  In this paper we study some conditional probabilities for the totally
asymmetric simple exclusion processes (TASEP) with second class particles. To
be more specific, we consider a finite system with one first class particle and
$N-1$ second class particles, and we assume that the first class particle is
initially at the leftmost position. In this case, we find the probability that
the first class particle is at $x$ and it is still the leftmost particle at
time $t$. In particular, we show that this probability is expressed by the
determinant of an $N\times N$ matrix of contour integrals if the initial
positions of particles satisfy the step initial condition. The resulting
formula is very similar to a known formula in the (usual) TASEP with the step
initial condition which was used for asymptotics by Nagao and Sasamoto [Nuclear
Phys. B 699 (2004), 487-502].
