We consider a class of mixed-integer optimization problems subject to N randomly drawn convex constraints. We provide explicit bounds on the tails of the probability that the optimal solution found under these N constraints will become infeasible for the next random constraint. First, we study constraint sets in general mixed-integer optimization problems, whose continuous counterpart is convex. We prove that the number of support constraints (i.e., constraints whose removal strictly improve the optimal objective) is bounded by a number depending geometrically on the dimension of the decision vector. Next, we use these results to show that the tails of the violation probability are bounded by a binomial distribution. Finally, we apply these bounds to an example of robust truss topology design. The findings in this paper are a first step towards an extension of previous results on continuous random convex programs to the case of problems with mixed-integer decision variables that naturally occur in many real-world applications.