  It is well known that the proportion of pairs of elements of
$\operatorname{SL}(n,q)$ which generate the group tends to $1$ as $q^n\to
\infty$. This was proved by Kantor and Lubotzky using the classification of
finite simple groups. We give a proof of this theorem which does not depend on
the classification.
  An essential step in our proof is an estimate for the average of
$1/\operatorname{ord} g$ when $g$ ranges over $\operatorname{GL}(n,q)$, which
may be of independent interest. We prove that this average is \[
  \exp(-(2-o(1)) \sqrt{n \log n \log q}). \]
