This paper presents robust optimization models for minimizing the required backup capacity while providing probabilistic protection against multiple simultaneous failures of physical machines under uncertain virtual machine capacities in a cloud provider. If random failures occur, the required capacities for virtual machines are allocated to the dedicated backup physical machines, which are determined in advance. We consider two uncertainties: failure event and virtual machine capacity. By adopting a robust optimization technique, we formulate six mixed integer linear programming problems. Numerical results show that for a small size problem, our presented models are applicable to the case that virtual machine capacities are uncertain, and by using these models, we can obtain the optimal solution of the allocation of virtual machines under the uncertainty. A simulated annealing heuristic is presented to solve large size problems. By using this heuristic, an approximate solution is obtained for a large size problem.