Analyzing the average behavior of conic programs on Gaussian random data, though not to be confused with analyzing their behavior on ``real-world" data, is arguably a first cautious step towards this goal. It turns out that this step finds a firm ground in the theory of intrinsic volumes of convex cones. We showcase this mathematical connection by answering the intriguing question: What is the probability that the solution of a random semidefinite program has rank $r$? More precisely, we will give closed formulas for this probability in terms of certain integrals that decompose Mehta's integral, but for which no simple expression is known yet. Along the way we will mention further results that hold for any cone program (under the Gaussian random model), a generality that counterbalances the restrictive character of the random model. These general results include estimates on the average condition of a cone program, a quantity that can be used to bound the running time of interior-point algorithms that solve this program.