  We show that if one selects uniformly independently and identically
distributed matrices $A_1, \ldots, A_s \in \mathrm{SL}_2(\mathbb{Z})$ from a
ball of large radius $X$ then with probability at least $1 - X^{-1 + o(1)}$ the
matrices $A_1, \ldots, A_s$ are free generators for a free subgroup of
$\mathrm{SL}_2(\mathbb{Z})$. Furthermore, to show the flexibility of our method
we do similar counting for matrices from the congruence subgroup $\Gamma_0(Q)$
uniformly with respect to the positive integer $Q\le X$. This improves and
generalises a result of E. Fuchs and I. Rivin (2017) which claims that the
probability is $1 + o(1)$. We also disprove one of the statements in their work
that has been used to deduce their claim.
