To the experts, the question is surely either an obvious yes or no: 

Can we explicitly describe all finitely generated free subgroups of the Artin braid groups $B_n$ (for $n=2,3,\ldots$) ? 

More specifically, seeing $B_n$ as the isotopy space of all $n$-pointed braids in the closed 2-disk, can we characterize all configurations which generate the rank $k$ free group?

In other words, do we know, say, that all free subgroups arise from ping-pong and can we describe all ping-pong games?