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The ever-growing insertion of intermittent energy sources requires to account for this uncertainty by precise models. Probabilistic constraints are an interesting technique to deal with the high fluctuations of such energy resources, while accounting for power flow and unit generating constraints. In the context of hydrothermal unit commitment problems, with a significant share of wind power generation, this paper suggests a novel mixed integer optimization model with joint probability constraints and continuous distributions that handles the probability constraints exactly. Thus, extending classical cutting plane methods, the approach provides at each iteration a feasible solution and a certificate on its gap to optimality. The efficiency of the algorithm is tested by numerical experiments when compared to two popular probability constrained approaches: individual and a sample-based method. A set of out-of-sample Monte-Carlo evaluations confirms that the solutions provided by the algorithm do indeed satisfy the a priori defined probability level with optimal costs, contrary to the solution given by individual probability constraints. The simulations also intend to highlight the advantages of the proposed optimization model since it does not depend on the set of sampled scenarios. The proposed method is also evaluated on a 46-bus based system, showing its capabilities on larger systems. [ABSTRACT FROM AUTHOR]