Irreducibility of random polynomials of large degree has been studied recently in works by several authors (in particular by Bary-Soroker, Kozma, Koukoulopoulos and by Varju and myself). We study analogous problems in the setting of word maps in matrix groups, such as SL2(C) or more general semi-simple Lie groups. Conditionally on GRH, we are able to determine the dimension and number of components of word varieties  with an exponentially small probability of exceptions. We also exhibit a phenomenon of Galois rigidity when the defect of the random presentation is equal to one. The proofs use effective Chebotarev type theorems and new spectral gap bounds  for Cayley graphs of finite simple groups. Joint work with Peter Varju and Oren Becker