Probability of $\mathcal{O}_{\mathbb{Q}(\alpha)}=\mathbb{Z}[\alpha]$
You are in luck. This exact problem has been considered in this paper: https://arxiv.org/abs/1611.09806 . They prove that $ p_n = 1/\zeta(2)$, in particular it is independent of $n$!
Read the introduction to the paper, it is very clear.
(Do note that they work with a different height function.)