Cloud reliance is critical to its success. Although fault-tolerance mechanisms are employed by cloud providers, there is always the possibility of failure of infrastructure components. We consequently need to think proactively of how to deal with the occurrence of failures, in an attempt to minimize their effects. In this work, we draw the risk concept from probabilistic risk analysis in order to achieve this. In probabilistic risk analysis, consequence costs are associated to failure events of the target system, and failure probabilities are associated to infrastructural components. The risk is the expected consequence of the whole system. We use the risk concept in order to present representative mathematical models for which computational optimization problems are formulated and solved, in a Cloud Computing environment. In these problems, consequence costs are associated to incoming applications that must be allocated in the Cloud and the risk is either seen as an objective function that must be minimized or as a constraint that should be limited. The proposed problems are solved either by optimal algorithm reductions or by approximation algorithms with provably performance guarantees. Finally, the models and problems are discussed from a more practical point of view, with examples of how to assess risk using these solutions. Also, the solutions are evaluated and results on their performance are established, showing that they can be used in the effective planning of the Cloud.