  We consider a one-dimensional run-and-tumble particle, or persistent random
walk, in the presence of an absorbing boundary located at the origin. After
each tumbling event, which occurs at a constant rate $\gamma$, the (new)
velocity of the particle is drawn randomly from a distribution $W(v)$. We study
the survival probability $S(x,t)$ of a particle starting from $x \geq 0$ up to
time $t$ and obtain an explicit expression for its double Laplace transform
(with respect to both $x$ and $t$) for an arbitrary velocity distribution
$W(v)$, not necessarily symmetric. This result is obtained as a consequence of
Spitzer's formula, which is well known in the theory of random walks and can be
viewed as a generalization of the Sparre Andersen theorem. We then apply this
general result to the specific case of a two-state particle with velocity $\pm
v_0$, the so-called persistent random walk (PRW), and in the presence of a
constant drift $\mu$ and obtain an explicit expression for $S(x,t)$, for which
we present more detailed results. Depending on the drift $\mu$, we find a rich
variety of behaviours for $S(x,t)$, leading to three distinct cases: (i)
subcritical drift $-v_0\!<\!\mu\!<\! v_0$, (ii) supercritical drift $\mu <
-v_0$ and (iii) critical drift $\mu=-v_0$. In these three cases, we obtain
exact analytical expressions for the survival probability $S(x,t)$ and
establish connections with existing formulae in the mathematics literature.
Finally, we discuss some applications of these results to record statistics and
to the statistics of last-passage times.
