By using ergodic theory of subadditive processes and variational convergence, we study the macroscopic behavior of a thin 3D composite made up of high-conductivity fibers that are randomly distributed according to a stochastic point process in a bounded open set of ℝ3. The thickness of the body, the conductivity and the size of the cross sections of the fibers depend on a small parameter ε. The variational limit functional energy obtained when ε tends to 0 is deterministic and depends on two variables: one is the solution of a variational problem posed in a 2D bounded open set and describes the behavior of the medium; the other captures the limit behavior of suitably rescaled solutions in the fibers when the thickness and the size section become increasingly thin and the conductivity of the fibers becomes increasingly large.