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

Counting embeddings of free groups into $\mathrm{SL}_2(\mathbb{Z})$ and its subgroups