  In this paper we obtain general integral formulas for probabilities in the
asymmetric simple exclusion process (ASEP) on the integer lattice with nearest
neighbor hopping rates p to the right and q=1-p to the left. For the most part
we consider an N-particle system but for certain of these formulas we can take
the limit as N goes to infinity. First we obtain, for the N-particle system, a
formula for the probability of a configuration at time t, given the initial
configuration. For this we use Bethe Ansatz ideas to solve the master equation,
extending a result of Schuetz for the case N=2. The main results of the paper,
derived from this, are integral formulas for the probability, for given initial
configuration, that the m'th left-most particle is at x at time t. In one of
these formulas we can take the limit as N goes to infinity, and it gives the
probability for an infinite system where the initial configuration is bounded
on one side. For the special case of the totally asymmetric simple exclusion
process (TASEP) our formulas reduce to the known ones.
