Pick your favorite two vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$, and let $X$ be the dot product of $\vec{u}$ with a random permutation of $\vec{v}$. What can be said about $\mathbb{P}(X=0)$? In this talk, we discuss this and related problems, and we provide various upper bounds on this probability. As a motivating application, throughout the talk, we will discuss the connections between this and the study of random polynomials, and we obtain new upper estimates on the expected number of critical points a random polynomial will have. This is a joint work with Aaron Berger, Ross Berkowitz, and Van Vu.