Given a profinite group $G$, one can always consider the (normalized) Haar measure on it. In this talk, we'll consider a few types of measurable sets which concern the $p$-elements of $G$ and see what can said about the structure of $G$ whenever these sets have positive measure. First, we show that if $p$ is odd and the probability of randomly choosing a $p$-element is positive, then $G$ is virtually prosolvable. We'll then exhibit a counterexample to this when $p$ is 2. Second, we prove that if $G$ has the property that, for every $p$-element $x$, it is positive the probability that a randomly chosen element $y$ of $G$ generates with $x$ a pro-$p$ subgroup, then $G$ contains an virtually pro-$p$. This is joint work with Andrea Lucchini.