has probability 0, and the total sum would also be 0. That cannot equal 1 either.
There is no third option.
Therefore, there is no uniform lottery over all positive integers.
Answer: no such lottery can be designed.