  We consider the telegraph process with two velocities,
$a_1>a_2\in\mathbb{R}$, and two rates of reversal, $\lambda_1,\lambda_2>0$. We
study some of its features with respect to the conditional probability measure
where both the initial speed and the number of changes of direction are known.
We exhibit a new proof by induction of the (conditional) probability law and a
detailed study of the distribution of the motion at time $t>0$ conditioned on
its position at a previous time $0<s<t$. In the case of a symmetric process, we
present some results on the joint distribution of the position of the motion at
time $t>0$, its maximum and its minimum up to that moment.
