  The connection between absorbing boundary conditions and hard walls is well
established in the mathematical literature for a variety of stochastic models,
including for instance the Brownian motion. In this paper we explore this
duality for a different type of process which is of particular interest in
physics and biology, namely the run-tumble-particle, a toy model of active
particle. For a one-dimensional run-and-tumble particle subjected to an
arbitrary external force, we provide a duality relation between the exit
probability, i.e. the probability that the particle exits an interval from a
given boundary before a certain time $t$, and the cumulative distribution of
its position in the presence of hard walls at the same time $t$. We show this
relation for a run-and-tumble particle in the stationary state by explicitly
computing both quantities. At finite time, we provide a derivation using the
Fokker-Planck equation. All the results are confirmed by numerical simulations.
