The number of distinct sites visited by a lattice random walker is a subject of continuing interest in both mathematics and physics. All previous investigations have used the assumption that the lattice is unbounded. An assessment of the amount of tissue interrogated by a photon in reflectance measurements for diagnostic purposes suggests analyzing properties of the average number of distinct sites visited by a random walker trapped by an absorbing plane at time t. We show that for sufficiently large t this number is the same as the average number of distinct sites visited for this time when the surface is not present. A more complete analysis is possible for a random walk on a line terminated by an absorbing point.