I know that number fields have been the object of many statistical experiments.
Is there some kind of heuristics for the following?


Fix a degree $d$ and fix a bound $N$ on the coefficients of a monic integral degree $d$ polynomial $P$ (a more natural choice would probably be to bound the discriminant of $P$). Among those such polynomials which are irreducible, what is the proportion of those for which $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$? 


I wrote a very basic and naive script in pari/gp to test this and it seems to me that the proportion is about 60%. Is anything known or conjectured about this?


At least, is it easy to see that this proportion is non-zero?


What about the similar but different question: for which proportion of $P$ does the ring of integers have a power basis?



Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
I summarize the first two pages of Kedlaya, [A construction of polynomials with squarefree discriminants](http://arxiv.org/abs/1103.5728)



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> When $P$ is irreducible and the discriminant $\Delta(P)$ is square free, the number field $\mathbb{Q}[x]/P(x)$ has ring of integers $\mathbb{Z}[x]/P(x)$... When the coefficients of $P$ are chosen randomly, this is expected to occur with probability $\prod\_p a\_p$ ... where $a\_p$ denotes the probability that $\Delta(P)$ is not divisible by $p^2$. These probabilities have been computed by [Brakenhoff](https://doi.org/10.1080/10586458.2007.10129001):
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I'll omit the table, but they are all of the form $1-O(1/p^2)$, so the product is nonzero.



> 
> Unfortunately, ... it seems quite difficult to prove that a polynomial takes squarefree values with the expected probability ...
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Keldaya then summarizes work of [Granville](http://www.dms.umontreal.ca/%7Eandrew/PDF/polysq3.pdf) and [Poonen](http://arxiv.org/abs/math/0203292) which, assuming the ABC conjecture, implies that $\Delta(P)$ is squarefree with probability $\prod a\_p$, where the limit is taken over boxes in $\mathbb{Z}[x]\_{\deg n}$ of a certain shape (roughly, much longer in one direction than the others.).



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> However, without assuming any conjectures, it is difficult to establish even the existence of infinitely many polynomials of a given degree with squarefree discriminant.
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Kedlaya then explains that constructing infinitely many such polynomials is his main result.




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In short, there is a good conjecture for the probability of squarefree discriminant, but people can't unconditionally show that it is even positive.
Squarefree discriminant is a bit more special than $\mathbb{Z}[x]/P(x)$ integrally closed, but I think this is suggestive.


