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We present a testing preorder for probabilistic processes based on the natural notion of a process passing a test with a certain probability. The theory enjoys close connections with the classical testing theory of Hennessy and DeNicola in that whenever a process passes a test with probability 1 (respectively some non-0 probability) in our setting, then the process must (respectively may) pass the test in the classical theory. In addition, we develop an alternative characterisation of the probabilistic testing preorder that is based on the “must sets” characterization of De Nicola. Finally, we extend our theory of testing to substochastic processes, in which the sum of the probabilities of a process's outgoing transitions may be strictly less than 1, with the deficit representing the process' capacity for undefined behavior. A simple example involving the construction of pipelines from faulty buffer cells is given to illustrate …