My question may be not interesting or easy to answer ! but I am really not familiar with proba.
Let $p$ be an odd prime number. and let $r\geq1$ an integer. choose an element $A\in\mathrm{GL}_2(\mathbb{Z}/p^{r+1}\mathbb{Z})$
What is the probability for which $Tr(A)\equiv p^{r}\pmod{p^{r+1}}$ and $Tr(A)\not\equiv 0\pmod{p^{r+1}}$ ?
Thanks for any comments.