Let A,B $\in GL(2,Z)$, then what is the probability of $<A,B>\cong F_2$?
By probability, I mean the haar measure on $GL(2,Z)^2$. I already know what if we replace $\mathbb{Z}$ with $\mathbb{C}$, then the answer is almost from here: Almost all subgroups of a Lie group are free (D.B.A. Epstein)
But $ GL(2,\mathbb{Z})$ is not connected, so the argument in the proof doesn't apply.
Since the only units in $\mathbb{Z}$ are $\pm 1$ we can just deal with "random elements" $SL(2, \mathbb{Z})$.
The Modular group is known to be a free product ([1], [2]) $SL(2, \mathbb{Z})\simeq \mathbb{Z}_2 \ast \mathbb{Z}_3 $. In fact the Hecke groups satisfy $ H_q \simeq \mathbb{Z}_2 \ast \mathbb{Z}_q$
We can talk about random elements of this tree. They are unlikely to satisfy any relation except possibly that one is a power of another.
See also Calkin-Wilk Tree and Stern-Brocot Tree.