  In this article, we study several probabilistic properties of polynomials
defined over the ring of $p$-adic integers under the Haar measure. First, we
calculate the probability that a monic polynomial is separable, generalizing a
result of Polak. Second, we introduce the notion of two polynomials being
strongly coprime and calculate the probability of two monic polynomials {being}
strongly coprime. Finally, we explain how our method can be used to extrapolate
other probabilistic properties of polynomials over the ring of $p$-adic
integers from polynomials defined over the integers modulo powers of $p$.
