We will discuss some recent results and open problems at the interface of combinatorics, probability, and analysis. One such question concerns the existence of Hadamard matrices, a very difficult problem. The problem should become much easier if we relax the condition a little bit, an idea recently proposed by Dong and Rudelson. Solutions to the "approximate Hadamard matrix" problem exist but they are far from simple. Nice, simple and explicit solutions should exist! All of this is also connected to so-called vector-balancing problems and the Komlos conjecture. Finally, we'll discuss the problem of a (morally) bad scientist trying to design a battery of (individually fair) statistical tests such that a fair unbiased coin is typically identified as biased by at least one test. Geometrically, this corresponds to finding n x n square matrices that send the 2^n vertices of the {-1,1}^n cube to points with at least one large coordinate on average. We discuss an asymptotic solution of the problem and the curious structure of low-dimensional extremal examples when n=2,3,4,...