We introduce a novel modal logic, namely the doxastic $\mu$-calculus with error control (D$\mu$CEC), and propose a formalization of probabilistic anonymity and oblivious transfer in the logic, and the validation of these formalizations on implementations formalized in probabilistic CCS. The distinguishing feature of our logic is to provide a combination of dynamic operators for belief (whence the attribute ``doxastic'') with a control on the possible error of apprehension of the perceived reality, and for internalized probability. Both operators are dynamic (non-monotonic) thanks to the possibility of combining them with temporal operators, and are parameterized with a lower and upper probability bound (the error control).