Our aim in this paper is to identify the limit behavior of the solutions of random degenerate equations of the form −div Aε(x′,∇Uε)+ρεω(x′)Uε=F with mixed boundary conditions on Ωε whenever ε→0, where Ωε is an N-dimensional thin domain with a small thickness h(ε), ρεω(x′)=ρω(x′/ε), where ρω is the realization of a random function ρ(ω) , and Aε(x′,ξ)=a(Tx′ /εω,ξ) , the map a(ω,ξ) being measurable in ω and satisfying degenerated structure conditions with weight ρ in ξ. As usual in dimension reduction problems, we focus on the rescaled equations and we prove that under the condition h(ε)/ε→0 , the sequence of solutions of them converges to a limit u0, where u0 is the solution of an (N−1) -dimensional limit problem with homogenized and auxiliary equations.