Fog Computing infrastructures are deployed in the immediate vicinity of users in order to meet the stringent delay requirements of some emerging IoT applications, which cannot be achieved with traditional Cloud Computing infrastructures. The latency gains of Fog Computing come however at the cost of a potentially larger total capacity. The duplication of ressources in many micro data centres may also lead to an explosion of energy and operations costs. In this paper, we consider the optimal capacity planning of Fog Computing infrastructures under probabilistic delay guarantees. Despite the non-linearity of the delay constraints, we show that the problem can be formulated as a Mixed Integer Linear Programming (MILP) problem. We first present a MILP formulation of the problem assuming that the infrastructure cost depends linearly on the capacities. To account for economies of scale in favour of large data centres, we then extend this MILP formulation to arbitrary concave objective functions. Empirical results show that the optimal capacity-planning solution can be determined efficiently even for large-size problem instances, and that it can results in significant gains with respect to the solution in which user requests are always processed in the nearest data centre.