Network motifs are at the core of modern studies on biological networks, trying to encompass global features such as all-world or scale-free properties. Detection of significant motifs may be based on two different approaches: either a comparison with randomized networks (requiring the simulation of a large number of networks), or the comparison with expected quantities in some well-chosen probabilistic model. This second approach has been investigated here. We first provide a simple and efficient probabilistic model for the distribution of the edges in undirected networks. Then,we give exact formulas for the expectation and the variance of the number of occurrences of a motif. Generalization to directed networks is discussed in the conclusion.