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This paper introduces an information theoretic model that allows to quantify the degree of anonymity provided by schemes for anonymous connections. It considers attackers that obtain probabilistic information about users. The degree is based on the probabilities an attacker, after observing the system, assigns to the different users of the system as being the originators of a message. As a proof of concept, the model is applied to some existing systems. The model is shown to be very useful for evaluating the level of privacy a system provides under various attack scenarios, for measuring the amount of information an attacker gets with a particular attack and for comparing different systems amongst each other.