Let A be an n by N real valued random matrix, and HN denote the N-dimensional hypercube. For numerous random matrix ensembles, the expected number of k-dimensional faces of the random n-dimensional zonotope AHN obeys the formula Efk(AHN)/fk(HN) = 1 − PN−n,N−k, where PN−n,N−k is a fair-coin-tossing probability: PN−n,N−k [\equiv] Prob{N−k−1 or fewer successes in N−n−1 tosses }.