The concept of anonymity comes into play in a wide range of situations, varying from voting and anonymous donations to postings on bulletin boards and sending mails. A formal definition of this concept has been given in literature in terms of nondeterminism. In this paper, we investigate a notion of anonymity based on probability theory, and we we discuss the relation with the nondeterministic one. We then formulate this definition in terms of observables for processes in the probabilistic $pi$-calculus, and propose a method to verify automatically the anonymity property. We illustrate the method by using the example of the dining cryptographers.