Kleene's recursion theorem states that for every total computable function $f$ there is an index $e$ such that $$\phi_{f(e)} = \phi_e,$$ where $\phi_n$ is a valid enumeration of the partial computable functions. Call such an $e$ a fixed point of $f$.
In Cutland's Computability, a question is proposed: would a chimpanzee sending each $n$ to a random $f(n)$ leave some $e$ fixed?
Although the question is mainly philosophical, there is a purely mathematical version:
If a random total function $f : \mathbb N \to \mathbb N$ is chosen, what is the probability that $f$ does not have fixed points in the above sense?
I conjecture that the measure of that set is going to be zero, so the particular probability measure used is not going to be important.
My reasons for the conjecture. I find it rather unlikely that computable functions all share a property that few non-computable functions share (except for properties that clearly have to do with $f$ being computable).
In fact, all the functions in Turing degrees below $\emptyset'$ are known to have this property, not only computable functions. (Edit: only the recursively enumerable degrees below it)
Nevertheless, it also appears that, for every $e$, almost no function fixes $e$, and so many, many functions won't have fixed points. This is counter evidence for my conjecture.
This question is probably simple, but I don't have much background in measure theory.