A fundamental and critical problem for Model Reference Adaptive Control (MRAC) systems is the characterisation of the system response during transients. This problem is strictly related to the estimation of the reachable set (RS) from a fixed set of initial conditions and it is typically tackled using the Lyapunov's direct method. One well-known drawback of this approach is the excessive conservatism in the estimation of the RS. To overcome this limitation the authors propose a novel probabilistic framework where uncertain parameters and control signals are considered as random variables. In this framework the RS design is translated into a stochastic convex optimisation problem. This brings the benefit that (probabilistic) LMIs with reduced conservatism can be worked out. The so-called scenario optimisation approach is then used to solve the stochastic optimisation problem with a-priori specified level of reliability. The novel approach is compared with an existing worst-case approach in determining the RS of MRAC systems in the presence of matched and input uncertainty via simulation studies. The proposed methodology can potentially be a useful tool for the probabilistic analysis and design of a broad category of existing adaptive control systems.