  We show that for any $n\geq 2$, two elements selected uniformly at random
from a \emph{symmetrized} Euclidean ball of radius $X$ in
$\textrm{SL}_n(\mathbb Z)$ will generate a thin free group with probability
tending to $1$ as $X\rightarrow \infty.$ This is done by showing that the two
elements will form a ping-pong pair, when acting on a suitable space, with
probability tending to $1$. On the other hand, we give an upper bound less than
$1$ for the probability that two such elements will form a ping-pong pair in
the usual Euclidean ball model in the case where $n>2$.
