We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform distribution. We analyze the average voting power of the $k$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of $n$ and a general theorem about the functional form of the relation between the average Penrose--Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.