In this talk I will describe a family of events arising in two related probability models, one having to do with uniformly random ``fully packed loops'' (a family of combinatorial objects which are in bijection with alternating sign matrices), and another appearing in connection with a natural random walk on noncrossing matchings. The connection between the two models is highly nonobvious and was conjectured by physicistsRazumov and Stroganov in 2001, and given a beautiful proof in 2010 by Cantini and Sportiello. Another intriguing phenomenon is that the probabilities of the events in question, known as ``connectivity events'', appear to be rational functions of a size parameter N (with the simplest such formula being $3(N^2-1)/2(4N^2+1))$, but this is only conjectured in all but a few cases. The attempts to prove such formulas by myself and others have led to interesting algebraic results on a family of multivariate polynomials known as ``wheel polynomials'', and to a family of conjectural constant term identities that is of independent interest and poses an interesting challenge to algebraic combinatorialists.