  We call an element of a finite general linear group $ \textrm{GL}(d,q) $
\emph{fat} if it leaves invariant, and acts irreducibly on, a subspace of
dimension greater than $d/2$. Fatness of an element can be decided efficiently
in practice by testing whether its characteristic polynomial has an irreducible
factor of degree greater than $d/2$. We show that for groups $G$ with $
\textrm{SL}(d,q) \leq G \leq \textrm{GL}(d,q) $ most pairs of fat elements from
$G$ generate irreducible subgroups, namely we prove that the proportion of
pairs of fat elements generating a reducible subgroup, in the set of all pairs
in $ G \times G $, is less than $q^{-d+1}$. We also prove that the conditional
probability to obtain a pair $(g_1,g_2)$ in $G \times G$ which generates a
reducible subgroup, given that $g_1, g_2$ are fat elements, is less than
$2q^{-d+1}$. Further, we show that any reducible subgroup generated by a pair
of fat elements acts irreducibly on a subspace of dimension greater than $ d/2
$, and in the induced action the generating pair corresponds to a pair of fat
elements.
