  We investigate crossing path probabilities for two agents that move randomly
in a bounded region of the plane or on a sphere (denoted $R$). At each discrete
time-step the agents move, independently, fixed distances $d_1$ and $d_2$ at
angles that are uniformly distributed in $(0,2\pi)$. If $R$ is large enough and
the initial positions of the agents are uniformly distributed in $R$, then the
probability of paths crossing at the first time-step is close to $ 2d_1d_2/(\pi
A[R])$, where $A[R]$ is the area of $R$. Simulations suggest that the long-run
rate at which paths cross is also close to $2d_1d_2/(\pi A[R])$ (despite marked
departures from uniformity and independence conditions needed for such a
conclusion).
