In [Math. Z. 110, 199–205 (1969; Zbl 0176.29901)], the reviewer proved a longstanding conjecture of Netto that the probability \(\Pr(\left\langle \pi,\sigma\right\rangle =A_{n})\) that two random permutations of the alternating group generate all of \(A_{n}\) tends to \(1\) as \(n\rightarrow\infty\). He also raised the question of whether the same was true for the other families of simple groups. Over the next two decades other mathematicians answered the latter question in the affirmative (see [W. M. Kantor and A. Lubotzky, Geom. Dedicata 36, No. 1, 67–87 (1990; Zbl 0718.20011); M. W. Liebeck and A. Shalev, Geom. Dedicata 56, No. 1, 103–113 (1995; Zbl 0836.20068)]). For most infinite families of finite simple groups these proofs depend on the classification of finite simple groups (CFSG) and it is not known whether there are more elementary proofs. For example a closely related result, also based on CFSG, states that for random elements \(\pi,\sigma\) of \(\mathrm{SL}(n,q)\) we have \(\Pr(\left\langle \pi,\sigma\right\rangle \neq \mathrm{SL}(n,q))=2q^{-n}+O(q^{-7(n-1)/6})\) (see [W. M. Kantor, Lond. Math. Soc. Lect. Note Ser. 165, 403–421 (1992; Zbl 0830.20033)]). In the present paper, the authors give a elementary proof (avoiding CFSG) of a weaker result: \(\Pr(\left\langle \pi,\sigma\right\rangle \neq \mathrm{SL}(n,q))\leq\exp(-c\sqrt{n}\log q)+\exp (-(2-o(1))\sqrt{n\log n\log q}\) for some absolute constant \(c>0\). A crucial lemma which is used in the proof may be of independent interest: \(\eta_{G}:=\left\vert G\right\vert ^{-1}\sum_{g\in G}1/ord(g)\leq\exp (-(2-o(1))\sqrt{n\log n\log q})\) whenever \(G=GL(n,q), \mathrm{PGL}(n,q),\mathrm{SL}(n,q)\) or \(\mathrm{PSL}(n,q)\) (note that \(1/\eta_{G}\) is the harmonic mean of the orders of the elements of \(G\)).