  In this paper we derive an explicit formula for the numerical range of
(non-self-adjoint) tridiagonal random operators. As a corollary we obtain that
the numerical range of such an operator is always the convex hull of its
spectrum, this (surprisingly) holding whether or not the random operator is
normal. Furthermore, we introduce a method to compute numerical ranges of (not
necessarily random) tridiagonal operators that is based on the Schur test. In a
somewhat combinatorial approach we use this method to compute the numerical
range of the square of the (generalized) Feinberg-Zee random hopping matrix to
obtain an improved upper bound to the spectrum. In particular, we show that the
spectrum of the Feinberg-Zee random hopping matrix is not convex.
