The Hausdorff dimension of the intersection of two randomly translated Cantor sets can be expressed in terms of the top Lyapunov exponent (growth rate) of products of random 2×2 matrices. Existing methods for computing this Lyapunov exponent either (1) handle degenerate cases where the stationary measures are discrete, or (2) suffer combinatorial explosion. We present a new method that computes the Lyapunov exponent when the stationary measures are continuous and also avoids combinatorial blow-up for a broad class of examples. As an application, we determine the Hausdorff dimension of the intersection of the middle-seventh Cantor set with a random translate of itself. Our method also yields growth rates for expanding random Fibonacci sequences.