In this talk we will consider a collection of piecewise monotone interval maps, which we iterate randomly, together with a collection of holes placed randomly throughout phase space. Birkhoff’s Ergodic Theorem implies that the trajectory of almost every point will eventually land in one of these holes. We prove the existence of an absolutely continuous conditionally invariant measure, conditioned according to survival from the infinite past. Absolute continuity is with respect to a conformal measure on the closed systems without holes. Furthermore, we prove that the rate at which mass escapes from phase space is equal to the difference in the expected pressures of the closed and open systems. Finally, we prove a formula for the Hausdorff dimension of the fractal set of points whose trajectories never land in a hole in terms of the expected pressure function.