This article studies and derives beamforming design in a dual-functional radar-communication (DFRC) multiple-input-multiple-output system. We focus on a scenario, where the DFRC base station communicates with downlink communication users, with imperfect channel state information knowledge, and performs target detection, all via the same transmit signal. Through careful relaxation procedures, we arrive at a suitable and novel optimization problem, which maximizes the radar output power in the Bartlett sense, under probabilistic outage signal-to-interference-and-noise ratio constraints. Theoretical analysis proves optimality of the solution given by the relaxed version of the problem, as well as closed-form solutions in certain scenarios. Finally, the achieved performances and trade-offs of the proposed beamforming design are demonstrated through numerical simulations.