  We study the splitting probabilities for a one-dimensional Brownian motion in
a cage whose two boundaries move at constant speeds $c_1$ and $c_2$. This
configuration corresponds to the capture of a diffusing, but skittish lamb,
with an approaching shepherd on the left and a precipice on the right. We
derive compact expressions for these splitting probabilities when the cage is
expanding. We also obtain the time-dependent first-passage probability to the
left boundary, as well as the splitting probability to this boundary, when the
cage is either expanding or contracting. The boundary motions have a
non-trivial impact on the splitting probabilities, leading to multiple regimes
of behavior that depend on the expansion or contraction speed of the cage. In
particular, the probability to capture the lamb is maximized when the shepherd
moves at a non-zero optimal speed if the initial lamb position and the ratio
between the two boundary speeds satisfy certain conditions.
